EINSTEIN METRICS OF R A N D E R S T Y P E by C O L L E E N ROBLES B.A. Smith College, 1996 M.S. University of Washington, 1998 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA April 2003 ©Colleen Robles, 2003 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d without; my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada Date 9L5v Arh»ri \ \ ^ 0 * 3 Abstract This thesis presents a study of Einstein Randers metrics. Initially introduced within the context of relativity, Randers metrics have a strong presence in both the theory and applications of Finsler geometry. The starting point is a new characterization of Einstein metrics of Randers type by three conditions. The conditions form a coupled, highly non-linear (due to the presence of a Riemannian Ricci tensor), second order system of partial differential equations. The equations are polynomial in the unknowns; a Riemannian metric a and differential 1-form b. Recently Z. Shen has generalized Zermelo's problem of navigation on the plane to arbitrary Riemannian manifolds. (The goal is to identify the paths of shortest time on a Riemannian manifold (M, a) under the influence of an external force W = W d i.) In this context, Randers metrics may be viewed as solutions to Zermelo's problem. The navigation structure yields the main result of the thesis, a succinct geometric description of Einstein metrics of Randers type. Explicitly, the Randers metric arising as the solution to Zermelo's problem on (a, W) is Einstein if and only if the Riemannian metric a is Einstein itself, and W is an infinitesimal homothety of a. The navigation description quickly yields a Schur lemma for the Ricci curvature of Randers metrics. It is a testament to the navigation description that this result, the first Schur lemma for Ricci curvature in (nonRiemannian) Finsler geometry, is obtained with relative ease. An extension of Matsumoto's Identity for Randers metrics of constant flag curvature to the Einstein setting then follows. Having established these general results, I then explore three scenarios: Einstein metrics on surfaces of revolution, constant flag curvature metrics, and Einstein metrics on closed manifolds. The thesis closes with a collection of open questions. l x Contents Abstract ii Acknowledgment v Dedication vi Chapter 1. Introduction 1. Preliminaries 2. Brief summary of results 3. Outline 4. Finsler metrics and flag curvature 5. Randers metrics 1 1 2 3 3 7 Chapter 2. Einstein metrics of Randers type 1. Ricci curvature and Einstein metrics 2. The Einstein criterion 3. Necessary conditions for Einstein: Preliminary form 4. The constancy of a 5. Necessary conditions for Einstein: Final form 6. Characterization of Einstein metrics of Randers type 11 11 13 14 16 17 18 Chapter 3. The Einstein navigation description 1. Randers metrics as solutions to Zermelo's problem 2. Covariant differentiation with respect to a 3. The Basic Equation 4. The E Equation 5. The Curvature Equation 6. The theorem 7. A Schur lemma 8. The Matsumoto Identity ' 21 21 22 23 25 26 28 30 31 Chapter 4. Case Studies 1. Randers metrics with K = K(x) in dimension 2 2. Spaces of constant flag curvature 3. Closed Einstein spaces 33 33 34 36 Chapter 5. Open Questions 1. A Schur lemma 2. Chern's conjecture 39 39 39 2 3 iii CONTENTS 3. Einstein rigidity 4. A generalized navigation problem Appendix A. 40 . 40 The Matsumoto Identity: Preliminary Form Appendix B. A Schur Lemma 1. The Riemannian Schur Lemma 2. A Finslerian obstacle 3. The Randers Schur Lemma 4. The Matsumoto Identity: Final Form Appendix. Bibliography Appendix. iv Index • 41 43 43 44 44 46 47 49 Acknowledgment I t is a p l e a s u r e t o t h a n k t h e p e o p l e w h o h a v e m a d e t h i s t h e s i s p o s s i b l e . F i r s t , I a m g r a t e f u l t o t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a for t h e f i n a n c i a l support provided b y the University G r a d u a t e Fellowship program. I a m fortunate to have two wonderful advisers i n R i c h a r d Froese a n d D a v i d B a o . I t h a n k R i c h a r d for h i s u n f a i l i n g e n c o u r a g e m e n t t o p u r s u e m y interests, even w h e n t h e y t o o k m e afield of B r i t i s h C o l u m b i a . T h e generous s u p p o r t h e h a s o f f e r e d i n t h e s e e n d e a v o r s goes a b o v e a n d b e y o n d t h e c a l l o f duty. D a v i d is a s e e m i n g l y e n d l e s s s o u r c e o f g u i d a n c e a n d i n s i g h t . H e p r e s e n t s an ethic a n d standard of scholarship I m a y only aspire to. B e y o n d these v i r t u e s , I a p p r e c i a t e D a v i d m o s t for i n t r o d u c i n g m e t o t h e j o y s o f c o l l a b o r a t i o n . T h e e x c i t e m e n t a n d p l e a s u r e I f o u n d i n o u r w o r k set t h e s t a n d a r d by w h i c h I shall compare future joint ventures. Dedication For Abuela, our family's first student of mathematics. CHAPTER 1 Introduction 1. Preliminaries This thesis is about Finsler metrics. A Finsler space is a manifold M equipped with a family of smoothly varying Minkowski norms; one on each tangent space. Riemannian metrics are examples of Finsler norms that arise from an inner-product. Indeed, when introducing the concept of a manifold and its structures, Riemann acknowledged that quadratic differentials comprise only a special case. Unfortunately, the door closed on Finsler geometry when Riemann claimed "The study of the metric which is the fourth root of a quartic differential form is quite time-consuming and does not throw new light to the problem." It was not until Paul Finsler's 1918 thesis under Caratheodory that the field was resurrected. Riemann's comment illuminates the two major obstacles facing the development of Finsler geometry. The first is the ubiquitous computational difficulty associated with the field. As challenging as Riemannian geometry is, the computations in Finsler geometry are considerably more daunting. This serves to frustrate and discourage many mathematicians. Nonetheless, recent advances have brought the computations of Finsler geometry to a more accessible level. The geometric description of Einstein Randers metrics, based on Zermelo's problem of navigation, I shall present offers a substantial improvement in this arena. Indeed, it is fair to say the computations of Einstein metrics of Randers type are now comparable to their Riemannian counterparts. The computational difficulty contributes directly to the second challenge to Finsler geometry. This is the lack of meaningful examples. As evidenced by Riemann's comment that "The study of the metric ... does not throw new light to the problem", the prevailing belief is that Finsler metrics do not capture geometric phenomena omitted by their Riemannian counterparts. Until recently this belief has been supported by a dearth of explicit examples illustrating non-Riemannian behavior. In 2002 D. Bao and Z. Shen [BS02] constructed a Finsler metric on the 3-sphere that is of constant flag curvature, but not projectively flat. The example breaks the rigidity of Beltrami's theorem which states that a Riemannian metric is of constant flag (read "sectional") curvature if and only if it is projectively flat. Illustrating a distinctly non-Riemannian geometry, the S example 3 l 2. B R I E F S U M M A R Y O F RESULTS 2 shows Finsler geometry may indeed "throw new light to the problem." The need for explicit examples motivates me to present several in §4. 2. B r i e f s u m m a r y of results In this short review of the contents I postpone defining terms and notation to subsequent sections. Randers metrics are constructed from two familiar objects; a Riemannian metric a,ij(x) and a 1-form bi(x) on the manifold M. The norm of a vector y = y di € T M in the tangent space at x € M is then defined to be x X F(x,y) = a(x,y) + P(x,y), where a(x, y) = aijytyi and (3(x, y) = hy . l Essentially we alter a Riemannian metric by adding a linear term. Conceptually the Ricci curvature is the average of the flag curvatures. (For the moment it suffices to think of flag curvature as the Finslerian analog of Riemannian sectional curvature.) A Finsler metric is Einstein if the Ricci curvature is a function of x € M alone, rather than the a priori (x,y) € TM. Theorem 2.3 provides the foundation for our analysis of Einstein Randers metrics. The theorem characterizes Einstein metrics of Randers type as solutions to a system of partial differential equations. The equations are polynomial in the unknowns: the Riemannian metric a and its Ricci tensor; the 1-form b and its first and second order covariant derivatives. The theorem is joint work with David Bao [BR03a]. Recently, Shen [She02a] has shown that Randers metrics arise as solutions to Zermelo's problem on navigation on a Riemannian manifold (M, a) under an external force W. The navigation structure is an enormously valuable tool for constructing examples of Einstein metrics. This utility is the motivation to rephrase the characterization of Einstein Randers metrics in terms of the underlying metric a and vector field W. We are rewarded by a succinct geometric description of Einstein metrics of Randers type. Explicitly, the Randers metric F is Einstein if and only if the underlying Riemannian metric a is Einstein, and W is an infinitesimal homothety of a. The navigation description yields two elegant results. First a Schur lemma, which states that the Ricci curvature of an Einstein Randers metrics must be constant in dimension greater than two. The second result is a generalization of Matsumoto's identity for Randers metrics of constant flag curvature to the Einstein setting. The identity further illuminates the relationship between the Ricci scalars of F and i l With the Einstein navigation description in hand I turn to three case studies. First, Einstein metrics are constructed on surfaces of revolution. The Ricci scalar of the non-Riemannian metrics is none other than the Gaussian curvature induced from Euclidean R . Second, I discuss a recent classification of constant flag curvature Randers spaces. Lastly, I consider 3 4. F I N S L E R M E T R I C S A N D F L A G C U R V A T U R E 3 closed manifolds. The results here include rigidity theorems akin to AkbarZadeh's for constant flag curvature spaces. For example, I show that any closed Einstein Randers space with negative Ricci scalar is necessarily Riemannian. 3. Outline The remainder of this chapter is concerned with introducing Finsler metrics, and their spray and flag curvatures. I then focus on Finsler metrics of Randers type. The underlying Riemannian metric d of a Randers space permits an expression, known as Berwald's formula, of the spray curvature in terms of a, its curvature, and covariant derivatives of b. Berwald's formula is integral to our treatment of the flag curvature. The material of this chapter is treated briefly, without the detailed derivations that may be found elsewhere. Chapter Two is devoted to a characterization theorem for Einstein metrics of Randers type. While refining the characterizing equations, I establish the constancy of an essential scalar. The material of Chapter Two is joint work with David Bao [BR03b]. The heart of the thesis is Chapter Three. Via Zermelo's problem of navigation, the characterization result of Chapter Two is parlayed into a breviloquent description of Einstein Randers metrics. At this point the Schur lemma and Matsumoto's Identity fall out neatly. Chapter 4 is devoted to the three case studies outlined above. The closing chapter details some open questions. 