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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 B r i t i s h Columbia, 1968  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF -SSIEM€E i n the Department of Mathematics WE accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1973  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 at 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 copying 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 by h i s r e p r e s e n t a t i v e s .  be  granted by  permission.  Department of The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  Date  Department or  I t i s understood t h a t copying or  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 written  the Head of my  s h a l l not be  publication  allowed without  my  Supervisor:  Dr. K. Hoechsmann  ABSTRACT In t h i s t h e s i s we consider a submanifold M of a Riemannian manifold R where M i s given as the i n t e r section of l e v e l sets of C, r e a l valued functions defined 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 a t x. In p a r t i c u l a r , we determine the Taylor expansion o f S  x  a t 0 i n TMj_ i n terms of the  Riemannian structure of R and the d e f i n i n g equations of/JYU.  CHAPTER I  Let  INTRODUCTION  R be a r e a l 6  Riemannian manifold. L e t M be a submanifold  of R which i s given as the intersection of l e v e l sets o f r C** r e a l valued functions defined on R. For a fixed point x i n M l e t U be a R open neighbourhood  of x such that the tangent bundle of R r e -  s t r i c t e d 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 Remannian structure on U induces a connection on M U. This induced connection i s not i n t r i n s i c , , that i s , i t depends on the t r i v i a l i zation chosen and does not i n general coincide with the induced Riemannian connection on M. Corresponding to the induced connection on M a diffeomorphism  S__:' TK^  vM  U there exists  U) where TMx i s the tangent space  to M a t x and S . i s the spray associated to the induced connection. x  ( 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 -  i z a t i o n 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 o f t h i s paper i s to analyze- This dependence i n the following way. We s h a l l compute the Taylor expansion of S  x  i n terms of the Riemannian structure of R  and the derivatives a t x of the defining equations f o r M. We then  2. use t h i s information to construct a p a r t i c u l a r l y simple S  x  i n the  following.case: Assume that R i s R  ft  m  y  e  M  so that M « R . Consider a C°° function Nl  consider the function f r e s t r i c t e d 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 ferent Riemannian structures f o r  the map  f • Sx.  I f we p i c k ' d i f -  Sx w i l l vary.. I t i s  therefore natural to inquire whether there exists a Riemannian structure on R^'  ( or at l e a s t i n an  open neighbourhood of x ) such  that S . has a p a r t i c u l a r l y simple form. x  ;  CHAPTER I I Let R be a r e a l C°° Riemannian manifold. Since a l l the results here are of a l o c a l nature we assume that R i s R^'.  