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Induced sprays on implicitly defined submanifolds of riemannian manifolds Fournier, David Anthony 1973

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INDUCED SPRAYS ON. IMPLICITLY DEFINED SUBMANIFOLDS OF RIEMANNIAN MANIFOLDS by DAVID ANTHONY FOURNIER B.Sc., University of British Columbia, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF -SSIEM€E in the Department of Mathematics WE accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April , 1973 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia Vancouver 8, Canada Date Supervisor: Dr. K. Hoechsmann ABSTRACT In this thesis we consider a submanifold M of a Riemannian manifold R where M i s given as the inter-section of level sets of C, real valued functions de-fined on R. For a fixed point x iniM we construct diffeomorphisms S^i-TK^ > M where TM^ i s the tan-gent space of M at x. In particular, we determine the Taylor expansion of S x at 0 i n TMj_ in terms of the Riemannian structure of R and the defining equations of/JYU. CHAPTER I INTRODUCTION Let R be a real 6 Riemannian manifold. Let M be a submanifold of R which i s given as the intersection of level sets of r C** real valued functions defined on R. For a fixed point x in M l e t U be a R open neighbourhood of x such that the tangent bundle of R re-stricted to U i s t r i v i a l . Choos© a t r i v i a l i z a t i o n of TU. The Re-mannian structure on U induces a connection on M U. This induced connection i s not intrinsic,, that i s , i t depends on the t r i v i a l i -zation chosen and does not in general coincide with the induced Riemannian connection on M. Corresponding to the induced connection on M U there exists a diffeomorphism S__:' TK^ vM U) where TMx i s the tangent space to M at x and Sx. i s the spray associated to the induced connection. ( The map Sx i s i n general defined only on an open neighbourhood of 0 i n TMj_. ) In the case where R i s ( Euclidean ni space ) and the t r i v i a l -ization of TR i s the obvious one the induced connection on M w i l l correspond to the induced Riemannian.connection. The function S^ depends on the Riemannian structure of R and on the defining equations forlM, The purpose of this paper i s to analyze- This dependence in the following way. We shall compute the Taylor expansion of S x in terms of the Riemannian structure of R and the derivatives at x of the defining equations for M. We then 2. use this information to construct a particularly simple S x in the following.case: Assume that R i s R M so that M«R N l. Consider a C°° function ft m y e consider the function f restricted to M and inquire whether a point x i n M i s a c r i t i c a l point of f, and i f so, what i t s nature as a c r i t i c a l point i s . This question can be solved by studying the derivatives of the