4. Finsler metrics and flag curvature The material in the remainder of the chapter is but a sketch of the elements of Finsler geometry I find useful. I shall not attempt a detailed treatment of this material which has been handled adroitly elsewhere. See, for example, [BCSOO, Run59, SheOla, SheOlb]. Let's begin with notation and some definitions. The following abbreviations are introduced to reduce some of the notational clutter associated with differential geometry: • The Einstein convention : repeated up-down pairs of indices imply a summation. For example, V -§ ~ = YA^IV ^• The partial derivatives ^ and Jj£ are given by f i and f i , respectively. % 1 X I x y Throughout the thesis M shall denote a manifold of dimension n. Points on M are denoted by x, and the tangent space to M at x is T M . The canonical coordinates on the tangent bundle T M are given by {x,y), where y = y £e T M. X i X 4. F I N S L E R M E T R I C S A N D F L A G 4 CURVATURE A Finsler space (M, F) is a manifold M equipped with a smoothly varying family of Minkowski norms F - one on each tangent space - defined in the following way. DEFINITION 1.1. A Finsler metric is a continuous function F : TM —• [0, o o ) with the following properties, • Regularity: F is smooth on TM\0 := {(x,y) e TM\y ^ 0} . • Positive homogeneity: F(x, Ay) = XF(x, y) for all A > 0. • Strong convexity: the fundamental tensor ga(x,y) •= {\ ) iyi F2 y is positive definite for all (x, y) € TM\0. Certainly, Riemannian metrics a = \faijy yi are Finsler. Notice however, Riemannian metrics are absolutely homogeneous, while Finsler metrics need only be positively homogeneous. Loosening absolute homogeneity to positive homogeneity allows such exciting examples as the Finslerian Poincare disc [BCSOO, Oka83] for which the travel time from the center to the rim is infinite, while the return trip takes only log(2) seconds. Also, the fundamental tensor of a Riemannian metric is a function of x alone. In general the fundamental tensor of a Finsler metric is a function of (x, y) € TM. Indeed, a Finsler metric is Riemannian if and only if the the fundamental tensor is a function of x alone. As in Riemannian geometry, the formal Christoffel symbols of F are given by l i = lis (z\__i_ z\%k , 9ks\ d where (g i) is the inverse of the fundamental tensor. Similarly, the geodesic spray coefficients are l The spray curvature of F is then defined by K\ = 2 (C7% - (&) - yi ( G * ) ^ + IV ( G % yk fc v . The expression above is known as Berwald's formula [Run59]. Let us take a moment to consider the case of a Riemannian metric. Note that if F = a = \J<iijy' yi is Riemannian, then K = y R£ jyi, where is the Riemann curvature tensor. The sectional curvature of the plane spanned by 0 ^ w, y € T M is given by l l h k k X w K (x,y)w a(y,y)a(w,w) - d(y,w) l k ik 2 ' where a(-, •) denotes the inner product associated to a. Because the Christoffel symbols are functions of x alone, the spray curvature K (x,y) is quadratic in y and w Kik(x,y)w = y Kik(x,w)y . l k l k l k 5 4. F I N S L E R M E T R I C S A N D F L A G C U R V A T U R E On the other hand, the formal Christoffel symbols of a general Finsler metric depend on both x and y. So, while K (x,y) is homogeneous of degree two in y, it is not quadratic. In particular, it will no longer be true that w Kih(x,y)w = y Ki (x,w)y . This necessitates the generalization of sectional curvature to Finslerian flag curvature. Begin by planting a flag pole y ^ 0 in T M. To this flag pole y we associate the spray curvature K = K (x, y). The flag - a two-dimensional subspace of T M - is then specified by selecting 0 ^ w G T M transverse to y. The flag curvature is given by l k l k z k k X % l k k X X N K(x, y, w) = — V L where K{ = gijK k ik -2 , r and g is the fundamental tensor. Explicitly, 3 k w K (x,y)w l — k g(y,y) = g(w,w) = g(y,w) = gij(x,y)y y ', l : gij(x,y)w w , l and J g (x,y)y w '. l : ij Notice that both Kik and are evaluated at the flag pole y, not the transverse edge w. Allow me to reiterate that in general K(x, y, w) ^ K(x, w, y). Though it is by no means obvious from the definition, the tensor is symmetric, Kik = K. ki The verification is tedious; details may be found in [BCSOO]. The flag curvature is invariant under positive re-scaling of y. To see this first note that the formal Christoffel symbols ^ - and the geodesic spray coefficients G = \ f^y are positive homogeneous of degree 0 and 2, respectively. Hence, the spray curvature K is positive homogeneous of degree 2 in y. Now suppose A > 0. Then l k l r k l k Kik(x,Xy) = X K (x,y), 2 ik g(Xy, Xy) = X g{y, y), g(Xy,w) = Xg(y,w). 2 Hence K(x,Xy,w) = K(x,y,w), as claimed. This says the flag curvature depends only on the direction of the flag pole y, not its length. We would also like to see that K is determined by the flag alone, and not the transverse edge w. That is, if both {y, w} and {y, v} determine the same flag (i.e. subspace) of T M, then K(x,y,w) = K(x,y,v). Happily, this is the case, as I shall illustrate in the next section. X 4. FINSLER METRICS AND FLAG CURVATURE 6 4.1. Euler's Theorem. Many of the functions and tensors I introduced above are positively homogeneous in y. We may take advantage of this structure with Euler's theorem for homogeneous functions. T H E O R E M 1.2 (Euler, [BCSOO]). Assume the function $ : R -> R is differentiable away from the origin. The following two statements are equivalent: n • $ is positively homogeneous of degree m. That is, $(Ay) = A $(y) VA>0. m • The radial directional derivative o / $ is m$. y $ (y) = i yi Explicitly, m$(y). Since F is positively homogeneous of degree 1, Euler's theorem gives yF = F, i yi Also, Qijy* = FF j and y g^y ^ 1 =F . 2 Let's turn to the spray curvature . Recall the geodesic spray coefficients G = \ f j yiy are homogeneous of degree two. By Euler's theorem , l r k k yG k = 2G>, j yk y {G ) k j = 2(G ) , y {G ) k j = (G ) j. k l y k x l y y l XJ l y These equalities, and the symmetry of K^, may be used to show that K (x, y)y = 0 = y K (x, k ik l ik y). This last equality implies the flag curvature K(x, y, w) depends on the flag alone, and not the transverse edge w. To see this, suppose both {y,w} and {y, v} determine the same flag in T M. Then v = [iy + TW, for some ( I , T ER, and X g(v, v) = v g{y, y) + 2^Tg(y, w) + r g(w, w), g(y,v) = ng{y,y) vK v i k ik = 2 2 +rg(y,w), T wK w . 2 i k ik It follows that K(x,y,v) = K(x,y,w). Let me summarize our discussion by reiterating that the flag curvature depends only on • the direction of the flag pole y, not its length, and • the flag, or subspace, determined by the transverse edge w, but not w itself. 5. RANDERS METRICS 7 5 . R a n d e r s metrics Randers metrics were introduced by G. Randers [Ran41] in 1941 within the context of relativity. They offer a smooth transition from Riemannian to Finsler metrics as they are built from objects well known to the geometer: a Riemannian metric a and a drift 1-form b. DEFINITION 1.3. A Randers metric is a Finsler function F =a+ (3, where a(x,y) = ya^-yV' ,' P(x,v) = by 1 and \\b\\ := b^bj < 1. 2 Let's consider the requirements in Definition 1.1 for the norm F to be Finsler. First, < 1 is necessary if F is to be positive. As it happens, the condition is also necessary and sufficient for the fundamental tensor gij to be positive definite [BCSOO]. Second, the Randers metric is positively homogeneous of degree one in y. The metric will be absolutely homogeneous (and Riemannian) if and only if b = 0. I shall say a Randers metric with nonzero b is non-Riemannian. Randers metrics are ubiquitous in Finsler geometry. Recently Mo and Shen [MS02b] have shown that if a compact Finsler manifold with n > 2 has flag curvature K(x,y) < —1, depending on position x and flag pole y alone (i.e. no dependence on the flag {y,w}), then F is Randers. Randers metrics also describe the solutions to Zermelo's problem of navigation. 5.1. Zermelo's p r o b l e m of navigation. Introduced and solved by Zermelo in 1931 [Zer31, Car99], the problem may be posed in the following way. Consider a ship moving with constant speed on the open sea in calm waters. Imagine a breeze comes up. How must the ship be steered in order to reach a given destination in the shortest time? If the wind is time-independent, then the paths of shortest time are geodesies of a Randers metric. Shen [She02a, She02b] has generalized Zermelo's problem to Riemann ian spaces. Consider a manifold M with Riemannian metric a = \ft\jtfyi. If a ball rolls about M with constant speed 1, then any geodesic is a path of shortest time. Next suppose a 'wind' W = W d i blows over M . The wind represents an external force acting on the ball. Assume additionally that a(W) < 1. Shen has shown [She02a] any path of shortest time for the ball is a geodesic of the Randers metric F = a + j3 given by 1 l x _ a [l-& (W)] 2 ( 1 ij + WW i j - -Wj 8 5. R A N D E R S M E T R I C S where Wi := a,ijW . The assumption a(W) < 1 implies that a is indeed strongly convex and a Riemannian metric. The condition also guarantees that 11511 < 1, and F is strongly convex. We say the Randers metric F solves Zermelo's problem of navigation on the Riemannian manifold (M, a) under the external force W. As we will see in Chapter Three, the navigation structure underlying a Randers metric is ideally suited to the study of Einstein metrics. 3 5.2. Some special tensors. In this section I define a few important tensors and establish some notation particular to Randers metrics. The Christoffel symbols 7* - of the Riemannian metric a define a covariant differentiation on M. Covariant derivatives are denoted by a vertical slash. For instance, the covariant derivative of the drift 1-form b is given by fe bj\k'-= j,x ~ ifjk > b k b where bj k means d kbj. There are three ubiquitous objects in this thesis: X x lieij '•= i\j + bj\i curly '•= i\j - bj\i b b @i := b cuTl . h hi Note that lie^- is the Lie derivative of the metric a along the vector field p := b%i := a bjd , ij xi whereas curljj is so named because d(t>idx ) — —(b j — bj i) \ dx A dx l l ix = 3 X —(b j—bj i)dx <g)dx l ix :i X and b%\j — bj\i = bi j — bj i. x x Observe lie is symmetric, curl is skew-symmetric, and liejj + curly = 2b^ . The third tensor O denotes a contraction of curl with b. Indices on b, lie, curl and 9 are lowered and raised by a^- and its inverse a . Contraction of any tensor index with the vector y is indicated by a 0 subscript. For example: 13 liejo = liey y lieoo = liejj y y° curljo = curl^ y curr = curPj y e 0 0 = 3 l 3 3 e\ iy 5. R A N D E R S M E T R I C S 9 5.3. The spray curvature of a Randers metric. In §1.4 I introduced the spray curvature of a Finsler metric K\ = 2 ( G % fc - (&) - yi ( G * ) ^ + 2& ( G % yk v . Following Bao and Shen [BS02, SheOla] I make two adjustments to this formula: • adapt it to the Randers setting, by expressing the data in terms of the Riemannian metric a and 1-form b, and • covariantize the expression. As a whole the formula is tensorial. However, individually the four terms on the right hand side are not. We want an expression that is manifestly tensorial in each term on TM. Recollect G — \*f j y^y are the geodesic spray coefficients of F. Similarly, let G := \^ j y^y denote the geodesic spray coefficients of a. The two geodesic spray coefficients differ by a perturbation term £ [BCSOO]: % k k 1 l k k G =& + C, i with ^ 2F ^ ° ° - ° ) + 2 « V By substituting this decomposition into Berwald's formula we may express the spray curvature as K = K + B . This fulfills our first goal by expressing K as the Riemannian spray curvature K , plus a perturbation. The perturbation term B involves x-partial derivatives of £. These are the terms we wish to covariantize. (The y-partial derivatives already transform tensorially. See [BCSOO].) Notice ( is a tensor over TM, not M. In order to covariantize the x-partial derivatives on TM we horizontally lift the vector ^ G TM to = He curl a 0 l l l k k k x l k k l k l 3? -5?-<*V£ : e TiTM). We may now define horizontal covariant differentiation on TM by lifting the Riemannian covariant derivative operator on M. The action on ( is given by % and is tensorial on TM. The action has two properties worth noting. • First, when a tensor on M (e.g. 6j) is lifted to TM, the action of the horizontal covariant derivative on TM agrees numerically with the Riemannian covariant differentiation on M. So there is no great abuse in notation in referring to both actions as "|". • Second, y is horizontally covariantly constant. That is, y ^ = 0 . In particular, we may contract a tensor with y either before or after covariant differentiation, with the same result. 