iH  Let  R ,R% N,  denote the k - l i n e a r  A Riemannian metric on R  Nl  i s a function  maps from (  ,  ): R  R  Ni  to  R .. J  > L2(R ,R1) M  M  which i s symmetric and p o s i t i v e d e f i n i t e . . The value of t h i s function, at a point z in-R^  i s denoted by  Let g_:R -—*R  , ) «. z  i = l , . . . , r be r C*°functions. Let M be the  M  set o f a l l y i n -  (  such that gi^y)=0  f o r each i i Let Dgi(y) denote  the (Frechet)) d e r i v a t i v e of g_- a t y. Then~Dg_(y) i s an element of L(TR|' „ R  1  ) . Let K | = kernel Dg^Cy))..  Let Ty= H K | . . We assu me  that dimension (Ty) = N-r.. Let TR denote the tangent bundle of R K  Nl  cotangent bundle.. The Riemannian metric  and TR * denote the  (  N|  )  induces an i s o -  morphism as followst For w i n TRJ_* l e t  w be the element of TR_f such that f o r  :  each u inTR__  !  w(u) = (  i s C* implies that i f 0  C°° section of TR  Nl  .  GRw,, u )  TR'* . N  . The assumption that  i s a C*° s e c t i o m o f TR I f g i s a G~  the map defined by z > of  z  N  ,  then  function from  > Dg(z)< i n TR_?  (  R  M  )  is a To R  1  ,  gives r i s e to a C** section-  Composition with GR gives a C°* section o f TR . For N  a function g,, t h i s section w i l l be denoted by Vg.. I t ' s value a t a point z i s an element of TR_; u s u a l l y called the gradient of g at z* I t follows immediately from the condition that that the vectors V g i ( y )  dimension(Ty)=E-r»  are l i n e a r l y independent f o r e a c h y i n M..  4.. By continuity there exists an R^" open set U containing y such thatthe vectors  Vg^(z)  are l i n e a r l y independent f o r each z i n U. Let NL^ be  the subspace of TRf  spanned by the  1  onal complement of N  z  i n TR^' •  that  i * : : TM  i*(TMy); =  > TR  T  .  z  77 (z), =m that  orthog-  I t i s also true that uhder^fche >  c  R  N  y  > TRz' 1  Fix x imM..  be the orthogonal projection  1  > L (R  :  ( u,u )  be the  T ..  Let7T: U  z  g  induced by the i n c l u s i o n i : M  Ni  For z i n U l e t T T ^ T R f onto  Let T  I f z i s i n M t h i s d e f i n i t i o n of Tg  agrees with the e a r l i e r d e f i t i o n . inclusion  ^gj,(z) .  R)  M  be the  N  fc  Let a ~ " N  r  1  0*° map defined by  be the set of u i n TM^  such  = 1. Each u i n S^"~ "^ determines a unique d i r e c t i o n oro r  x  M at x.. A standard r e s u l t of d i f f e r e n t i a l geometry i s that f o r each such u there exists a unique curve a n d ^ ^ ) ( W/(t)  ) = 0.  (  such that  <K0)  i s i n general not defined f o r a l l t ,  but there exists an c. independent of u such that ^ Since T  u  (-c,c) .  = x, 3-/(0) = u„  i s ai curve i n K,  ^u(t)^  u  i s defined on (-c,c>).  * ^ ( t ) f o r each t i n  Since '7^1(t)^ ^ ( t ) ) = 0 i t follows that (omitting the u  subscript f o r s i m p l i c i t y ) : r ( t ) = 0 7 ^ ) ( 7'(t), r (t)) n  2.1  f  t  Now  using  7T^.( ^( t  ^ ' ( t ) ) = "f(t) again we  7r"(t) = D^ct)^ 7 7 "?