composition f • S x . I f we pick'dif-ferent Riemannian structures for the map Sx w i l l vary.. It is therefore natural to inquire whether there exists a Riemannian struc-ture on R^ ' ( or at least i n an open neighbourhood of x ) such that S x . ; has a p a r t i c u l a r l y simple form. CHAPTER I I Let R be a real C°° Riemannian manifold. Since a l l the results here are of a lo c a l nature we assume that R i s R^'. Let iH RN,,R% denote the k-linear maps from RNi to RJ.. A Riemannian metric on RNl i s a function ( , ): RM > L2(R M , R 1 ) which i s symmetric and positive definite.. The value of this function, at a point z in-R^ i s denoted by ( , )z«. Let g_:RM-—*R i=l,...,r be r C*°functions. Let M be the set of a l l y in- such that gi^y)=0 for each i i Let Dgi(y) denote the (Frechet)) derivative of g_- at y. Then~Dg_(y) i s an element of L(TR|' „ R1 ). Let K | = kernel Dg^Cy)).. Let Ty= H K | . . We assu me that dimension (Ty) = N-r.. Let TR K denote the tangent bundle of RNl and TRN|* denote the cotangent bundle.. The Riemannian metric ( ) induces an iso-morphism as followst For w i n TRJ_:* l e t w be the element of TR_f such that for each u inTR__! w(u) = ( GRw,, u ) z . The assumption that ( , ) i s C*0 implies that i f i s a C*° sectiomof TR then i s a C°° section of TRNl . I f g i s a G~ function from R M To R1 , the map defined by z > > Dg(z)N< i n TR_? gives rise to a C** section-of TRN'* . Composition with GR gives a C°* section of TRN. For a function g,, this section w i l l be denoted by Vg.. It's value at a point z i s an element of TR_; usually called the gradient of g at z* I t follows immediately from the condition that dimension(Ty)=E-r» that the vectors Vgi(y) are linearly independent for eachy i n M.. 4.. By continuity there exists an R^ " open set U containing y such thatthe vectors Vg^(z) are linearly independent for each z i n U. Let NL^  be the subspace of TRf1 spanned by the ^gj,(z) . Let T g be the orthog-onal complement of N z i n TR^ ' • I f z i s i n M this definition of Tg agrees with the earlier defition. I t i s also true that uhder^fche inclusion i * : : TM > TRNi induced by the inclusion i : M c > R N that i*(TMy); = Ty.. For z in U l e t TT^TRf > TRz'1 be the orthogonal projection onto T z . Let7T:: U > L 1(R M f c R N) be the 0*° map defined by 77 (z), =mz Fix x imM.. Let a N~ r" 1 be the set of u in TM^  such that ( u,u ) x = 1. Each u i n S^"~r"^ determines a unique direction oro M at x.. A standard result of di f f e r e n t i a l geometry i s that for each such u there exists a unique curve such that <K0) = x, 3-/(0) = u„ a n d ^ ^ ) ( W/(t) ) = 0. ( i s in general not defined for a l l t , but there exists an c. independent of u such that ^ u i s defined on (-c,c>). Since Tu i s ai curve in K, ^u(t)^ * ^ ( t ) for each t in (-c,c) . Since '7^1(t)^ ^ ( t ) ) = 0 i t follows that (omitting the u subscript for simplicity) : r n ( t ) = 0 7 ^ t ) ( 7'(t), r f(t)) 2.1 Now using 7 T ^ . ( t ^ ( ^'(t)) = "f(t) again we get: 7r"(t) = D^ct)^77"?