1 5. R A N D E R S M E T R I C S 10 Substituting the formula (C)xJ = C\j + {C)yh{G )yi ~ C, {G )yhyj h into the expression for B covariantized form: l = k\ K \ k h l we obtain Berwald's formula in split and + {2CV - y {C\j) * j y - + 2C '(CW} J , where K is the spray curvature of a. The split and covariantized form of Berwald's formula is enormously useful, playing a substantial role in both the characterization theorem of the next chapter, and the navigation description of §3. We now have all the tools necessary to begin the thesis proper. L K CHAPTER 2 Einstein metrics of Randers type The two objects under consideration, the Randers metric and its Ricci tensor, are related by their histories in physics. The well-known Ricci tensor was introduced in 1904 by G. Ricci. Nine years later Ricci's work was used to formulate Einstein's theory of gravitation. (Details and references may be found in [Bou79].) Almost thirty years later the physicist Gunnar Randers remarked that: ...the Riemannian metric has one property which does not seem quite appropriate for the application to physical space-time, and that is the perfect symmetry between opposite directions for any coordinate interval. Perhaps the most characteristic property of the physical world is the uni-direction of time-like intervals [Ran41]. It was Randers aim to introduce asymmetry to the metric while retaining a quadratic indicatrix (the set of unit vectors in the tangent space). This is done by displacing the center of a Riemannian indicatrix; the result is a Randers metric. It is the aim of this chapter to characterize Einstein metrics of Randers type. Einstein metrics are defined in the following section. Loosely, though, we will say a Finsler metric F is Einstein if the average of its flag curvatures at a flag pole y is a function of position x alone, rather than the a priori position x and flag pole y. After definitions I begin the derivation of a characterization theorem for Einstein metrics of Randers type. The theorem characterizes Einstein Randers metrics by three conditions - the Basic, E23 and Curvature Equations - which form a tensorial system of partial differential equations, polynomial in the unknowns, the Riemannian metric a and the 1-form b, with independent variable x. The Curvature Equation is of particular interest because it describes the Riemannian Ricci tensor Ric^ in terms of a, b and their derivatives. The material in this chapter is joint work with David Bao [BR03b]. 1. R i c c i c u r v a t u r e a n d E i n s t e i n metrics Let's begin with a discussion of the Ricci tensor and Einstein metrics. 11 1. RICCI CURVATURE AND EINSTEIN METRICS 12 The R i c c i scalar of F is given by the function Ric(x,y) := ^K . s s In §1.4 we saw that the spray curvature K is positive homogeneous of degree 2 in y. Therefore, the Ricci scalar is positive homogeneous of degree 0 in y. This means Ric(x, y) depends on the direction of the flag pole y, but not its length. The Ricci tensor of a Finsler metric F is defined in [BCSOO] as Ricy := [\K° ] = [\F Ric(x,y)] . . l k 2 s yiyi yiyj When F = a is Riemannian, the curvature tensor depends on x alone and the definition is equivalent to the familiar Ric^ = Ri js S D E F I N I T I O N 2.1. f[BCSOO]j A Finsler metric is Einstein if the Ricci scalar Ric is a function of x alone. Equivalently, RiCij = Ric(x)gij or, Ricoo — Ric(x)F . 2 Recollect (§1.5.2) the 0 subscript denotes contraction with y. This means Ricoo = RiCij2/*2/ . When F = a is Riemannian the definition yields Ri j = Ric(x)dij. Notice that our Einstein metrics are distinguished from constant Ricci curvature metrics, characterized by Ricoo = A F , where A is a constant. However, the Schur lemma of §3.7 says the Ricci scalar Ric of an Einstein metric of Randers type in dimension n > 2 must be constant. So, among Randers spaces of dimension greater than two, there is no distinction between Einstein metrics and constant Ricci curvature metrics. The Ricci scalar Ric(x, y) of a Finsler metric may be realized as the sum of n — 1 appropriately chosen flag curvatures. To see how this is done, pick a flag pole 0 ^ y <E T M. If necessary normalize y to have norm F(y) — 1. The pole determines an inner product on T M via the fundamental tensor of F,~ gij(x,y). Recall (§1.4.1), gijy y = F (y). This means y has norm 1 with respect to the inner product. Use the inner product to select an orthonormal basis {ei} € T M with e = y. It follows from our discussion of flag curvature in §1.4.1 that s J S 2 X X l X 3 2 n K(x v e) - I K i i [ i i ^ U since K = y Kijy = 0 (§1.4.1). With respect to the basis {ej} the fundamental tensor is given by gij(x,y) = Sij, the Kronecker delta, at (x, y). Therefore, the spray curvature K = g Kj^ is numerically equal to K^. In particular, l J nn l tJ k n-l K* a = Y,Kiii=l We may conclude that the Ricci scalar Ric(x, y) = -p^K = K of n — 1 flag curvatures. a s s s is the sum 13 2. T H E EINSTEIN CRITERION 2. The Einstein criterion With this section I begin to derive necessary conditions for a Randers metric (§1.5) to be Einstein. The first step is to re-express the Einstein criterion via Berwald's split and covariantized formula (§1.5.3). Assume the Ricci scalar Ric of the Randers metric F = a + /3 is a function of x alone. We have F Ric(x) = K . With Berwald's formula this is re-written as 2 l i 0 = K\-F Ric(x) 2 = k \ + {2c -y (oy - (c%i(c'v+2c(c''w} - ^R^) j j H b I a = Ricoo + acurl*,,^ + \{n - l)^©o|o - ?(n - l)-^lie |o 00 2 . + \{n - 1)—curl lie - \{n - 1)—© curl + ^curl curlj z l o i0 l i0 0 0 + |a curFcurly + ^ ( n - 1)-^(lie00) - |'(n— l ) - ^ l i e © o 2 2 00 + f(n-l)|J(©o) 2 -F ^c(x). 2 Recollect iY* is the spray curvature of the Riemannian metric dij. The relation K = Ricoo is a consequence of the definition of Ric - and Euler's theorem. The 0 subscripts indicate contraction with y . Since y is horizontally covariantly constant (§1.5.2), those contractions can be carried out either before or after the covariant differentiations (vertical slash), with the same result. Multiplying the re-expressed Einstein criterion by F removes y from the denominators. The criterion for a Randers metric to be Einstein then takes the form fc % i y l 2 Rat + a Irrat = 0, where Rat and Irrat are, respectively, degree 4 and degree 3 polynomials in y, whose coefficients are functions of x. We think of Rat as the rational part of the equation, and Irrat as the irrational part of the equation. Analogous to complex numbers a + ib, which vanish if and only if both the real, a, and imaginary, b, parts vanish, we have the following lemma. 2.2. LEMMA A Randers metric is Einstein if and only if both Rat — and Irrat = 0 hold. 0. P R O O F . In view of homogeneity, it suffices to prove this for all y ^ First note that a can never be polynomial in y. Otherwise the quadratic o-ij(x)y y = would have been factored into two (identical) linear terms. Its zero set would then consist of a hyper-plane, contradicting the positive definiteness of a^. Now suppose the polynomial Rat were not zero. The displayed equation would imply that it is the product of a polynomial Irrat with a non-polynomial factor a. This is not possible. So Rat must vanish l 3 ft2 0 3. N E C E S S A R Y CONDITIONS F O R EINSTEIN: P R E L I M I N A R Y F O R M 14 and, since a is positive at all y ^ 0, we see that Irrat must be zero as well. • The formulas for Rat and Irrat are, Rat = (a + (3 ) R i c 2 + \a + 2 a / ? c u r l | + \ {a + /? ) c u r l V u r l 2 2 0 0 i 2 0 2 i i 0 (a + (3 ) c u r F c u r l y - ( a + 6a (3 + /? ) Ric(x) - 2 2 + \(n- 2 4 2 2 4 1){ a 6 | o - 5/?lie o|o + a c u r F l i e j 2 2 0 0 0 0 §a (e ) } 2 - a ^ e W l i o + |(lieoo) + 2 2 0 and, Irrat = 2/?Ric o + ( a + /? ) c u r l , ^ + /ScurpQCurljo 2 2 1 0 + £ a / 3 W l curly - 4/? ( a + /? ) iftc(z) 2 y H - I ^ - 1 2 2 ^®o|o - |lie o|o + /3curr lie ) ! 0 0 -c^GWlio - i0 |lie oOo}0 From these two expressions we will derive the preliminary form of the three necessary conditions for a Randers metric to be Einstein. 3. N e c e s s a r y c o n d i t i o n s for E i n s t e i n : P r e l i m i n a r y f o r m For convenience I abbreviate Ric(x) by Ric. 3.1. T h e B a s i c E q u a t i o n . Assume F is Einstein. That is, Rat = 0 and Irrat = 0. Hence, 0 = Rat -0 = (a 2 Irrat -/? )JRic o PcuT\ 0 + ^curlVurk) l + 2 oli + i a c u r l c u r l » j - ( a + 3f3 )Ric 2 y 2 + i ( n - 1) [ c u r i y i e 2 + |(9 ) i0 + 6 | ]} 2 0 0 0 + ^ ( n - l ) ( l i e o o + 2/36o) . 2 Fix x. Considering the above expression as a polynomial in y, we see that a — 0 divides (lieoo + 2/?0o) . The polynomial a - (3 is irreducible. To see why it can not be factored, recollect that < 1, an assumption that holds for all our Randers metrics. This implies the non-negative a — ft is zero only at the origin. Were the polynomial to factor non-trivially, it must do so as two linear terms. In this case the zero set would contain a hyper-plane; a contradiction. Because a — ft is irreducible it must divide, not just the square, but lieoo + 2/?0o itself. Namely, there exists a scalar function a(x) on M such that 2 1 2 2 2 2 2 2 lie o + 0 2/?e 0 = cT( )(a -/3 ). 2 a; 2 2 3. NECESSARY CONDITIONS FOR EINSTEIN: PRELIMINARY FORM 15 This is our Basic Equation. Differentiating with respect to y and y gives an equivalent version: % k liejfc + bi 0fe + bk ©i = a(x)(d - bibk) • ik To recover the original version, just contract this with y y . l k 3.2. T h e C u r v a t u r e E q u a t i o n . Return to the expression for 0 = Rat — f3Irrat. Use the Basic Equation to replace lieoo + 2/30rj with a(x)(o? — (3 ). We may divide off by a uniform factor of o? — fi . The result reads 2 2 Ricoo = (oc + 3 P ) Ric — /Jcurr ^. — |a curl' - 'curly — ^curr'ocurljo 2 2 -\{n 7 2 1 7 - 1) {§cr 0r)(a - /? ) + curP lie + 1(6 ) + 6 | } • 2 2 2 2 0 j0 0 0 0 This is the Curvature Equation, so named because it describes the Ricci tensor of a. We obtain the indexed form by differentiating with respect to y and y , and making use of the symmetry of Ricjfc. l k RiCi f c (aik + 3b~ib )Ric - \(hicurr ^. = 7 k - \a curl' - 'curly - ^curl^ curl 1 ik b curr '^.) + 7 k 1 jfc - \(n - l)||cr (a;)(aifc - b~ih) + \(curl^lie^ + curl^lie.,;) 2 + lQi&k + ^{&i\ + ®k\i)}k 3.3. T h e E 2 3 E q u a t i o n . Here I derive the final characterizing equation, the E23 Equation. (The number 23 is chosen because it is of some chronological significance in our research notes.) Two pieces of information from the Basic Equation are required. To reduce clutter, I abbreviate a(x) as a. First, differentiate to obtain lie o|o = cT\ (a - (3 ) - l i e o o M + 6 ) 2 0 2 0 0 2/?e | • 00 Next, contract the indexed form of the Basic Equation with y^curl^Q for curl lie = -pQ cvLx\ j - (9 ) j 0 j0 jQ 0 2 - tT/39 . 0 Return to the expression 0 = Irrat. Replace the term Ricoo with the expression given by the Curvature Equation. Then, wherever possible, insert the expressions for lieoo, heoo|o d curl liejo given by the Basic Equation. After dividing off a factor of a — ft we have the E23 Equation: a n J 0 2 curl j 2 = 2Ric(3 + (n - 1) {\a p 2 0b + ±CT0 + | e c u r l + ^cr, } . J o j0 0 Again, differentiating by y produces the indexed form of the E23 Equation l curr .,. = 2Ricbi + (n - 1) {§cr 6; + 7 2 + ^'curl,-* + \a\A . 16 4. T H E C O N S T A N C Y O F a 4. The constancy of a In this section I will show , If the Basic, E23 and Curvature Equations are satisfied, then cr must be constant on each connected component of the manifold M. I do so by deriving a second formula for curP |j from the Curvature Equation. A comparison of this expression with that given by the E23 Equation will show a must be constant. Utilizing Ricci identities and the definition of lie^ we have 0 = b i\k\j + b \i\j = ^ ik\j k\i\j + h\j\i = b Rksij k\j\i - bj\k\i = bj\k\i ~ j\i\k = b Rjski h\j\k ~ i\k\j b b k h h b Risjk e Summing the five equalities above and applying the first Bianchi identity produces curl Contracting with ij|fc = s ife ks fc:/ we have ay lk + lie |j - lie |j. -2b R ij 3 curP^ - 2b mE + lie\| - • i i0 0 The second and third terms on the right are computed with the Basic Equation. The formulas are i|0 = (n - |H| ) CT - a (1 - ||6|| ) (a/? + Oo), 0 | i = a| -/?6^-ia (n-2||6|| lAV lie' 2 2 |0 2 2 0 + \(5Q0 + iO'curlio - P & + l)/3 + i a ( 2 | | 6 | | - n ) e 2 { i - . The Curvature Equation produces 2feRiCjo l = O curljo l 11^10 + ^(0,10 + 00^)) + /?{2(1 + || | )Ric - ^curFcurLj 2 -(n-1) [K(3-||cbr) At this point it is convenient to compute b Q^Q, l + i6Vj } 6 0o|i and ©V. l 0 5. N E C E S S A R Y C O N D I T I O N S F O R EINSTEIN: F I N A L F O R M 17 Three intermediary computations. In these computations I make frequent use of the facts bi\j = 5 (lie^j + curly) and 6j|jCurF = ^curly-curl - . 1 7 Notice that b @i = 6*tVcurly = 0, because curly is skew-symmetric. Whence, the first term is given by the Basic Equation as l tVe^o = \®&P - icrGo - ie'curlio. The computation of the second term &*©o|i is involved. We require both the Basic Equation and the expression for curly ^ derived at the beginning of this section. I will make use of the facts < 1 and b b Rhijk = 0. (The last equality is a result of the skew-symmetry of the Riemann curvature tensor in (h,i).) The result is h tVe |i = \®0P l - ^ c u r h . 0 + ^crOo + ||6|| CT|O - b a^. 