^)  ( r , ( t ) )  Following t h i s cue l e t ^ T T i U  ^n (. z  Assume now  »  T I  V(t)  get: 7  (  *  ( t ) ) )  •  2  z  that we have defined  2  > L ( R y R ) be defined by 2  N  N  ) = D7T (7r -,Tr . ) z  -  2.3  z  :U  •L (R ,R ) . k  K  Ni  5. * 7T:U  Define  k  J 7T (  »L * (R '# k  1  M  ) by  1  • •» *) = B U " 7 T z )  k  k  z  ... . > V  1  )  2.4  I t follows immediately by induction that:  r ^ D ( t ) = £ ^ ) ( ?'(t) k +  T'(t))  k  ( t  2.5  In p a r t i c u l a r :  V _ k  It  X )  (o)  follows that i f S ( t u ) = \(t) x  * S : TM x  *> X  2.6  >M i s the map given by  that the n^* Taylor polynomial of S 1  x  at 0 i s  given by: u, ... , u ) ^ j j )  remainder 2.7  6. CHAPTER III.  CALCULATION OF S 7 l ^  I t i s c l e a r that  ^^TT  D377^  k  j^ki  <T rr (. 2  For example: ,  ^77^. x  can be expressed i n terras of the objects  • )  = D7T (7T . X  )  .^x  X  3-1  .^V. " V)  =D ^ (7T 2  X  • >  X  +DTT ( D 7 T ( 7 t . , 7 T . . ) , - A - ) X  x  +D7r (7T. 3  X  X  „ D T \ ( 7 »  X  <£ ^ (-.-  x  2  X  <riyr ( •  x  +  (  ^  + 2 D 7 T  X  (  S^TX^s  + D T T  7 I  (rr2«  X  ). S - ) 7  X  A  (  \  •  „ •  ) ,  „• ), ^  ! £  2  7 T  X  (  .  ))  )  ( « „•'))  x  „  C  * *-  x  X  2  x  „ ^7r (  x  D 7 T  . 2  ,rr . )  *  1  +2D27T (n x  3  X  . Z *^  x  )  ),TT - „ n x . )  x  +2D n (n . 2  X  x  x  + D TT (  ,7T . )  .  X  ,rr .  „• ) = D3- (TT .• , T r . x  X  •  ,  *  )  3-3  )  I t therefore s u f f i c e s to calculate the the D>i71 . x  We can without l o s s o f generality assume that the vectors ^ g ^ ( x ) are orthonormal.  I t follows that7T (u) = u -£E(u , Vg^Cx) ) ^ g^(x) r  x  Now assume that we have calculated  x  D ^ ^ x f o r j ^ k - 1 . I t i s then  s u f f i c i e n t to give an expression f o r  D^TT  i n terras of  D J 7 T  for  N Let  u^,, . . . , U j ^  s u f f i c e s to determine  be k + 1 vectors i n TR^ .  By l i n e a r i t y i t  D ^ r T ( u^+j,. ••• #• ^ ) when u^ i s a normal or U  x  a tangent vector ( i e . a member o f N  x  or T  x  ) . Assume u^ i s a normal  vector. Again by l i n e a r i t y we can assume that u^ = Vg^(x) f o r some i . In order to proceed we introduce the following notation. The map z v — — J g u C z ) i s a C * map. We denote the k - 1 d e r i v a t i v e of this map  7. by V  k  ( x ) . then V g _  :: R  k  g i  In order to evaluate  ,R  )  Ni  7"T ( u^ ^ „ ... , Vg_(x) ) we consider x  +  7 T Vg_(z) = 0  the i d e n t i t y  3«A  Z  Differentiating D"z<  NK  ^ ( H *  M  3 . 4 we get::  u „ V g i ( z ) ) = -7T ( v 7 2  Z  2  g i  ( )(u2)):  3-5  z  D i f f e r e n t i a t i n g 3»5 we get: D  2  ^  U3 „ u , , V ( ) ) = - D7T ( u 2  g i  z  z  -Drr_( u —D (^  3  z  g i  ,V gi(z)(u3) 2  2  )  „v gi(z)(u_); ) 2  3  (z)  3^  (u *u ) ) 3  2  D i f f e r e n t i a t i n g k- times and evaluating the resulting expression at x we obtain an expression f o r D^7T ,  terms of  j ^ k - 1 and  X  ... ^Vg_(x) ); i n  Uj^,,  D 7V__( K  V^g_(x<)  J4  where the value of  k + 1  V**g_(x)'' depends on the p a r t i c u l a r Riemannian m e t r i c . Now consider  ^_ ) when  u^+^t. • ••• t.  D 7T ( K  X  i s a tangent  vector,. /We calculate f i r s t the tangential'component Tr_^ 7Tj_( u k  u_ =  k + 1  »  ...  .  TT^XLI  u_^ ) .  