^) ( r , ( t ) ) » T I V ( t ) ( 7 * ( t ) ) ) • 2 - 2 Following this cue l e t ^TTiU >L 2(R NyR N) be defined by ^ n z ( . ) = D7Tz(7rz-,Trz. ) 2.3 Assume now that we have defined :U •L k(R K,R N i) . 5. Define * k 7T:U »L k* 1(R M'# 1 ) by J k 7 T z ( • •» *) =BU k" 17Tz) ... . >V ) 2.4 It follows immediately by induction that: r^ k +D(t) = £ k^ ( t)( ?'(t) T'(t)) 2.5 In particular: V k _ X ) (o) * *> 2.6 I t follows that i f S x: TMX >M i s the map given by S x(tu) = \(t) that the n^*1 Taylor polynomial of S x at 0 i s given by: u, ... , u ) ^ j j ) remainder 2.7 6. CHAPTER III. CALCULATION OF S k 7 l ^ I t i s clear that ^^TT can be expressed in terras of the objects D377^ j ^ k i For example: ^ 7 7 ^ . , • ) = D 7 T X ( 7 T X . .^x • > 3-1 <T2rrx(. ) =D 2^ X(7T X- . ^ V . " V ) +DTTX( D 7 T x ( 7 t x . ,7T X..),-A X- ) + D 7 r X ( 7 T . „ D T \ ( 7 » X . , 7 T X . ) ) 3 . 2 <£ 3^ X(-.- „• ) = D3- x(TT x.• , T r . ,rrx. , r r x . ) + D 2TT X( <riyrx( • ),TTX- „n x . ) + 2 D 2 n x ( n x . . Z1*^ * ). 7SC- ) + 2 D 2 7 T x ( n x - „ ^7r x ( * * - )) + D 7 T X ( ^ 2 7 I X ( \ • „ • ) , ) + 2 D 7 T X ( S^TX^s „• ), ^ x ( « „•')) + D T T X (rr2« A ! £ 2 7 T X ( . „ • , * ) ) 3-3 I t therefore suffices to calculate the the D>i71x . We can without loss of generality assume that the vectors ^g^(x) are orthonormal. I t follows that7T x(u) = u -£E(u , Vg^Cx) )x^rg^(x) Now assume that we have calculated D ^ ^ x for j ^ k - 1 . It i s then sufficient to give an expression for D ^ T T in terras of D J 7 T for N Let u^,, . . . , U j ^ be k + 1 vectors i n TR^ . By linea r i t y i t suffices to determine D^rT x( u^+j,. ••• #• U ^ ) when u^ i s a normal or a tangent vector (ie. a member of N x or T x ) . Assume u^ i s a normal vector. Again by linea r i t y we can assume that u^ = Vg^(x) for some i . In order to proceed we introduce the following notation. The map z v——JguCz) i s a C * map. We denote the k - 1 derivative of this map 7. by V k g i ( x ) . then V kg_ :: R M ^ ( H * ,RNi ) In order to evaluate N K7"T x( u^ +^ „ ... , Vg_(x) ) we consider the identity 7T Z Vg_(z) = 0 3«A Differentiating 3 .4 we get:: D"z< u 2„Vgi(z) ) = -7T Z( v7 2 g i ( z ) ( u 2 ) ) : 3-5 Differentiating 3»5 we get: D 2 ^ U3 „ u 2 , , V g i ( z ) ) = - D7Tz( u 2 ,V 2gi(z ) ( u 3 ) ) -Drr_( u 3 „v 2gi(z)(u_); ) — D z ( ^ 3 g i ( z ) (u 3*u 2) ) 3 ^ Differentiating k- times and evaluating the resulting expression at x we obtain an expression for D K 7V__( U j ^ , , ... ^Vg_(x) ); i n terms of D^7T X, j ^ k - 1 and V^g_(x<) J 4 k + 1 where the value of V**g_(x)'' depends on the particular Riemannian metric. Now consider D K 7 T X ( u^+^t. • ••• t. ^_ ) when i s a tangent vector,. /We calculate f i r s t the tangential'component given:by Tr_^ k7Tj_( u k + 1 » ... u_^  ) . Since u± i s a tangent vector u_ = TT^XLI . Therefore we get: ^xP^xX uk+l»- ••• »'ul ) = ^xP* 7 7*^ uk+l»- ••• ^x^i ) • 3*7 We can evaluate the right side of 3«7 via-, the following "trick", since the value of z i s a projection 7TZ 7 T Z = 7T^ 3«8 Differentiating 3 . 