2 0 l With the E23 equation we may express the final term as = icurlycurF - {2Ric + |(n - 1) a } ||6|| 2 + i(n-l)e 6 i i 2 - i(n-l)6V . N The final maneuver. Substituting the three intermediary computations into the expressions for tVRicjo, ne ^ d *o|i produces 1 curl* |i 0 = 26 mc + lie%| - h V a n ne i i0 = 0 o N 2Ric p + (n - 1) \a p + i e W l i o + 2 5 t T 6 0 + CT| 0 - f ||6|| cT| 2 0 A comparison of this expression with that given by the E23 equation indicates | ( 1 - ||cb|| )cT| = 0. Since the norm of b is strictly less than 1, we must have tT|o = 0. In particular, all covariant derivatives vanish. Since cr is a scalar (i.e. a function of x), this means all the partial derivatives of a are zero. Therefore a is constant on each connected component of M. 2 0 5. Necessary conditions for Einstein: F i n a l form We may now state the final form of the three necessary conditions for a Randers metric to be Einstein. 5.1. The Basic Equation. The Basic Equation undergoes little cosmetic alteration. I emphasize below the constancy of a: lieoo + 2/? 9 = (const.cr) (a 2 0 p ). 2 Equivalently, lie + bi 0 + h 0j = (const.cr)(a - hih ). ifc fc ik k 6. C H A R A C T E R I Z A T I O N O F E I N S T E I N M E T R I C S O F R A N D E R S T Y P E 18 5.2. T h e C u r v a t u r e E q u a t i o n . To derive the final form of the Ricci Curvature Equation we replace all appearances of lieoo, li oo|o d CUT\ \iejo with the formulas given by the Basic Equation in §2.3.1 and §2.3.3; and curl ^. with the expression given by the E23 equation of §2.5.3. e a n j 0 7 Ricoo = (a + P ) Ri — |a curl^curl/ j — ^curFgCurljo 2 2 c 2 l - ( n - 1) {±a (3a - /3 ) 2 2 2 + i(9 ) + ie 2 0 0 | } • 0 The indexed form of the final version of the Curvature Equation is Ricjfc = (aik + hi b )Ric - \a curl^curl/y - icurl^curL;*. k ik -(n - 1 ) {±a (3d 2 5.3. ifc - k b ) + ISiQk + \(e k + e )} i]k kli . T h e E23 E q u a t i o n . The constancy of a updates the E23 Equation to curl j = 2Ricf3 + (n - 1) {gtr /? + \a@ + ie^'curljo} 2 0|j 0 or , curr ^ = 2Ricbi + (n - 1) | | c r 6 i + \aQi + ^ c u r l j i j . 7 6. 2 C h a r a c t e r i z a t i o n o f E i n s t e i n metrics o f R a n d e r s t y p e I have shown that the Basic, E23 and Curvature Equations are necessary for a Randers metric to be Einstein. The are also sufficient, as we shall see in the following section. 6.1. T h e three necessary conditions are also sufficient. Recollect the Randers metric F is Einstein if and only if both Rat and Irrat vanish (§2.2). Assume the Basic, E23 and Curvature Equations hold as given in §2.5.1-2.5.3. I will show these three conditions imply Rat = 0 = Irrat. First, note that the preliminary (§2.3) and final (§2.5) forms of the three equations are equivalent. To see this, it suffices to show the final forms of the Basic, E23 and Curvature Equations imply the preliminary forms. • Thanks to the constancy of a the final forms of the Basic and E23 Equations are immediately equivalent to their preliminary forms. • The final form of the Curvature Equation was deduced from the preliminary form by replacing the terms lieoo, lie | , curl liejo and curr ^. with the expressions given by the Basic and E23 Equations. Certainly, we may reverse this algebraic substitution to resurrect the preliminary form of the Curvature Equation from its final form. Having seen that the preliminary and final forms of the necessary equations are equivalent, it remains to show that the preliminary forms imply Rat = 0 = Irrat. To that end, assume the preliminary forms of the Basic, E23 and Curvature equations hold. Recall, from §2.3.3, that we deduced the preliminary E23 Equation from Irrat = 0 by: J 00 7 0 0 6. CHARACTERIZATION OF EINSTEIN METRICS OF RANDERS T Y P E 19 • replacing Ricoo with the expression given by the preliminary form of the Curvature Equation (§2.3.2), • substituting the three terms lieoo, he o|o ^ curr liejo with the formulas given by the Basic Equation, • and dividing by a uniform factor of a — (3 . a n ( 7 0 0 2 2 By reversing the three algebraic steps above we may derive Irrat = 0 from the E23 Equation. Similarly, we derived the preliminary form of Curvature Equation from Rat - /flrrat = 0 by: • using the Basic Equation to replace lieoo + 2/?0o with a(a — ft ), • and dividing by a — (3 . Again, reversing the two step above will give us Rat — /3Irrat = 0. Hence, Rat = 0 since Irrat = 0. It follows that the Basic, E23 and Curvature Equations of §2.5.1-2.5.3 are necessary and sufficient conditions for Rat = 0 = Irrat. The vanishing of Rat and Irrat characterizes Einstein metrics, allowing us to formalize the previous three sections in the following theorem. 2 2 2 2 6.2. The theorem. THEOREM 2.3 (Einstein Characterization). Let F — a + P be a Randers metric on a smooth manifold M of dimension n > 2, with a = •\jd j{^ y y^ i i j and (3 — bi^y . Then (M, F) is Einstein with Ricci curvature Ric(x) if and only if the Basic Equation, the Curvature Equation, and the E23 Equation of §2.5.1-2.5.3 are satisfied. 1 Explicitly, a Randers metric is Einstein with Ricci curvature Ric{x) if and only if there exists a constant cr, such that the following three equations are satisfied: liejfc + bi 6fe + bk Oi = cr(d - bib ) ifc Rki = fe (dik + h bk)Ric - \d curl' - curly - icurl^curljfc 1 ik k 7 - ( n - 1) {^tr (3d - k b ) + \GiQk + I (e 2 ik k ilk + e |i)} fc curr .^. = 2Ricb~i + (n - 1) | | c r 6 i + ±CT0; + ^G-'curljjj . 7 2 Tracing the Basic Equation tells us a takes the geometrically significant value n-||6|| ' 2 (Recall, b ^ is the divergence of the vector field b d i.) I would like to emphasize a few properties of the Basic, Curvature, and E23 Equations. 1 l x 6. C H A R A C T E R I Z A T I O N O F E I N S T E I N M E T R I C S O F R A N D E R S T Y P E 20 • The equations are tensorial and highly non-linear (see Ricjfc) second order partial differential equations. • They are polynomial in the tangent space coordinates y , whereas the Einstein criterion is not (unless b = 0). The polynomial structure of the three characterizing equations substantially reduces the computational complexity. Recall the Einstein criterion (§2.2) is K — F Ric(x). While testing Einstein examples we found cases in which Maple (the mathematical software) was unable to complete computations of K . So for these examples we could not verify the Einstein criterion. Maple was able, however, to verify the three characterizing equations. • The E23 Equation is redundant. That is, given the Basic and Curvature Equations, the E23 Equation necessarily holds. The redundancy is almost evident from the proof that a is constant. In §2.4 we derive a second expression for curP |j which, with the constancy of a , is identical to the E23 Equation. During the computation we use the E23 Equation only to compute G ^ . Now that we know a is constant, it is possible to recompute 0 ^ via the formula b ^ = 26 RiCjo + hV^Q — lie* j , introduced at the beginning of §2.4, and requiring only the Basic and Curvature Equations. With this alteration, the argument that illustrates the constancy of a, now shows the Basic and Curvature Equations imply the E23 Equation. Although superfluous, the E23 Equation is enormously useful in computations. For this reason it is included in the Einstein characterization theorem. • A similar characterization for those special Einstein Randers metrics of constant flag curvature has also been derived. The interested reader might consult [BR03a, MS02a]. The next step is to parley the characterization into an explicit geometric description of Einstein metrics of Randers type. This is done in Chapter Three, where we re-express the characterizing equations within the context of the perturbation technique based on Zermelo's problem of navigation. It happens that this is the ideal setting to discuss Einstein metrics of Randers type. l l 2 t l i 0 2 l Q l 0 i CHAPTER 3 The Einstein navigation description Given the straightforward construction of a Randers metric as the sum of a Riemannian metric a and a linear term tb, it seems most natural to derive an Einstein characterization based on a and b. Such reasoning motivates the characterization theorem of Chapter 2. However, an overwhelming majority of the Einstein examples known today are constructed as solutions to Zermelo's problem of navigation (§1.5.1). This suggests the structure underlying the perturbation technique may be well-suited to a study of Einstein metrics of Randers type. So guided, I rewrite the characterizing equations of §2 in terms of the Zermelo data (il, W). I find that a Randers metric is Einstein with Ricci curvature Ric = (n — 1)K if and only if the Riemannian a is Einstein with Ricci scalar (n — 1)(K + and the Lie derivative of a by W is —aa. With the navigation description in hand a Schur lemma and Matsumoto's Identity follow quickly. The Schur lemma constrains the geometry in dimensions three and higher by declaring the Ricci scalar of an Einstein Randers metric to be constant. The Matsumoto Identity, describing the constant cr, generalizes a result of Matsumoto for Randers metrics of constant flag curvature [MatOl], to the Einstein setting. J Q O ~ 2 ) , 1. Randers metrics as solutions to Zermelo's problem We know solutions to Zermelo's problem are given by Randers metrics (§1.5.1). The converse must be checked before deriving a classification based on a and W: can every Randers metric F — a + (3 be realized as a solution to Zermelo's problem of navigation on a Riemannian manifold (M, il) under an external force Wl Happily, the answer is 'yes'. The desired Riemannian metric il and vector field W may be constructed as follows. Given a = dijy y and (3 = b~iy , let A := 1 — ||t>|| . Consider the Riemannian metric and vector field defined by 2 l 3 l aij := A(oij - bibj), 2 W = -— . l Recall ||tb|| < 1 is required of all our Randers metrics. This means A > 0, and W is well-defined. Also a consequence of < 1, we have ily-yV = A ( a - / ? ) > 0 , 2 21 2 2. C O V A R I A N T D I F F E R E N T I A T I O N W I T H R E S P E C T T O a 22 with equality only when y = 0. Hence a is positive definite, and is therefore a genuine Riemannian metric. After checking W ciijW —: \\W\\ — \\b\\ , a straightforward computation confirms F = a + (3 solves Zermelo's problem for (a, W). Explicitly, l J 2 2 ciij [1 - a (W^ + WjWj [l-a (W)} ' - _ 2 13 2 2 1 -Wj \-a\W)' where Wi = aijW . Equivalently, 3 _ a (y) [l-a (W)] 2 2 +W 2 ' [l-a (W)] 2 _ 2 0 2 -W 0 l-a (W)' H 2 with a (y) := dijy y . In particular, F solves Zermelo's problem for a and W. Finally, with some linear algebra, the inverse b? of a is given by 2 x 3 J a = To A • 2 Again, ||tb|| < 1 guarantees a is well-defined. %d 2. Covariant differentiation w i t h respect to a This section is the first of four containing the computations to express the characterizing Basic, E23 and Curvature Equations in terms of the Riemannian metric a, the vector field W, and covariant derivatives of W with respect to a. Let the colon denote covariant differentiation with respect to o. For example, the covariant derivative of W is given by W'-.j := W\ +Wf s xi sj . Here, 7* - denotes the Christoffel symbols of a. Indices on W , Wi and their covariant derivatives are lowered and raised by the metric ciy and its inverse 0. Because the covariant derivatives of the characterizing equations are with respect to a we need to understand the relationship between the Christoffel symbols of a and a. To wit, the connection coefficients of d and d are related by l fc 7 jk 7 jk "t" V jk ' 3. T H E B A S I C E Q U A T I O N 23 with + 2A +• 3. The Basic Equation I n this section I shall re-express the B a s i c E q u a t i o n of §2.5.1 lieoo + 2 / 3 6 o = c r ( a i n t e r m s o f a a n d W. 2 - (3 ), 2 T h e e x p r e s s i o n s for a a n d /3 a r e g i v e n i n § 3 . 1 ; w e n e e d o n l y c o m p u t e lieoo a n d © o F i r s t , c o n s i d e r lieoo = 2t>o|o- W i t h t h e f o r m u l a for 6, i n § 3 . 1 w e h a v e = b - i:j bn s s i:j N o w lieoo is g i v e n b y c o n t r a c t i n g t h i s e x p r e s s i o n w i t h M o v i n g o n t o 0Q, a q u i c k c a l c u l a t i o n curly = bi\j - 2y y . l 3 yields bj\i 4( - A2 (WiW W a a:j - WWr.iWj) . W h e n c e © o = tVcurljo is g i v e n b y ©o = 1 + ^WW 2 a - W W :s s 8:0 0 -^W'W'Ws-.tWo. I t f o l l o w s n o w , t h a t t h e B a s i c E q u a t i o n is e q u i v a l e n t t o a expression -2WQ:0 - ^a W W W 2 s t s:t = £ a 2 . preliminary 3. T H E BASIC E Q U A T I O N 24 This formula may be refined. First differentiate by y and y for l -w d - w - z\& w w W ..t =^ij • a i:j j:i t ij 9 Contract this with W W to obtain l J w ww s = t s:t ~\\w\\ . 