Since u±  given:by  i s a tangent vector  Therefore we get:  ^ x P ^ x X k+l»- ••• »' l ) u  u  =  ^xP* *^ k+l»- ••• ^x^i 77  u  )  • 3*7  We can evaluate the right side of 3«7 via-, the following " t r i c k " , since the value o f  i s a projection 7T 7 T  z  Z  Z  = 7T^  3«8  D i f f e r e n t i a t i n g 3 . 8 we get: D7f (u z  2  „Tr )  + 7T D7T ( u  zni  z  2  „ u_ ) =  u , ,u_ ) 2  3*9  D i f f e r e n t i a t i n g 3»9 we get*  + D7r ( u ,, D n ( u ,u_) ) + z  z  3  2  A  ^  J  T  .  C  u^ug,,^ )  = r ^ r ^ C u ,u :„u > 3t  2  1  In general,, d i f f e r e n t i a t i n g k times we obtain an expression of the formt D  3.10  8; D >T ( u  , ... iXfoOi  k  z  )  9  k + 1  + Terms involving only *j  +  7T D  =  D T7 (  T E  ujg^,, ....  7T ( A  a  u  K  Z  k + 1  ,  ....  ):  Multiplying on the l e f t by 7 T JT^7T (  •••  X  ) 3*11  ..  and evaluating at x we obtain:  z  ) =  Terms involving only  ^7T  X  i;$k-l 3.12  DT^C  The normal component o f i=l  u  D7T ( X  k + 1  „ ... „ u  , ...,u^ ) i s equal to  )„^ .(x)  x  )  gi  T7 (x)  x  3-13  g i  I t follows that we need only compute the expression (  V* *^ 7  u  » l ^ fVgiU)  k+l»*'*  )x  u  .  Consider the i d e n t i t y :  ( 77^^)  „ Vg (z)  ) . =0  ±  3.15  z  Dg. (z)> (,7T u-i ) ) = 0 an expression  This can be written as  Z  A  l  which i s independent of the Riemannian metric. D i f f e r e n t i a t i n g 3.15 we get: Dgi,(z)( D ^ C u g , ^ )  ) + T^giCzXug.^U!  ) = CP  3-16  D i f f e r e n t i a t i n g 3.16 we get: D +  G  D  I  (  2 g i  Z  )  (  DV (  U „ U „ U ) ) 3  Z  (z)(u  2  D7T (u  2 t  z  3 f !  In general d i f f e r e n t i a t i n D ( z ) ( D r r ( k+i»k  g i  +  1  u ) ) + 1  D  2 g i  ( z ) ( u , . D-r^Cug,,^) ) 3  D^g^iuy^n^}  k times we e t : » l) )  g  ) = 0  3.17  g  u  u  z  =  T e r r a s  involving only  ®^7T D  J  g i  Z  U)  j $ k - l and  j^k+1  Evaluating at x we get: u  k+l»'  * l^ u  only QJ7\  X<  !  ^?&±^ ^ )'x x  j^k-1  ~  T e r r a s  and D^g.^x)  involving j^k+1  In p a r t i c u l a r t h i s expression i s independent of the metric.  3*18  9i.  CHAPTER IV/  CALCULATION OF D ' k  x  In the following formulas u denotes a tangent vector ^7g^(x)^ denotes i t s e l f and  denotes an a r b i t r a r y vector.,  Employing the techniques:-of the previous chapter we obtain by a series o f tedious calculations the following formulas:  D7T ,( v X  D7T (v_ x  i t  , Vg (x)  :  ±  4.1  )' = - - ^ V g (x).(v ) 2  i  1  4.2  „ u ) = - _ £ D g ( x ) ( v_„ u ) V g . f r ) ; 2  i  i  ilk D 7T ( v 2  X  y „ V g i ( x ) )) - - DTT-^C V2„V g (x.)(v ) ) 2  i t  2  i  1  - D7X^( v_ „ v g i ( x ) ( v ) ))  4.3  2  2  D 7T (v__,v_ „u)> = - • ^iPgj/x) ( v i ,y „u) ^ ( x ) 2  x  2  + i ^ g , . ( x ) ( v „ V g ( x ) ) D g (x)(v ,u) V g ( x ) 2  2  i  i  2  ±  + * ^ D g , (x) ( v „ V g ,(x) >D g (x) (V ,u) V g , (x) 2  2  2  -  1  S: (^ g (x)(y ))0 g .(x)(y ,u> 2  ±  - i^  2  x c  i  2  i  (V g (x):(v ))D g (. )(y „u); 2  x  1  2  i  1  i  x  2  4.4  D 3 7 r  xi  3^2» i"  ( v  v  i  V s  ( x ) )  }  =  - D 7T (v v ,'9 g (x)(v ) 2  x  2  3t  i  1  )  2  D 7r (v ,v , ^ g (x))'  -  2  x  3  - D 7r (v „v.  2  2  i  V ^ U X v j ) ))  2  x  2  )  lfc  - D r r ( v , V g (x)(y ,^|) x  3  3  i  2  )  - D7T .