8 we get: D7f z(u 2 „Trzni) + 7TzD7T ( u 2 „ u_ ) = u2, ,u_ ) 3*9 Differentiating 3»9 we get* + D7rz( u3,, Dn z( u2,u_) ) + A ^ J T . C u^ug,,^ ) = r ^ r ^ C u3t,u2:„u1> 3.10 In general,, differentiating k times we obtain an expression of the formt D 8; D k>Tz( u k + 1 , ... 9iXfoOi ) + Terms involving only *j + 7 T aD T E 7 T A ( ujg^,, .... ) = D K T 7 Z ( u k + 1 , .... ): .. 3 *11 Multiplying on the l e f t by 7T z and evaluating at x we obtain: JT^7TX( ••• ) = Terms involving only ^7TX i;$k-l 3.12 The normal component of D T ^ C , ...,u^ ) i s equal to i=l D 7 T X ( u k + 1 „ ... „ u x )„^ g i.(x) ) x T 7 g i ( x ) 3-13 It follows that we need only compute the expression ( V*7*^ uk+l»*'* »ul ^ f V g i U ) )x . Consider the identity: ( 77^^) „ Vg±(z) ) z. =0 3.15 This can be written as Dg. (z)> (,7T u-i ) ) = 0 an expression l Z A which i s independent of the Riemannian metric. Differentiating 3.15 we get: Dgi,(z)( D ^ C u g , ^ ) ) + T ^ g i C z X u g . ^ U ! ) = CP 3-16 Differentiating 3.16 we get: D G I ( Z ) ( D V Z ( U 3 „ U 2 „ U 1 ) ) + D 2 g i ( z ) ( u 3 , . D-r^Cug,,^) ) + D 2 g i ( z ) ( u 2 t D 7 T z(u 3 f !u 1) ) + D^g^iuy^n^} ) = 0 3.17 In general d i f f e r e n t i a t i n g k times we g e t : D g i ( z ) ( D k r r z ( uk+i»- » ul) ) = T e r r a s involving only ®^7TZ j $ k - l and D J g i U ) j^k+1 Evaluating at x we get: uk+l»' * u l ^ ! ^?&±^ x ^ )'x ~ T e r r a s involving only QJ7\X< j^k-1 and D^g.^x) j^k+1 3*18 In particular this expression i s independent of the metric. 9 i . CHAPTER IV/ CALCULATION OF Dk' x In the following formulas u denotes a tangent vector ^ 7g^(x)^ denotes i t s e l f and denotes an arbitrary vector., Employing the techniques:-of the previous chapter we obtain by a series of tedious calculations the following formulas: D7TX,( v i t, Vg ±(x) : )' = - - ^ V 2g i(x).(v 1) D7Tx(v_ „ u ) = - i_£D 2g i(x)( v_„ u)Vg.fr); ilk D 27T X( v i t y 2 „ Vgi(x) )) - - DTT-^ C V2„V 2g i(x.)(v 1) ) - D7X^( v_ „ v 2gi(x)(v 2) )) 4.3 D 2 7Tx(v__,v_ „u)> = - • ^ iPgj/x) (vi ,y2„u) ^ ( x ) + i ^ 2g,.(x)(v i„Vg i(x))D 2g (x)(v 2,u) Vg ±(x) + * ^ D 2 g , (x) (v 2 „Vg ,(x) >D2g (x) (V1 ,u) Vg, (x) - ± S: x c(^ 2g i(x)(y 2))0 2g i.(x)(y 1,u> - i ^  x(V 2g i(x):(v 1))D 2g i(. x)(y 2„u); 4.4 4.1 4.2 D 3 7 r x i ( v 3 ^ 2 » v i " V s i ( x ) ) } = - D 27T x (v 3 t v 1,'9 2 g i (x)(v 2 ) ) - D 2 7 r x ( v 3 , v 2 , ) ^ 2 g i ( x ) ) ' - D 27r x(v 2„v. l f c V ^ U X v j ) )) - D r r x ( v 3 , V 3 g i (x ) (y 2 , ^| ) ) - D 7 T x . ( v 2 * ^ g ^ x ) ( y 3 „ v i ) ) - E )7t ! C ( v 1„V 3 g i ( x ) ( y 3 t^2)) ) - 7TX,( ^ g ^ C x X v ^ „ v 1 ) )) In particular i f Vj_ = v 2 = vy = u we D 3-n x (u,u,ji^Vgi(x') ' y = - 3 D 27T x <(u t ,ui,v 2g i l(x):Cu)) )) - 3 D7r x(u, >V 3 g i.