2 Substituting this equality into the preliminary expression yields The indexed form, obtained via differentiation by y and y \ describes the Lie derivative of ay by W x 3 Wi-j + Wj-.i = -0-CLij . This the Liew Equation, so named because Wi-j+Wj-i is the Lie derivative of a by W. It says W is an infinitesimal homothety of a. When a vanishes, W is Killing (an infinitesimal isometry). A quick calculation shows the Lievt/ Equation is equivalent to the preliminary expression above. This means, The Lievv Equation is equivalent to the Basic Equation. The Lievi/ Equation provides a number of useful identities that I shall apply to the derivation of the perturbation versions of E23 and Curvature Equations. These include W s :s W W ^ t W W s s:t W t + Wo: ) s :0 s:0 W W 0:S S - W W 3 s 0: :0 2 = \no = s:0 W {w -^CJ\\W\\ 2 s (Wo-.s + W ) s -\na = + W W\ s:t = s:0 = -aWo \o a 2 2 = 0 W - f c (W* + W- *) - -aW W W W\ = -o-w w .. :0 0: (Wf.o + S S W (W s s:t tf>b:t) s s:0 + W :s) ft -o = t w i:j:k +w j±k W'W'W^ = 0 = 0. s 0 -°-W W s s s:0 4. T H E E 2 3 EQUATION 25 4. The E23 Equation Next we turn our attention to the E23 Equation of the Einstein characterization (§2.5.3) curP = 2Ric(3 + (n - 1) {\a p + ±CT0 + ^ c u r l / o } . 2 0|j o Two terms require consideration: curr ^. and ©•'curL/o- Let's start with 7 curl . = a^curlioij- = a (curl j - curl rfy- - curl ?f ) . J y 0|j i0: s0 is 0j The tensors a - and curly are evaluated in §3.1 and §3.3, respectively. After simplifying with the identities derived from the Liew Equation we have 1 7 curl J 0|i = W a - W a 0: :a :0:a + ^" ^(2 + ^W W W A a 0 k H^H ^0 2 2 a t + ACT WW S S:0 + 2 A W W ^ W t o +2W W W W Vr t^ s a t . r 0 :a r: Much of the work for the second term © 'curljo = ©ja^curljo has been done: d is given in §3.1; ©j and curljo are computed in §3.3. The Liew Equation identities imply J lJ 6>'curljo = ^ ( o W W W W t 4W a t r :a r: + AL\W W\ W .o - tr #b) • s s 2 t Now the E23 Equation is expressed in the preliminary form A (W s - W , %) = (n - 1) (2K + ger ) W + 2W W W , * , 2 :0:s S 0 0 0 S t where K:=-±rRie. n—1 The algebraic derivation may be reversed to recover the E23 Equation from the Liew Equation and the preliminary form above. As was in the case in the computation of the Liew Equation, the preliminary expression may be refined. To do so differentiate by y and contract with W to obtain -2W W \ = (n - 1) (2K + \a ) \\Wf . f l s 2 s: t Then, a straightforward computation shows the preliminary form is equivalent to W .o: ~ W % = (n- 1) {2K + la ) W . s 2 S 0: 0 This is the E23 Equation , the navigation version of the E23 Equation. To ensure the navigation description we are deriving is both necessary and sufficient for a Randers metric to be Einstein, it is important that we keep track of the relationships between the characterizing equations (which are necessary and sufficient by Theorem 2.3) and their navigation forms. Assume the Liew Equation holds. Above I mentioned that the E23 Equation and the preliminary form are equivalent. Since the preliminary form holds 5. T H E C U R V A T U R E E Q U A T I O N 26 if and only if the E23 Equation does, we see the E23 Equation is equivalent to its navigation version. At the end of §3.3 I observed that the Basic and Lievi/ Equations are equivalent conditions. Taken together these remarks imply The Basic and E23 Equations hold if and only if the Lievv and E(23) Equations do. I close the section with an identity derived from the Lien/ and E(23) Equations W = (n-1) {K + ±a )W . This formula will come in handy during our computation of the Curvature Equation. s 2 :0:s 0 5. T h e C u r v a t u r e E q u a t i o n In this section I turn to the characterizing Curvature Equation of §2.5.2 Ricoo = (a + (3 ) Ric - |a curFcurly - icurl curL;o 2 2 2 J 0 _ ( „ _ i) {^a 2 {3a - P ) + l O + i9 |o} • 2 2 2 0 Four terms of the term above require our attention. They are curFcurly, curpQCurljo, @o|o> d Ricoo- Throughout these computations I assume that the Liew Equation, 623 Equation, and related identities hold (§3.3, 3.4). Let's begin with curF curly. In §3.1 and §3.3 we computed a and curly, respectively. We have a n lJ curF = a tV c\ix\hk ih = k A (w - W ^ , j:i i]j and curFcurly = ^ | ( n + 2) a A - 2a - 4 A W W\ 2 2 s t s +8 r^r ^ } . s :( Similarly, curlVurlj-o = ^ { - < T ( ^ O + A a ) + 4A 2 2 2 + 4A W W .. 2 S 0 + 4W 2 0 S 0 2 + 4CT W^W^Wr-.t (W W ) s 2 s:Q AW W W S 0 S:0 + SAW0W W .sWP .oY 8 t This takes care of the curl terms. Next up is the covariant derivative of 0. With the expression for 0 computed in §3.3 we have @o|o = ©0:0 - ®sV oo s = ^ 3 j 5 (4 W W W , Wr:t ~ cr ) ( A d + 2 W Q ) s l r 2 2 2 s 2 A W .QW O + iAWoW'W^Wf.o 2 S S: + 2A W W 2 s s:0:0 5. T H E C U R V A T U R E E Q U A T I O N 27 The fourth and final term of the Curvature Equation is more involved: to compute Ricoo with respect to a and W I return to Berwald's formula (in split and covariantized form) of §1.5.3. Recall, Berwald's formula allowed us to compare the spray curvature K of F to the spray curvature K of the 'background' metric a. This is but one case of a more general comparison formula (see [BR03b]). The general comparison formula allows us to compare the spray curvature K j of a to the spray curvature K of the 'background' metric a. First define l l k k l l k <? = WjkV y : J • k Then the geodesic spray coefficients are related by & = & +f . Berwald's formula tells us the Ricci tensor Ricy of a is related to the Ricci tensor RiCy of a by RkoO =- RiCQO + {2f:i - y (C:j)y< " (fV^V + 2 ^ % ^ } J = Ricoo + AT( {(n-l)A (l-A)iY + A ^ 2 2 - (n + 1) A + : T T : S W^W^Wr-.t - B(7n-3)A - + W i ( n _ 1 ) A 2 ] ( 7 2 A j a + J 2 (n - 1) A K - 2(n + 1) W WW W + AW W + [i(n + l) - | A ( 5 n + l ) ] c T } l t o s l r :s s r:t t :t 2 2 :s - (n + 1) a A Wo W W - {n + 1) A {W W .. ) s s s:0 - (n + 1) S S 2 W W 2 a - 2(n + 1) A Wo W W\ W - (n - 1) A 2 W 0 AW W 2 S 0 S:0 s s : 0 : 0 Taken together, the four expressions computed above allow us to miraculously rewrite the the Curvature Equation as Ricoo = {n-l)(K + ^cr ) a . 2 2 Call this the Einstein Equation, because it says the Riemannian metric a is Einstein, with Ricci scalar (n — l)(K + T^CT ). Let's take a moment to review. We assumed the Liew and E23 Equations hold, and then algebraically derived the Einstein Equation from the Curvature Equation. We may reverse the process and resurrect the Curvature Equation from Einstein Equation. In particular, given the Liew and E23 Equations, the Curvature and Einstein Equations are equivalent. 2 6. T H E THEOREM 28 At the end of §3.4 we remarked that the Liew and E23 Equations hold if and only if the Basic and E23 Equations do. Therefore, the Liew, E(23) and Einstein Equations are equivalent to the characterizing Basic, E23 and Curvature Equations. The characterization theorem (Theorem 2.3) implies The Randers metric given by perturbing a by W is Einstein with Ricci scalar Ric = (n — 1)K is if and only if the Lie^y, E(23) and Einstein Equations are satisfied. 6. The theorem We have just seen that a Randers metric with underlying a and W is Einstein if and only if the Liew, E(23) and Einstein Equations hold. The E23 Equation is redundant. (This is not a surprise. I remarked, in §2.6, that the Basic and Curvature Equations imply the E23 Equation.) LEMMA 3.1. The Liew and Einstein Equations imply the E23 Equation. P R O O F . The implication may be seen in the following way. With the Ricci identities for W, and the Liew Equation identity Wi j + Wj± = 0, we have the following 5 equations : :k W i:j:k - W W i:k:j +W = i:k:j k WR s isjk = 0 k±j - f e y + fe:i = -W"Rksij ' -Wk:j:i - W k:i = 0 j: W :i ~ fe:fc = WR , S j:k jski which sum to produce Wj-.i-.k = ~2W R with a little help from the first Bianchi identity. Trace on (i, k) and apply the Einstein Equation to derive . fe:fc W^s-Wj,', s - ksij = 2W Ric s sj = W (n - 1)(2K + |cr )a = (n - 1)(2K + \o- )Wj . s 2 sj 2 This is the indexed form of the E23 Equation. Contracting with y produces the E23 Equation. • 3 We are now ready to state the navigation description for Einstein metrics of Randers type. Before I do so, recall the perturbing vector field W must satisfy a(W) < 1, if the resulting Randers metric is to be strongly convex. T H E O R E M 3.2 (Einstein navigation description). Suppose the Randers metric F solves Zermelo's problem of navigation on the Riemannian manifold (M,a) under the external force W, a(W) < 1. Then (M,F) is Einstein with Ricci scalar Ric =: (n — 1)K if and only if 6. T H E THEOREM 29 • The Einstein Equation Ricoo = (n - l)(K + ^a )a 2 2 holds. That is, the Riemannian metric a is Einstein with Ricci scalar (n — l)(K + • The vector field W is an infinitesimal homothety of a. To be precise, the Liew Equation J Q C T Wj i: + 2 ) . Wj i = : -LTCLij is satisfied. I refer to this theorem as a 'description' and distinguish it from the 'characterization' theorem of §2.6.2 because it provides a concise description of Einstein metrics of Randers type, in which the underlying geometry is explicit. One consequence of the theorem is that the Ricci scalars of the Einstein F and ci agree when W is Killing. Matsumoto's Identity of §3.8 describes in greater detail the values a may take, and allows us to refine the relationship between the Ricci scalars. Constant flag curvature and 3-D rigidity. We say a Finsler metric is of constantflagcurvature if the flag curvature K(x, y, w) = K is constant (§1.4). Since the Ricci scalar is the sum of n — 1 flag curvatures, we see that constant flag curvature metrics are Einstein with Ricci scalar (n — 1)K. These metrics are the Finslerian analogs of Riemannian metrics of constant sectional curvature. Recollect that a three-dimensional Riemannian metric is Einstein if and only if it is of constant sectional curvature. It is not known if this rigidity holds for arbitrary Finsler metrics. However, the rigidity does hold for Randers metrics. The proof rests on a navigation description for Randers metrics of constant flag curvature similar to the Einstein description above [BRS03]. T H E O R E M 3.3 (Constant flag curvature navigation description). Suppose the Randers metric F solves Zermelo's problem of navigation on the Riemannian manifold (M, a) under the external force W. Then (M,F) is of constantflagcurvature K if and only if • The Riemannian metric a is of constant sectional curvature (K + j^cr ). That is, 2 Rhijk = (K + ^ C T ) (aijiihk - Clikdhj) • 2 • The vectorfieldW is an infinitesimal homothety of a. To be precise, the Liew Equation • Wj i: is satisfied. + Wj-.i = -LTCLij 30 7. A SCHUR L E M M A As in the Einstein setting, the derivation begins with a characterization of Randers metrics of constant flag curvature in terms of a and b. The description is then produced by re-writing the characterizing equations within the context of Zermelo navigation. This is joint work with David Bao. Z. Shen has solved the Liew Equation and obtained a list of W for each of the three standard Riemannian space forms it; see also §4.2. The detailed classification of constant flag curvature Randers metrics is derived in [BRS03]. The three dimensional rigidity is immediate. A Randers metric is Einstein if and only if the Liew Equation holds, and it is Einstein with Ricci scalar (n — 1)(K + Tgcr). In three dimensions, all Riemannian Einstein metrics are of constant sectional curvature. Hence it is of constant sectional curvature K + -j^cr , and the conditions of Theorem 3.3 are met. Interestingly, the two navigation descriptions also tell us any Einstein Randers metric arising as a solution to Zermelo's problem of navigation on a Riemannian space form is necessarily of constant flag curvature. 