(v * ^ g ^ x ) (y „vi)) x  -  E)  2  3  7t (v „V g (x)(y ^2)) ) 3  !C  1  i  3t  - 7T ,( ^ g ^ C x X v ^ „ v ) X  1  In p a r t i c u l a r i f Vj_ = v  2  ))  = vy = u we  D -n (u,u,ji^Vgi(x')' y = 3  x  - 3 D 7T (u ,ui,v g (x):Cu)) )) 2  x<  2  t  il  - 3 D7r (u, V .(x.)Cu,u) ) x  3  >  gi  —7^.( V ^ C x X u . u . u ) ; )  D rr (vyy ,y_,u) = 3  x  2  - ^2J> g ( x ) ( y , V 2 ^ „ u ) ^ ( x ) ' 3  i  +  i  ^ 5rD .(x.) ( v , y „ V g ^ x ) 3  ;  + i  t  gi  3  2  )D g .(x>Cv „u) V ( x > 2  >  1  g i  2 D g ( x v < v , y „ V g ( x > ) D g X x ) ( v , ) Vg.(x) 3  J  i  3  i  2  j  2  u  + i 2 D g i ( x ) ( v , y „ V g .(x); )D g (x)(y ^u) Vg..(x) 3  2  i  - i ^ g i U K v ^  2  1  3  D 7T (v ,v „u) ) ^ (x) 2  x  2  1  g i  - lD g ( )!(y „D 7T (v ,,v ,!i)'  ))V .(x)  -  ) Vg^x)  2  i  i  i :  X  2  2  x  3  1  gi  ZD g (x)(y „D7r (v ,v „u) 2  """  D7I  i  x  1  x< 3" (y  X  D 2 : T  ( v  2» 1» v  3  2  u )  *  - D 7 T ( v , D 7 r ( v y y „ u ) )) x  - D7T.(y - °  2 r T  x  2  2 f  x  D 7T (v y „-u)i ) 2  lt)  x  <> ,y ., 3  2  -ifnjv'yy^ "  D  v  2  D7r (y u) ) x  it)  x  (  g  3fc  D7T <v u) )  ^xi v2» i»'  Settin  1  y  2>  ^ 3»' ) >  D 7  v  u  = v , = v^ = u we obtain::  3  2  3  D ^ ,tu,u,u,,u) = x  -i-ZD + 3  gi  3  -3  g i  lD g (x)(u,u„Vg 3  ±  i  - 3 iZD "  (x)(u„u,u,u) V ( x )  2  g i  (x) )D g .(x)(u,u)Vg..(x) 2  i  (x))(.u,,D 7T 2  X i L  ( u , u ) )<7 (x> U f  D7T (u„ D 7T (li,u,vi) ) x<  D27T  x  2  jCu„u„  x  D T T ^ C U . U )  ):  g±  12. CHAPTER V  CALCULATION OF  £ 7T k  x  We combine the results of chapter IV with to obtain expressions for  i " 7T (u„u) = - 2  D  1  x  ±  2 gi  3.1  3.2 and  3.3  .  (x)(u,u) V  gjL  (x>  5-1  <f 7T (u.u.u) = - . 2 D . ( x ) (u.u.u) V . (x) 2 2  3  :  g  g  5.2  - DTT (u, D7r (u„u) ) x  x  5.2 canbe expressed as:: c ^  2  = -  ±  - ^ 3 i D (x)(u,u,u) gi  g i  (x)  + 3 .. ,XD g..Ox)(u„V 2  - ^2; r r v x  2 g i  (x)( )D u  2 g i  g  (x)  )D g.(x)( ,u)V (x) 2  u  gi  5.3  (x)(u,u)  where only the l a s t term of 5«3 depends on the i n f i n i t e s i m a l behaviouE of the Riemannian m e t r i c , the simplest expression f o r 5.3 w i l l occur when 7T V*g.(x)  = 0 .  We now determine the form of 77^ v ^ ( x ) f o r p a r t i c u l a r g i  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 .  Ni for TR  i s the vectors ( 1 , 0 , , . . . , 0 )  That i s , the basis  , (0,1„0,..,0)  „ ...  , ( 0 , ...  t  lfr  With respect to t h i s the standard Riemannian i s represented by the i d e n t i t y matrix. Since we have a fixed coordinate structure the Ni Ni \ Dg-:R  — — > L ( R ,R ) are represented by 1XN arrays.of C  r e a l valued  functions-. The vectors represented by these arrays at a point z  are just the Vg^(z) • With t h i s i d e n t i f i c a t i o n i n mind can say that  D g.(x) =  ;.(x)i  ..  one  I t i s well known that i n this  case the induced connection corresponds to the induced Riemannian connection so that the map S  XJ  :. TM^  >M  corresponds to the  exponential map.  