(x.)Cu,u) ) —7^ .( V ^ C x X u . u . u ) ; ) D 3rr x ( v y y 2 , y _ , u ) = - ^2J> g i ( x ) ( y 3 , V 2 ^ „ u ) ^ ( x ) ' + i ; ^ 5rD3 g i.(x.) ( v 3 , y 2 „ Vg ^x) )D2g>.(x>Cv1„u) V g i ( x > + i t J 2D 3 g i ( x v < v 3 , y i „ V g j ( x > ) D 2 g X x ) ( v 2 , u ) Vg.(x) + i i 2 D 3 g i ( x ) ( v 2 , y 1 „ V g .(x); )D2g (x)(y3^u) Vg..(x) - i ^ g i U K v ^ D 2 7 T x ( v 2 , v 1 „ u ) ) ^ g i ( x ) - ilD 2 g i ( X ) ! ( y 2 „ D 2 7 T x ( v 3 , , v 1 , ! i ) ' ))V g i .(x) - i : Z D 2 g i ( x ) ( y 1 „ D 7 r x ( v 3 , v 2 „ u ) ) V g ^ x ) """D 7 Ix< ( y3" D 2 : T X ( v 2 » v 1 » u ) * - D 7 T x ( v 2 f , D 2 7 r x ( v y y 1 „ u ) )) - D7T.(y l t ) D 2 7T x (v 3 f c y 2 „-u ) i ) - ° 2 r T x < > 3 , y 2 . , D7r x (y i t ) u) ) -ifnjv'yy^ D7Tx<v2>u) ) " D ^ x i ( v 2 » y i » ' D 7 ^ v 3 » ' u ) > S e t t i n g v 3 = v 2 , = v^ = u we obtain:: 3 D ^ x , tu,u,u,,u) = - i - Z D g i (x)(u„u,u,u) V g i ( x ) + 3 ± i l D 3 g i ( x ) ( u , u „ V g (x) )D2g .(x)(u,u)Vg..(x) - 3 i Z D 2 g i ( x ) ) ( . u , , D 2 7 T X i L ( U f u , u ) )<7 g ± (x> " 3 D7T x < (u„ D 2 7T x (li,u,vi) ) - 3 D 2 7 T x j C u „ u „ D T T ^ C U . U ) ) : 12. CHAPTER V CALCULATION OF £ k 7 T x We combine the results of chapter IV with 3.1 3 .2 and 3 .3 to obtain expressions for . i " 1 7Tx(u„u) = - ± 2 D 2 g i(x)(u,u) V g j L(x> 5-1 <f27T (u.u.u) = - . :2D 3 g.(x) (u.u.u) V g. (x) 2 - DTT x(u, D7rx(u„u) ) 5 .2 5.2 canbe expressed as:: c 2 - ^ 3 ^ = - ± i D g i(x)(u,u,u) g i ( x ) + 3 .. ,XD2g..Ox)(u„Vg (x) )D 2g.(x)( u,u)V g i(x) - ^2 ; r r x v 2 g i ( x ) ( u ) D 2 g i ( x ) ( u , u ) 5 . 3 where only the last term of 5«3 depends on the infinitesimal behaviouE of the Riemannian metric, the simplest expression for 5.3 w i l l occur when 7T V*g.(x) = 0 . We now determine the form of 77^  v ^ g i ( x ) for particular Riemanniarr. metrics on R . Example 1.. Ni Choose the standard t r i v i a l i ^ a t i o n of TR . That i s , the basis Ni for TR i s the vectors ( 1 , 0 , , . . . , 0 ) , ( 0 , 1 „ 0 , . . , 0 ) „ . . . , ( 0 , ... t l f r With respect to this the standard Riemannian i s represented by the identity matrix. Since we have a fixed coordinate structure the Ni Ni \ Dg-:R ——>L(R ,R ) are represented by 1XN arrays.of C real valued functions-. The vectors represented by these arrays at a point z are just the Vg^(z) • With this identification i n mind one can say that D g.(x) = ;.