2 2 7. A Schur l e m m a It is my goal in this section to prove a Schur lemma for the Ricci scalar of a Randers metric. Essentially, the lemma constrains the geometry of Einstein metrics in dimension greater than two, insisting that the Ricci scalar be constant. Historically, this is the second Schur lemma in (non-Riemannian) Finsler geometry. In 1973 del Riego [dR73] proved a Schur lemma for the flag curvature of Finsler metrics. Matsumoto [Mat86] presented a second proof in 1986. In two dimensions, the Ricci scalar Ric of a Riemannian metric is the Gaussian curvature of the surface. That is, all Riemannian surfaces are Einstein. Since the Gaussian curvature is not constant in general, we see the Schur lemma fails for Riemannian, and therefore Randers, metrics in two dimensions. At this point it is natural to ask if the Schur lemma must also fail for non-Riemannian (b ^ 0) Randers surfaces. The answer is 'yes'. Section 4.1 develops a class of non-Riemannian Randers metrics with Ricci scalar a non-constant function of x alone. In particular, these non-Riemannian metrics are Einstein, but fail the Schur lemma. L E M M A 3.4 (Schur). The Ricci scalar Ric(x) of any Einstein Randers metric in dimension greater than two is necessarily constant. P R O O F . The lemma follows immediately from the Riemannian Schur lemma. To see this, suppose F is an Einstein metric of Randers type, solving Zermelo's problem on the Riemannian manifold (M, it). The Einstein Equation says it is Einstein with Ricci scalar (n — 1)(K + By the Riemannian Schur lemma (see Appendix B), K + j^o is constant when n > 2. Since a is constant (see §2.4), we see K = -TIT-Ric must be as well. • J Q O ~ 2 2 ) . 31 8. T H E M A T S U M O T O I D E N T I T Y Discussion. One proof of the Riemannian Schur lemma is based on the second Bianchi Identity. Unfortunately, the Finslerian second Bianchi Identity does not yield itself to a similar argument. This has led geometers to expect the Schur lemma to fail in general. However, the establishement here of a Randers Schur lemma permits some optimism for arbitrary Finsler metrics. The interested reader may find a discussion of these issues in the second appendix. There I include • a proof of the Riemannian Schur lemma, • a discussion of the obstacle presented by the Finslerian second Bianchi identity, and • a second proof of the Randers Schur lemma based on the characterization theorem of §2.6.2. 8. The Matsumoto Identity Matsumoto has shown [MatOl] that any Randers metric of constant flag curvature K satisfies a{K+±a )=0. 2 The identity may be extended to Einstein metrics. (Recall, constant flag curvature metrics are Einstein, with Ricci scalar Ric = (n — 1)K.) I will show any Einstein metric of Randers type, with Ricci scalar Ric =: (n — l)K, satisfies the Matsumoto Identity for Einstein Randers metrics a(K + ^ C T ) = W K , 2 when n = 2 l :i a(K + jetr ) = 0 when n > 2. 2 The proof rests on the Ricci Identity for Wi-j — Wj-i where R£ - is the curvature tensor of a. Trace the Ricci Identity on (i, k) and (h, j) to obtain k (w . i:j - W .^ ri A = (\V i - Ricij . h The Ricci tensor Ric^ of a is symmetric. Since W —W is skew-symmetric the right hand side must vanish. With the help of the E23 Equation this i:j J:l 8. T H E M A T S U M O T O 32 IDENTITY implies o= = (w^ -w^ :i :j 2(n-l){w^K + (K + :j ^a )W } 2 j :j = 2(n - 1) {w K - \o(K + ^a )} . 3 2 :j The identity now follows from the Schur lemma. A second proof of the Matsumoto Identity, based on the Einstein characterization theorem of §2.6.2, may be found in Appendix A. 8.1. T h e refined description for constant Ric. The navigation de- scription tells us that every Einstein Randers metric arises as the solution to Zermelo's problem of navigation on an Einstein Riemannian manifold (M, a) under an infinitesimal homothety W. Matsumoto's identity further refines this description when Ric is constant. To see how this is so, suppose the Ricci scalar Ric = (n — 1)K of F is constant. The Matsumoto Identity reads cr(fY + icT ) = 0. 2 J Recollecting that (n — 1) (K + j ^ t r consider the following three cases: 2 ) is the Ricci scalar of the Einstein a, (+) If Ric > 0, then a = 0. In particular, the Ricci scalar of a is Ric, and W is Killing. (Equivalently, W is an infinitesimal isometry.) (0) If Ric = 0, then a — 0, and F solves Zermelo's problem of navigation on a Ricci-flat Riemannian metric under an infinitesimal isometry. (-) When Ric < 0, either a = 0 or a = ±^y/\K\. o If a = 0, the Ricci scalar of a is Ric and W is an infinitesimal isometry. o If cr = ±4:y/\K\, then a is Ricci-flat, and W is not Killing. Alternatively, any solution of Zermelo's problem of navigation on a Ricci-flat Riemannian manifold under a infinitesimal homothety with tr / 0, is an Einstein Randers metric with negative Ricci scalar Ric = — ^(n — l ) t r . 2 CHAPTER 4 Case Studies In this chapter I apply the Einstein navigation description of Theorem 3.2 to examine some special classes of Einstein metrics of Randers type. I begin in dimension two, constructing non-trivial Randers metrics with Ricci scalar a non-constant function of position x alone. The metrics are Einstein, and provide non-Riemannian counter-examples to the Schur lemma in two dimensions. Next, in Section 4.2, I discuss a recent classification of constant flag curvature Randers spaces. The result is analogous to the Hopf classification of Riemannian geometry. Last, Section 4.3 investigates closed Einstein manifolds. I will demonstrate that any Einstein Randers metric of negative Ricci scalar on a closed manifold is necessarily Riemannian. Also, any Ricci-flat Einstein Randers metric on a closed manifold must be Berwald. 1. R a n d e r s metrics w i t h K = K(x) in d i m e n s i o n 2 The first case study is of surfaces of rotation in M . We shall see that solutions to Zermelo's problem of navigation under infinitesimal rotations are Einstein with a non-constant Ricci scalar Ric = Ric(x). These metrics, constructed jointly with D. Bao [BR03a], are non-Riemannian counter examples to Shur's lemma in dimension 2. To begin take any surface of revolution M , obtained by revolving a profile curve 3 (0, g(ip)) /(</>), in the right half of the yz-plane around the z axis. The ambient Euclidean space induces a Riemannian metric a on M. Parameterize M as follows: (6,<p) - (/(</>)cos(t9), /Msin(t9), g(<p)). Now consider an external force acting on M represented by the infinitesimal isometry W := ede- Here e is a constant to be specified momentarily. By limiting the size of our profile curve if necessary, there is no loss of generality in assuming that / is bounded. Choose e so that e|/| < 1 for all LO. Then the solution to Zermelo's problem is the following Randers metric on M: A a ~ / n 2 2 / + ^ 2 ( 1 _ l - e e 2 2 f / 2 ) ( / 2 + g 2 ) _ ' r 33 ^ ~ l - e e u f 2 2 / 2 • 2. S P A C E S O F C O N S T A N T F L A G C U R V A T U R E 34 Here, U8Q + vd^ represents an arbitrary tangent vector on M , and / , g abbreviate the derivative of / , g with respect to <p. Because the vector field W is Killing a vanishes, and the Einstein navigation description says the Ricci curvature K of F is identical to the Gaussian curvature of the Riemannian metric a K HfS-fg) = f(f +9 ) 2 2 2 where the dots indicate derivatives by (p. A R a n d e r s m e t r i c o n the elliptic p a r a b o l o i d . Specialize to the surface of revolution z = x + y in R , and set the multiple e in W to be 1. The resulting Randers metric lives on the x + y < 1 portion of the elliptic paraboloid, and has Ricci curvature 4/(1 + 4x + 4y ) . It reads: 2 2 3 2 2 2 _ 2 2 y/ (-yu<+ xv) + {(1 + 4a; ) u + 8xy uv + (1 + Ay ) v }T> 2 0= W ~ X V 2 2 , \\b\\ = x +y , 2 2 2 2 where V := 1 - x - y . 2 2 2 A R a n d e r s m e t r i c o n the torus of revolution. Specialize to a torus of revolution with parameterization (ti,ip) H-> ( [2 + cos(cp)]cos(tf), [2 + cos(ip)]sin(tf), sin(cp) ). Set the multiple e in W to be | . The resulting Randers metric on the torus has Ricci curvature cos(</?)/[2 + cos(</>)]. It is given by: 4 y/16 [2 + cos(y)] u + {16 - [2 + cos(y)] } v 16 - [2 + cos((p)] 2 a 2 2 —4 [2 + COS(cp)l It 2 " 2 2 = 16-[2 cS)P ' . W l t h ..r.,9 , 11611 + i =^ r „ [ 2+ C O S . . l 9 ^ • )] 2. Spaces of constant flag curvature Constant flag curvature spaces form a distinguished subclass of Einstein spaces. As indicated in Section 3.6, all Randers metrics of constant flag curvature arise as solutions to Zermelo's problem on an Riemannian space of constant curvature under an infinitesimal homothety. This result, along with the Hopf classification of constant curvature Riemannian spaces, allows a complete classification of constant flag curvature Randers metrics. The work is joint with D. Bao and Z. Shen; the details and additional results may be found in [BRS03]. My primary goal in presenting the classification theorem here is to provide the reader with an explicit recipe for the construction of these special Einstein Randers metrics. First, some notation • Q = (Qij) is a skew-symmetric matrix and C = (C ) a vector, both constant; • Qx denotes (Q jX ), and x := (x ); l % J l 2. SPACES OF CONSTANT FLAG CURVATURE 35 • all indices on Q, C, x are manipulated by the Kronecker deltas ciy and e5*J; • is the standard Euclidean dot product. T H E O R E M 4.1 (Classification [BRS03]). Let F(x,y) be a Randers metric on a smooth manifold M of dimension n>2, solving Zermelo's problem of navigation for the Riemannian metric a and vectorfieldW. Then F is of constantflagcurvature K if and only if a is of constant curvature, and W is an infinitesimal homothety. Moreover, up to local isometry, the Riemannian space form a and the vectorfieldW must belong to one of the following four families. (+) When K > 0: a is the standard metric on the n-sphere of radius ^= and W = Qx + C + (x • C) x, with -{(Qx + C)-(Qx + C) + {x-CY} < K. l + (x-x) (Here W is given with respect to the projective coordinates parameterizing a hemisphere. See [BRS03].,) (0) When K = 0: a is the Euclidean metric on R and W = Qx + C, with (Qx + C) • (Qx + C) < 1 . n (-) WhenK<0: (—) either a is the Euclidean metric on R , and W = —\ox + Qx + C satisfies n e (Qx + C) • (Qx + C) + ax • (\<JX - C) < 1 with a = ±A^ /\K\•, (-)k or a is the Klein model of sectional curvature K on the unit ball in R , and W = Qx + C — (x • C)x satisfies t n 1 r,~ , -{(Qx + C)-(Qx + C)-(x-Cy} < \K\. • x) Furthermore, if M is simply-connected and a is complete, then the said local isometry is in fact a global isometry. 1 — (x The perhaps mysterious inequalities for Q and C are determined by the restraint that ||W|| < 1. Often times W will satisfy this restraint only on an open subset U of M. In these cases the Randers metric solving Zermelo's problem is defined only on U. In fact, the Klein model admits no non-trivial globally defined W satisfying the necessary inequality. It does however admit many locally defined Killing fields of norm less than one. In constrast, both Euclidean space and the sphere admit non-trival global infinitesimal isometries with 1111 < 1 ([BRS03]). In addition to describing the Randers metrics of constant flag curvature, the classification may be used to construct Einstein metrics of non-constant flag curvature. To do so, I recall the familiar fact that the product of two 3. CLOSED EINSTEIN SPACES 36 Einstein Riemannian spaces with a common Ricci scalar A is itself Einstein with Ricci scalar A. Fix A. Let M , be an n^-dimensional Riemannian manifold of constant curvature i = 1,2. Then Mi is Einstein with Ricci scalar A. Let Wi be an infinitesimal homothety on Mi. Modulo local isometry, Wi must correspond to one of the four families listed in the classification theorem. The product space M = Mi x Mi is Einstein, with infinitesimal homothety W = (Wi,W ). Theorem 3.2 tells us the Randers metric F on M solving Zermelo's problem of navigation under W is Einstein. Note that when A is nonzero, the Riemannian metric on M is not of constant sectional curvature. In particular F, though Einstein, is not of constant flag curvature (Theorem 3.3). 