Example 2.. Consider an i m p l i c i t l y defined submanifold M of R^.A Fix x i n Mi. Determine T  and N  z  with respect to the standard inner  z  product on R^. There exists an R^' open neighbourhood  U o f x such  that f o r each z i n U, TR^ i s spanned by the subspaces N  and T .  x  z  N (We use the standard t r i v i a l i z a t i o n of TR tangent spaces.) Define ( (. u  * ) ::U v; )  z  » L ( R , R> 2  to i d e n t i f y the d i f f e r e n t  by  M  f o r u, v, i n TR  N:  = The ordinary inner product i f u„v are either both i n N  Xi!  ,, or both i n T and z  0 otherwise. With respect to t h i s new metric on U,, the subspace N'  z  i s equal to the o l d  f o r each z i n Ui..  Calculation o f V g ^ ( x ) . k  Since N„ = KL,  i ; j  r e a l valued functions a., .  Vg,.(z> = £T& £ )Vg..(x),>  on U such that Then a  there exist C  ±  ( ) = z  (V  g ±  5.^  Z  ( z ) „Vg..(x)) \ = Dg.j(z)CVg.(x)  )  5-5  14  D i f f e r e n t i a t i n g 5*5 k times D a  (z)(u  k  i;j  we get::  uj_ ) = D ^ g ^ C z K u ^ ,  k  .u^Vg^x) )  5.6  Therefore y  g ( z ) ( u _ i , ... , i  )=  k  g (z)(u _ „ ... ,u , g..(x))'Vg..(x); ±  k  1  lt  5.7  Fortunately (or unfortunately from the point of view o f the value o f the general theory) there i s an easier way to calculate the 5^7T . f o r the above S . X  I t depends on the f a c t that  x  ^ 7 ^ ( u , ... ,u) i s i n N k  Let 50^:: R^ the map S  Then  by the compositions ?*S sTM --*R* • <  x<  x  s (tu)) = ( ^(s^Ctu););, ... , ^ ( s ^ C t u ) ) )•  i n fact and  f o r a l l k.  *R^be the coordinate functions on R^.  i s determined  x<  x  5.8  x>  d * (S .(tu») = ( ^ ' ( ^ ( S j t u ) ) ) ) , . . . . d ^ ( #.(S ,(tu))) ) cit* dtk dt ( x  x  K  Let \ .  = (  <p : R  Let  ±  ^ ( x ) , Vg.(x)  - rM ±  )  x  >R be given by: -  Then V cp {x.) i s i n TJ^and ±  .  ^ ( y ) = 9^.(y) f o r each y i n M.  Sinc«J ^ 7 T ( u , ....„u)' i s i i i N k  x  .  t  x  ,  ( ^^(x),  ^ T T ^ u , ... ,u) ) = 0  f o r a l l k.. Therefore:: D ^ ( x ) ( ( f ^ T T ^ u , ... ,u) > = 0 ) f o r a l l k .  5.9  i  C l e a r l y 5*8 remains v a l i d i f we replace ^  by  .  15.  This y^eldsi: d ^ C s ^ t u ) ) = ( d ^ C ^ C s ^ t u ) ) : ) , .... ,d> ( ^(s^Ctu))) ) dt* dt dt  5.icr  K  Since  d^CsCtu))  .  ... ,u>  ^ T T C U , ,  =  ^ we get that the l e f t hand side of 5.10 i s  5.11  k-1 £ 7 \ and the x  right side of 5.10 canbe expanded by the chain rule and using 5.9 yields an expression i n $^7T^ The derivatives of the ^ ^  and derivatives of the  can then be expressed i n terras of the  and derivatives of the g^. carrying out this program we obtain  the following: c^rTxCu.u) = - ^  D ( x ) ( u , u ) ^g (x)  5.12  2  gi  <f 7T (u,u,u) = -  ±  D .(x)(u,u u) V ( x )  2  3  x  gi  t  g ±  + 3 i ^D g.(x)(u,.Vg.(x»D g..(x)(u„u>Vg.(x) 5.13 2  2  <S* rr (u,u,u,u) = — i^Lv g (x)(u,u,u,u) Vg^Cx) 3  x  i  +6  i l f  ^lED  -3  l f j f k  3 g j  ( x ) { t t , u , V (x»D gi  2 gi  . ( x > ( u , u ) Vg^Cx)  2: D \ ( X ) ( <7g.(x) „ V j(x) ) D g  2 g i  (x)  (u,u)D  2 g j  (x)(u,u^  V S ^  +4 -12  l D g ( x ) ( „ g .(x))D g (x)(u»u„u) Vg (x) 1 j J J2  3  u  iD g Jx).(u,7g (x)')D g. (x)(u,,Vg.(x))D|.ix) (u,u 2  iiik  2  k  j  j  ;  X  )  -  

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