(x)i .. I t i s well known that in this case the induced connection corresponds to the induced Riemannian connection so that the map SXJ :. TM^ >M corresponds to the exponential map. Example 2.. Consider an implicitly defined submanifold M of R^ .A Fix x in Mi. Determine T z and N z with respect to the standard inner product on R^ . There exists an R^ ' open neighbourhood U of x such that for each z in U, TR^ i s spanned by the subspaces N x and T z. N (We use the standard t r i v i a l i z a t i o n of TR to identify the different tangent spaces.) Define ( * ) ::U »L 2( RM, R> by for u, v, i n TRN: (. u v; ) z = The ordinary inner product i f u„v are either both i n NXi! ,, or both i n T z and 0 otherwise. With respect to this new metric on U,, the subspace N'z i s equal to the old for each z i n Ui.. Calculation ofV kg^(x). Since N„ = KL, there exist C real valued functions a., . on U such that Vg,.(z> = £T&±£Z)Vg..(x),> 5.^  Then a i ; j( z) = ( V g ± ( z ) „Vg..(x)) \ = Dg.j(z)CVg.(x) ) 5-5 14 Differentiating 5*5 k times we get:: D ka i ; j(z)(u k uj_ ) = D^g^CzKu^, .u^Vg^x) ) 5.6 Therefore y g i(z)(u k_i, ... , ) = g±(z)(uk_1„ ... ,ult, g..(x))'Vg..(x); 5.7 Fortunately (or unfortunately from the point of view of the value of the general theory) there i s an easier way to calculate the 5^7TX. for the above S x. I t depends on the fact that ^ k 7 ^ ( u , ... ,u) i s i n N x for a l l k. Let 50 :^: R^ *R^be the coordinate functions on R^ . Then the map S x < i s determined by the compositions <?*Sx<sTMx--*R* • in fact sx>(tu)) = ( ^ (s^Ctu););, ... , ^(s^Ctu)) )• 5.8 and d* (Sx.(tu») = ( ^ ' ( ^ ( S j t u ) ) ) ) , .... d ^ ( #.(Sx,(tu))) ) cit* dtk d t K ( Let \ . = ( ^ ( x ) , Vg.(x) )x . Let <p± : R >R be given by: - r±M - . Then V cp±{x.) i s in TJ^and ^ ( y ) = 9^.(y) for each y in M. Sinc«J ^ k 7 T x ( u t , ....„u)' i s i i i N x , ( ^^(x) , ^ T T ^ u , ... ,u) ) = 0 for a l l k.. Therefore:: D^ i(x)( ( f ^ T T ^ u , ... ,u) > = 0 ) f o r a l l k . 5.9 Clearly 5*8 remains valid i f we replace ^ by . 15. This y^eldsi: d^Cs^tu)) = (d^C^Cs^tu)):), .... ,d> ( ^(s^Ctu))) ) 5.icr dt* dt K dt Since d^CsCtu ) ) . = ^ T T C U , , ... ,u> 5.11 ^ k-1 we get that the left hand side of 5.10 is £ 7 \ x and the right side of 5.10 canbe expanded by the chain rule and using 5.9 yields an expression in $^7T^ and derivatives of the The derivatives of the ^  can then be expressed in terras of the ^ and derivatives of the g^. carrying out this program we obtain the following: c^rTxCu.u) = - ^ D 2 g i(x)(u,u) ^ g ±(x) 5.12 <f27Tx(u,u,u) = - D3gi.(x)(u,utu) V g ±(x) + 3 i ^D2g.(x)(u,.Vg.(x»D2g..(x)(u„u>Vg.(x) 5.13 <S*3rrx(u,u,u,u) = — i^Lv gi(x)(u,u,u,u) Vg^Cx) +6 i l f ^ l E D 3 g j ( x ) { t t , u , V g i(x»D 2 g i.(x>(u,u) Vg^Cx) - 3 l f j f k 2: D \ ( X ) ( <7g.(x) „ V g j(x) ) D 2 g i ( x ) ( u , u ) D 2 g j ( x ) ( u , u ^ V S ^ X ) -+4 l D 2 g ( x ) ( u „ g .(x))D3g (x)(u»u„u) Vg (x) 1 j J J--12 i i i kiD 2g kJx).(u,7g j(x)')D 2g. j(x)(u,,Vg.(x))D|.ix) ;(u,u 

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