2 3. Closed Einstein spaces In the final case study, I restrict attention to closed (compact and boundaryless) manifolds. Assume throughout the present section that F is an Einstein Randers metric, w i t h Ricci scalar Ric, on a connected, closed manifold M. In particular, F solves Zermelo's problem of navigation for a Einstein Riemannian metric a on M under an infinitesimal homothety W. It is my goal in this discussion to determine the constraints placed on W, and therefore F, by the hypothesis that M is closed. I begin by observing that W must be an infinitesimal isometry. (Equivalently, W is a Killing vector field.) This is easily seen with the following divergence lemma. L E M M A 4.2 ([BCSOO]). Let V be any globally defined vector field on a closed Riemannian manifold (M, g). Let V denote the Riemannian connection. Then Setting g — a and V — W establishes the following lemma. LEMMA 4.3. The infinitesimal homothety W is Killing. Equivalently, a vanishes. P R O O F . By the divergence lemma we have Hence a is zero, and W is Killing. It follows now, from the Einstein navigation description, that the Ricci scalar of a is Ric. The following result of Bochner addresses the case Ric < 0. • 3. CLOSED EINSTEIN SPACES 37 T H E O R E M 4.4 (Bochner [Boc46, K N 9 6 ] ) . Let M be a connected Riemannian manifold whose Ricci tensor is negative definite everywhere on M. If the length of an infinitesimal isometry V attains a relative maximum at some point of M, then V vanishes identically on M. Because a is Einstein, the Ricci tensor of a is negative definite when Ric < 0. By Lemma 4.3 W is an infinitesimal isometry. The norm ||W"|| must attain a maximum as M is compact. Hence W = 0, and F — a is Riemannian. I have proven the following P R O P O S I T I O N 4.5. Let F be an Einstein Randers metric with negative Ricci scalar on a closed manifold M. Then F is Riemannian. Let's now turn our attention to the 1-form LO = Widx dual to W. The Laplacian of is given by the Weitzenbock formula [BCSOO] 1 LO Au = ( R V WJ ~ WV )r/x'. :j By the Einstein navigation description Ricj- = Ric cV • 7 Similarly, the Liew and E(23) equations of Sections 3.3 and 3.4 imply W = j i: :j -RicWi. Whence we compute ALO —2 Ric co . (I have made use of the fact that a = 0 in these computations.) Assume Ric = 0. Then the Ricci tensor of a vanishes and LO is harmonic. Another well-known result of Bochner implies to is parallel. = T H E O R E M 4.6 (Bochner [BCSOO]). Let 6 be a globally defined 1-form on a closed Riemannian manifold (M,g). Suppose the Ricci tensor of g is non-negative. Then 9 is harmonic if and only if it is parallel, that is, V0 = 0. The 1-form LO is parallel precisely when W is. In particular, W is parallel whenever Ric = 0. The converse holds as well, so long as F is non-Riemannian (ie. W ^ 0). (To see why we need to assume F is nonRiemannian, notice that F is Riemannian precisely when W = 0. In which case W is certainly parallel, but the Ricci tensor of F = a may assume any value.) To prove the converse assume that W is parallel, and not identically zero. Since a vanishes the E(23) Equation reads 0 = 2 Ric W. Because W is not identically zero on M, the Ricci scalar Ric must be zero. We have established the following P R O P O S I T I O N 4.7. Assume F is a non-Riemannian Einstein Randers metric on a closed manifold M. Then Ric = 0 if and only if W is parallel. 3. CLOSED EINSTEIN SPACES 38 It now follows as a corollary that a non-Riemannian Einstein Randers metric is Berwald if and only if Ric — 0. Assume (M.,J ) is an arbitrary Finsler space. Let Q denote the geodesic spray coefficients of T (cf. §1.4). l Then (M.J ) is said to be a Berwald space if the quantities (G )yjyk do not r l 7 depend on y [BCSOO]. For the moment, let us consider an arbitrary Randers metric F = a + (3, solving Zermelo's problem for (a, W). In particular, M need not be closed, nor F Einstein. It is known that F is Berwald if and only if b is parallel [BCSOO]. Look back at the expression for b^j in §3.3. Recollect our computation of this formula involves no assumptions on F. Notice that Wi-j = 0 implies b^j = 0. Conversely, if we assume b is parallel, and successively contract the right-hand side of the expression for b^j in §3.3 with W W , W" , and W , we may deduce that Wi-.j — 0. In particular, l 3 1 3 Let F be any Randers metric. Then b is parallel with respect to a if and only if W is parallel with respect to a. Now return to the realm of Einstein Randers metrics on closed manifolds. Proposition 4.7 and the discussion above imply C O R O L L A R Y 4.8. Assume F is a non-Riemannian Einstein Randers metric on a closed manifold M. Then Ric = 0 if and only if F is Berwald. Finally, let me close this section by observing that the rigidity of Proposition 4.5 does not hold when Ric = 0. P R O P O S I T I O N 4.9. The flat torus admits non-Riemannian Randers metrics of constantflagcurvature K — 0. P R O O F . A flat torus is simply Euclidean M modulo n linearly independent translations. The parallel vector fields on R are the constant vector fields. A constant vector field on R is certainly invariant under translation, and therefore defines a global, parallel vector field W on the torus. In particular, Zermelo navigation on the torus under W defines a non-trivial Randers metric. . • n n n CHAPTER 5 Open Questions It is natural to wonder which of the results established here, among Randers metrics, may be extended to Finsler spaces in general. To that end I have selected a few problems to discuss. 1. A Schur lemma Does the Einstein Schur lemma hold for arbitrary Finsler metrics? One proof of the lemma for Riemannian spaces rests on the second Bianchi identity. As I mention in §3.7 (and discuss in Appendix B), the Finslerian second Bianchi identity does not lend itself to an analogous proof. The intractable nature of the identity has led geometers to doubt the existence of a Schur lemma for Finsler metrics. However, my success in the Randers setting encourages a sanguine speculation. 2. Chern's conjecture S.S. Chern has conjectured that Every manifold admits a Finslerian Einstein metric. It is known that topological obstructions prevent some manifolds from admitting Einstein Riemannian metrics [Hit 74, LeB99]. According to the navigation description (Theorem 3.2) any manifold that admits an Einstein Randers metric must also admit a Einstein Riemannian metric. Hence the same topological obstructions restrict Einstein metrics of Randers type. To understand why the conjecture fails among Randers metrics, yet may still be expected to hold in the larger class of Finsler metrics, it is helpful to consider the indicatrix. The indicatrix of a Finsler manifold (M,!F) is the set of points in the tangent space T M of norm 1, S (T) := {y € T M. : — 1}. Given the Randers metric F solving Zermelo's problem of navigation for (a, W) the indicatrices are related by S (F) = S (a) + W . In particular, the Randers indicatrix is simply an ellipse centered at W . In this context, it seems asking a metric to be both Einstein and elliptical (ie. Randers) restricts the topology of M. In pursuing Chern's conjecture we are asking if relaxing the elliptical assumption on the indicatrix to strict convexity (this is, moving from Randers to Finsler metrics) is sufficient to lift the topological restraints. X X X X x x x 39 4. A GENERALIZED NAVIGATION PROBLEM 40 3. Einstein rigidity Consider closed Einstein manifolds and, in particular, Proposition 4.5 and Corollary 4.8 of §4.3. Together the results imply the following rigidity theorem. (Ricci rigidity). Suppose (M,F) is a connected compact boundaryless Einstein Randers manifold with Ricci scalar Ric. • If Ric < 0, then (M, F) is Riemannian. • If Ric = 0, then (M, F) is Berwald. T H E O R E M 5.1 This statement generalizes a theorem of Akbar-Zadeh's for Finsler metrics of constant flag curvature [AZ88]. 5.2 (Akbar-Zadeh). Suppose (M,F) is a connected compact boundaryless Finsler manifold of constantflagcurvature A . • If X <0, then (M, F) is Riemannian. • If A = 0, then (M, F) is locally Minkowski. THEOREM The Ricci rigidity theorem is a straightforward extension of AkbarZadeh's result when Ric < 0. To appreciate the generalization when Ric = 0, it is helpful to note that locally Minkowski spaces are the Berwald spaces with constant flag curvature K — 0 [BCSOO]. Notice though, that Akbar-Zadeh's theorem holds for arbitrary Finsler metrics, while the Ricci rigidity theorem above is restricted to the Randers setting. So, towards a complete generalization of Akbar-Zadeh's result: may we replace 'Randers' with 'Finsler' in the Ricci rigidity theorem? 4. A generalized navigation problem The Zermelo navigation structure of a Randers space is the sine qua non of this thesis. As such I am indebted to Z. Shen for realizing Randers metrics as solutions to Zermelo's problem of navigation [She02a, She02b]. The pivotal role of the navigation structure is made clear by the Einstein navigation description of §3.6. The concise, geometrically transparent nature of Theorem 3.2 is in stark contrast with the unintuitive characterization of Theorem 2.3. The sleek format of the navigation description is ideally suited to the theory of Einstein metrics. This is nowhere more evident than in the Schur lemma, in which an unexpected result is realized as a corollary to Riemannian geometry. It is desirable then, to extend Zermelo's problem of navigation as a means of parameterizing Finsler metrics in general. Given the integral role of the navigation structure in the study of Randers metrics, we may reasonably hope that such a parameterization will provide an arena to approach the questions outlined above. APPENDIX A The Matsumoto Identity: Preliminary Form Here I present a second proof of Matsumoto's identity. The argument is based on the characterization of § 2 . 6 . 2 . Unlike the proof of the Matsumoto Identity in §3.8 the following computation is considerably more involved. It is my hope that the juxtaposition of the two proofs, one straightforward, the second abstruse, illuminates the merit of the navigation description over the characterization of § 2 . 6 . 2 . The final form of the Matsumoto Identity requires the Schur lemma. For now, I show any Randers metric, for which the Basic Equation ( § 2 . 5 . 1 ) and the E23 Equation ( § 2 . 5 . 3 ) hold, with n > 2, satisfies the following preliminary identity: 0 = na{l - \\b\\ } {K + ^ c r ) + 2b%i 2 2 Here K = zrhiRic, and Ric is given by the E23 Equation. In Appendix B, I will show, via the Einstein characterization theorem of § 2 . 6 . 2 and the preliminary identity above, that the Ricci scalar Ric(x) must be constant when n > 2. Once we see Ric is constant the identity is updated to its final form, CT{1 - ||cb|| } 2 (K + i<7 ) VK^ = + 2 when n = 0 a(K + ^cT ) = 0 2 2, when n > 2 . To obtain the preliminary version I begin with the Ricci identity for curly, curly| | - curly|^| = curl R ^ fc h fe sj + curl #/ is fc/l . Contract this expression with d d and apply the skew-symmetry of curly to generate curl |^. = -curFRicy . lk h] U Since curl is skew-symmetric, and Ricy is symmetric, the right-hand side of this equation is zero. Hence, u curF N j = 0. Let us compute curF ^.. Virtue of the E23 Equation we have curF . |ib = 2b Ri l Cli + {2Ric+l(n-l)a }b 2 i li + \{n - l){curF6 - + curF, 9* + crG^} . ib 41 A. T H E M A T S U M O T O I D E N T I T Y : P R E L I M I N A R Y F O R M 42 We need to address the terms 1?^, curiae*, ©^ and curF©^ above. Note that b ^ = glie^, and the Basic Equation implies 1 = icr(n-||6|| ). ^ 2 Next, we compute the terms curF^.©, and © ^ with the E23 Equation. In these computations I make frequent use of the facts b^j = ^(Liey + curly) and 6j|jCurP = ^curly-curl - . The two terms are given by 1 7 J • c u r l ^ . 0 ; = - i ( n - 1) tr 6 i 0* , and • 0 ^ = ^curly curF - {2Ric + | ( n - 1) a } ||6|| + \{n - 1) 0; 6* . 2 2 The calculation of the fourth term, curP- ©^-, is more involved. Notice, by the skew-symmetry of curl - , that 7 1 7 curFe^j = £curl« (e - i{j . While it is difficult to compute 0^-, the computation of 0^- — Qj^ is relatively straightforward. The calculation proceeds as follows. In §2.4 I presented a formula for curly \ k curly j = -2b R + lie \j - s fc ksij ik lie \i. kj With this expression we have ~ = (^curhjj)^ - (^curl^)^ = (ft^curlhi + b curl ) = i(lie^-l-curl^OcurUj - ^(lie^ + curl^Jcurl/y h hi]j - (t^curl/y + b cm\ ) h hj{i b {-2b (R -R ) h s jshi ishj + (lie/y|i = liej-ijij) - (lie |j - liey| )} W h —crcurly . The third equality follows from the expression for curly ^ above. It can be shown, via the symmetries of the Riemann curvature tensor, that b"b (Rj hi — Rishj) = 0. So the curvature tensors to not contribute to the computation. The last equality is then a result of the Basic Equation. It follows now that s a curF0j|j = ^curF ( 0 ^ - 9j|j) = - ^ cr curly curl - . 1 7 Recollect curP- ©^- is the fourth, and final expression computed for c u r F ^ . . 7 We may now obtain the identity by substituting the derived formulas for b ^ 1 , curl '- .©j , 0 V and curF©^- into curF,.. = 0. 0 APPENDIX B A Schur Lemma In §3.7 the proof of the Schur lemma is based on the Einstein navigation description of §3.6. This appendix contains a proof of the Schur lemma based on the Einstein characterization of §2.6.2. Two purposes are served by the appendix. First, to confirm the first proof of §3.7. As in the case of the Matsumoto Identity in Appendix A, the reader will find this proof considerably more complex than the one based on the navigation description. The dichotomy serves the second purpose - to illustrate the strength of the navigation description over the characterization. 1. The Riemannian Schur Lemma Before addressing the Randers case, it is helpful to review the proof of the Riemannian Schur lemma. L E M M A B . l . Schur Suppose n > 3 and that the Ricci scalar Ric of a is a function of x alone. Then Ric is constant. P R O O F . Since Ric is independent of x, the Riemannian metric a is Einstein. In particular, Ricy = Ric(x)aij. Tracing on we have S := R i c \ = Ric{x) n. Now rewrite the Einstein condition as Recollect the second Bianchi identity for a, 0 = h jk\l + h kl\j + h lj\k • R R Contract the identity with S a 3 hl i R to obtain o = = = => 0 = 2mc -S\ k{i k 2 ( n ^ ) | i _ 5 l f c ^S|fc - S\ k (n-2)S, . fc Hence, when n > 2, S\ = nRic\ = 0. Since Ric is a scalar, Ric\ — Ric k, and the partial derivatives of Ric vanish. Hence, Ric is constant. • k k k 43 x 3. T H E R A N D E R S S C H U R L E M M A 44 2. A F i n s l e r i a n obstacle Above we saw that the Riemannian Schur lemma relies on the second Bianchi identity. It is reasonable then to consider the Finslerian second Bianchi identity n h jk:l + h lj:k + h kl.j ~ h js U U r n kl + - H i ks Ij + U jk > ls K where R - := -py Ri jk- Here R and P are respectively the hh- and /it>- curvature tensors of the Chern connection for F; and the colon ':' denotes the horizontal covariant derivative generated by the Chern connection. Details may be found in [BCSOO]. When F is Riemannian, the identity reduces to the Riemannian second Bianchi identity. For general Finsler metrics however, the right hand side is non-zero. In particular, we cannot mimic the Riemannian proof. We shall see in the following section that the Einstein characterization allows us a way around this obstacle. s l S k 3. T h e R a n d e r s Schur L e m m a Given the intractable nature of the Finslerian second Bianchi identity we can not hope to mimic the Riemannian proof with the Finsler Ricci tensor Ricy. Instead we turn to the Curvature Equation of the Einstein characterization (§2.6.2), and apply the Riemannian second Bianchi identity to the given expression for Ricy. In our discussion of the Riemannian Schur lemma we saw, via the second Bianchi identity for a, 0 = Ric ^ — 2Ric ^ . With the Curvature Equation (§2.5.2) we have k 0 k = Ric\|fc - 2Ric\|i = (2 O - \ie i b ) Ric + (n + ||6|| - 2) Ric\ - 2 b'Ric^ b l 2 fc k k k —curF^curljfc — 5 n curF curly ^ + curF curl^ij +(n - 1) { i a (2 9 2 2 ( fc|i 0 i 0 fe - - lie% b ) + £ 6 * 1 ^ + k ©i|fc) + K V|J 0 + k @ \i ~ \i\k)} @i We shall refer to this as the second B i a n c h i equation. Many of the terms above we may directly evaluate with the Einstein characterization or the (preliminary) Matsumoto Identity (§A). A few of them, however, require special attention: • — \ n curF curly 1 & + curFcurljfciJ , • © ^ f c i t - ©i|fc) , and 3. T H E RANDERS SCHUR L E M M A 45 The first term. In the computation of the first term we will make use of the formula kj\k = -2b Rksij + lie-jfcij - lie/y^, initially presented in §2.4. With this expression for curly| and the skewsymmetry of curly it is straightforward to check that cur s fc curl curljfc|j = ^curP curly | . y 7 fc Whence we may rewrite the first term as — 5 ncurFcurlyi*. + curFcurl^ij = — i (n — 1) curf curly ^ . 3 The second term. In Appendix A we computed ©fc|i - @i|fc = - c c u r l i . fc It is now easy to see that the second term is given by e^e^ - e^) = cre curl . i ifc The t h i r d term. We begin with two observations. • First, the Ricci identity for 6 implies = ® \i\k + 9 Ric,fc. l • The second observation, 9 '* ^ = © | | + 6 R i c j / from the first as a consequence of 9^^ — ®k\i Now we see the last term may be re-expressed as l fc l i fc c = l +tTcurP | , fe i follows ~ccurLfc. 5 + k \i - &\i\k) = 5 © W + 5 *fc|*Notice we have a good understanding of the right-hand side of this equation. We computed Q ^ in Appendix A. The computation of 9 ^ ^ is then & @ i W c i k + ffcurl 1 straightforward. The quantity 9 RiCjfc may be computed with the Curvature Equation. Similarly, curP^ is given by the E23 Equation. l The finale. We have reached the final step in the proof of the Randers Schur lemma. Substitute our expressions for the three terms computed above into the second Bianchi equation. Simplify the result with • the Basic, Curvature and E23 Equations of the Einstein characterization (§2.5), • the preliminary form of the Matsumoto Identity in Appendix A, • tyj = 5(lie^j- + curly), a tautology, and • 9j|fc — 9jt|j = —crcurljfc, computed in §A. The result is 0 = = 5| -2Rk f c f c K (n-2)(l-\\bf)Ric\ . k We have proven the following L E M M A B.2 (Schur). The Ricci scalar Ric(x) of any Einstein Randers metric in dimension greater than two is necessarily constant. 4. T H E MATSUMOTO IDENTITY: FINAL FORM 46 4. T h e M a t s u m o t o Identity: F i n a l F o r m In light of this result, we may update the Matsumoto Identity, as computed in Appendix A, for Einstein Randers metrics to CT{1 - ||£b|| } (K + J^CT ) + b K\i = 0 2 2 l a(K + ^ ) 2 a = 0 when n = 2, when n > 2. The identity now agrees with the navigation version computed in §3.8. Bibliography [AZ88] [BCSOO] [Boc46] [Bou79] [BR03a] [BR03b] [BRS03] [BS02] H . Akbar-Zadeh. Sur les espaces de Finsler a courbures sectionnelles constantes. Acad. Roy. Belg. Bull. Cl. Sci., 74:281-322, 1988. D . Bao, S. S. Chern, and Z. Shen. An Introduction to Riemann-Finsler Geometry, volume 200 of Graduate Texts in Mathematics. Springer, 2000. S Bochner. Vector fields and Ricci curvature. Bull. Amer. Math. Soc, 52:776797, 1946. J . P. Bourguignon. Ricci curvature and Einstein metrics. In Global Differential Geometry and Global Analysis, pages 42-63, 1979. D . Bao and C . Robles. O n Randers metrics of constant flag curvature. Rep. on Math. Phys., 51(l):9-42, 2003. D . Bao and C . Robles. O n Ricci and flag curvature in Finsler geometry. In Some perspectives in Finsler geometry, M S R I Series. Cambridge University Press, 2003. In preparation. D . Bao, C . Robles, and Z. Shen. Zermelo navigation on Riemannian manifolds. In preparation, 2003. D . Bao and Z. Shen. Finsler metrics of constant positive curvature on the L i e group S . J. London Maths. Soc, 66(2):453-467, 2002. C . Caratheodory. Calculus of variations and partial differential equations of the first order. A M S Chelsea Publishing, 1999. L . del Riego. Tenseurs de Weyl d'un spray de directions. P h D thesis, Universite Scientifique et Medicale de Grenoble, 1973. N . H i t c h i n . O n compact four-dimensional Einstein manifolds. Journal of Differential Geometry, 9:435-442, 1974. S. Kobayashi and K . Nomizu. Foundations of Differential Geometry, volume 1. John W i l e y and Sons, Inc., 1996. Claude L e B r u n . Four-Dimensional Einstein Manifolds, and Beyond. In Surveys in Differential Geometry: Essays on Einstein Manifolds, volume V I , pages 247285. International Press of Boston, Inc., 1999. M . Matsumoto. Foundations of Finsler Geometry and Special Finsler Spaces: Kaiseisha Press, Japan, 1986. M . Matsumoto. Epistolary communication w i t h D . Bao, 2001. M . Matsumoto and H . Shimada. The corrected fundamental theorem on the Randers spaces of constant curvature. Tensor, N.S., 2002. To appear. X . M o and Z . Shen. O n isotropic Finsler manifolds of negative flag curvature. Preprint, 2002. T . Okada. O n models of projectively flat Finsler spaces of constant negative curvature. Tensor, 40:117-124, 1983. G . Randers. O n an asymmetric metric in the four-space of general relativity. Phys. Rev., 59:195-199, 1941. H . R u n d . The Differential Geometry of Finsler Spaces. Springer-Verlag, 1959. Z. Shen. Differential Geometry of Sprays and Finsler Spaces. Kluwer Academic Publishers, 2001. Z. Shen. Lectures on Finsler Geometry. World Scientific, 2001. 3 [Car99] [dR73] [Hit74] [KN96] [LeB99] [Mat86] [MatOl] [MS02a] [MS02b] [Oka83] [Ran41] [Run59] [SheOla] [SheOlb] 47 BIBLIOGRAPHY 48 [She02a] Z . Shen. Finsler metrics w i t h K = 0 and 5 = 0. Canadian J. Math., 2002. In press. [She02b] Z . Shen. Two-dimensional Finsler metrics with constant curvature. Manuscripta Mathematica, 2002. In press. [Zer31] E . Zermelo. Uber das Navigationsproblem bei ruhender oder veranderlicher Windverteilungn. Z. Argrew. Math. Mech., 11:114-124, 1931. r Index 3-D rigidity, 29 a, a, a, a, 7, 21 7 7 7 F, 4 fx* > 3 Finsler space, 4 Finslerian Poincare disc, 4 flag, 5 pole, 5 flag curvature, 5-7 constant, 29 fundamental tensor, 4 of a Riemannian metric, 4 ll*M, 7 b, 7 Basic Equation, 15, 17 Berwald space, 38 Berwald's formula, 4 9a, 4 Yjk, split and covariantized, 10, 13, 27 22 P, 7 geodesic spray coefficients, 4 Chern's conjecture, 39 Christoffel symbols, 4 connection coefficients, 4 constant flag curvature, 29 navigation description, 29 curly, 8 Curvature Equation, 18 Curvature Equation, 15 homogeneity absolute, 4 positive, 4, 6 horizontal covariant differentiation, 9 indicatrix, 39 infinitesimal homothety, 24 infinitesimal isometry, 24 Irrat, 13 A , 21 drift 1-form, 7 K \ , 10 K, 25 K(x,y,w), 5 K\, 4 K i l l i n g , 24 E23 Equation, 18 E23 Equation, 25 E23 Equation navigation version, 25 Einstein characterization, 19 convention, 3 Equation, 27 metric, 11, 12 navigation description, 28 liey, 8 Liew Equation, 24 M, 3 Matsumoto Identity for constant flag curvature, 31 for Einstein metrics, 31, 41, 46 Euler's theorem, 6 49 preliminary, 41 n, 3 navigation description constant flag curvature; 29 Einstein, 28 refined, 32 non-Riemannian, 7 Randers metric, 7 R a t , 13 regularity, 4 R i c y , 27 R i c y , 12 R i c « , 15 Ric, 12 Ricci scalar, 12 tensor, 11, 12 Riemannian metric, 4, 7 S, 1 Schur lemma Randers, 30, 45 Riemannian, 43 sectional curvature, 4 cr, 14 spray curvature, 4, 6, 9 strong convexity, 4 3 T M, 3 6«, 8 TM, 3 TM\0, 4 X 22 W, 7, 21 W^j, a;, 3 y, 3 Zermelo's problem of navigation, 7
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Einstein metrics of Randers type Robles, Colleen 2003
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Title | Einstein metrics of Randers type |
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Robles, Colleen |
Date Issued | 2003 |
Description | This thesis presents a study of Einstein Randers metrics. Initially introduced within the context of relativity, Randers metrics have a strong presence in both the theory and applications of Finsler geometry. The starting point is a new characterization of Einstein metrics of Randers type by three conditions. The conditions form a coupled, highly non-linear (due to the presence of a Riemannian Ricci tensor), second order system of partial differential equations. The equations are polynomial in the unknowns; a Riemannian metric ã and differential 1-form b. Recently Z. Shen has generalized Zermelo's problem of navigation on the plane to arbitrary Riemannian manifolds. (The goal is to identify the paths of shortest time on a Riemannian manifold (M, ă) under the influence of an external force W = Wi∂xi.) In this context, Randers metrics may be viewed as solutions to Zermelo's problem. The navigation structure yields the main result of the thesis, a succinct geometric description of Einstein metrics of Randers type. Explicitly, the Randers metric arising as the solution to Zermelo's problem on (ă, W) is Einstein if and only if the Riemannian metric ă is Einstein itself, and W is an infinitesimal homothety of ă. The navigation description quickly yields a Schur lemma for the Ricci curvature of Randers metrics. It is a testament to the navigation description that this result, the first Schur lemma for Ricci curvature in (non- Riemannian) Finsler geometry, is obtained with relative ease. An extension of Matsumoto's Identity for Randers metrics of constant flag curvature to the Einstein setting then follows. Having established these general results, I then explore three scenarios: Einstein metrics on surfaces of revolution, constant flag curvature metrics, and Einstein metrics on closed manifolds. The thesis closes with a collection of open questions. |
Extent | 2714915 bytes |
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Thesis/Dissertation |
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Language | eng |
Date Available | 2009-11-11 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0079500 |
URI | http://hdl.handle.net/2429/14764 |
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Doctor of Philosophy - PhD |
Program |
Mathematics |
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Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2003-05 |
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Scholarly Level | Graduate |
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