2 PTR W. MICHOR So we see that for xed (y; v) the transition fnctions are linear in (; w) 2 R m V. This describes the vector bndle strctre of the tang

Transcription

1 TH JACOBI FLOW Peter W. Michor rwin Schrodinger Institt fr Mathematische Physik, Pastergasse 6/7, A-1090 Wien, Astria November 19, 1996 For Wlodek Tlczyjew, on the occasion of his 65th birthday. It is well known that the geodesic ow on the tangent bndle is the ow of a certain vector eld which is called the spray S : T M! T T M. It is maybe less well known that the ow lines of the vector eld T M T S : T T M! T T T M project to Jacobi elds on T M. This cold be called the `Jacobi ow'. This reslt was developed for the lectre corse [5], and it is the main reslt of this paper. I was motivated by the paper [6] of Urbanski in these proceedings to pblish it, as an explanation of some of the ses of iterated tangent bndles in dierential geometry. 1. The tangent bndle of a vector bndle. Let (; p; M) be a vector bndle with ber addition + : M! and ber scalar mltiplication m t :!. Then (T ; ; ), the tangent bndle of the manifold, is itself a vector bndle, with ber addition denoted by + T and scalar mltiplication denoted by m T t. If (U ; : U! U V ) 2A is a vector bndle atlas for, sch that (U ; ) is a manifold atlas for M, then ( U ; ) 0 2A is an atlas for the manifold, where 0 := ( Id V ) : U! U V! (U ) V R m V: Hence the family (T ( U ); T 0 : T ( U )! T ( (U ) V ) = (U ) V R m V ) 2A is the atlas describing the canonical vector bndle strctre of (T ; ; ). The transition fnctions are in trn: (?1 )(x; v) = (x; (x)v) for x 2 U (?1 )(y) = (y) for y 2 (U ) ( 0 ( ) 0?1 )(y; v) = ( (y); (?1 (y))v) (T 0 T ( ) 0?1 )(y; v; ; w) =? (y); (?1 (y))v; d( )(y); (d(?1 )(y)))v + (?1 (y))w : 1991 Mathematics Sbject Classication. 53C22. Key words and phrases. Spray, geodesic ow, Jacobi ow, higher tangent bndles. Spported by `Fonds zr Forderng der wissenscahftlichen Forschng, Projekt P PHY'. Typeset by AMS-TX 1

2 2 PTR W. MICHOR So we see that for xed (y; v) the transition fnctions are linear in (; w) 2 R m V. This describes the vector bndle strctre of the tangent bndle (T ; ; ). For xed (y; ) the transition fnctions of T are also linear in (v; w) 2 V V. This gives a vector bndle strctre on (T ; T p; T M). Its ber addition will be denoted by T (+ ) : T ( M ) = T T M T! T, since it is the tangent mapping of +. Likewise its scalar mltiplication will be denoted by T (m t ). One may say that the second vector bndle strctre on T, that one over T M, is the derivative of the original one on. The space f 2 T : T p: = 0 in T Mg = (T p)?1 (0) is denoted by V and is called the vertical bndle over. The local form of a vertical vector is T 0 : = (y; v; 0; w), so the transition fnctions are (T 0 T ( 0 )(y; v; 0; w) = )?1 ( (y); (?1 (y))v; 0; (?1 (y))w). They are linear in (v; w) 2 V V for xed y, so V is a vector bndle over M. It coincides with 0 M (T ; T p; T M), the pllback of the bndle T! T M over the zero section. We have a canonical isomorphism Vl : M! V, called the big vertical lift, given by Vl ( x ; v x ) t j 0 ( x + tv x ), which is ber linear over M. We will mainly se the small vertical lift vl :! T, given by vl (v x ) t j 0 t:v x = Vl (0 x ; v x ). The local representation of the vertical lift is (T 0 vl ( ) 0?1 )(y; v) = (y; 0; 0; v). If ' : (; p; M)! (F; q; N) is a vector bndle homomorphism, then we have vl F ' = T ' vl :! V F T F. So vl is a natral transformation between certain fnctors on the category of vector bndles and their homomorphisms. The mapping vrp := pr 2 Vl?1 : V! is called the vertical projection. 2. The second tangent bndle of a manifold. All of 1 is valid for the second tangent bndle T T M of a manifold, bt here we have one more natral strctre at or disposal. The canonical ip or involtion M : T T M! T T M is dened locally by (T T M T T?1 )(x; ; ; ) = (x; ; ; ); where (U; ) is a chart on M. Clearly this denition is invariant nder changes of charts (T eqals 0 from 1). The ip M has the following properties: (1) N T T f = T T f M for each f 2 C 1 (M; N). (2) T ( M ) M = T M and T M M = T ( M ). (3)?1 M = M. (4) M is a linear isomorphism from the vector bndle (T T M; T ( M ); T M) to the bndle (T T M; T M ; T M), so it interchanges the two vector bndle strctres on T T M. (5) It is the niqe smooth mapping T T M! T T M which s c(t; s) = t c(t; s) for each c : R 2! M. All this follows from the local formla given above. A qite early se of M is in [4].

3 TH JACOBI FLOW 3 3. Lemma. For vector elds X, Y 2 X(M) we have [X; Y ] = vrp T M (T Y X? M T X Y ); T Y X? T M T T X Y = Vl T M (Y; [X; Y ]) = (vl T M [X; Y ]) T (+ T M ) (0 T M Y ): See [3] 6.13, 6.19, or for dierent proofs of this well known reslt. 4. Linear connections and their crvatres. Let (; p; M) be a vector bndle. Recall that a linear connection on the vector bndle can be described by specifying its connector K : T!. This notions seems to be de to [2]. Any smooth mapping K : T! which is a (ber linear) homomorphism for both vector bndle strctres on T, T K p w p T T p K M wm T M w w p M and which is a left inverse to the vertical lift, K vl = Id :! T!, species a linear connection. Namely: The inverse image H := K?1 (0 ) of the zero section 0, it is a sbvector bndle for both vector bndle strctres, and for the vector bndle stctre : T! the sbbndle H trns ot to be a complementary bndle for the vertical bndle V!. We get then the associated horizontal lift mapping C : T M M! T ; C( ; ) = which has the following properties (T p; ) C = Id T MM ;?1 T pj ker(k : T! p()) C( ; ) : T p()m! T is linear for each 2 ; C(X x ; ) : x! (T p)?1 (X x ) is linear for each X x 2 T x M: Conversely given a smooth horizontal lift mapping C with these properties one can reconstrct a connector K. For any manifold N, smooth mapping s : N! along f = p s : N! M, and vector eld X 2 X(N) a connector K : T! denes the covariant derivative of s along X by (1) r X s := K T s X : N! T N! T! : See the following diagram for all the mappings. (2) T 44 hj T 4K 46 h hh s T N N X s NP ' ' ') N' NN '''' r X s N wm f

4 4 PTR W. MICHOR In canonical coordinates as in 1 we have then C((y; ); (y; v)) = (y; v; ;? y (v; )); K(y; v; ; w) = (y; w?? y (v; )); r (y;)(id; s) = (Id; ds(y)?? y (s(y); )); where the Christoel symbol? y (v; ) is smooth in y and bilinear in (v; ). Here the sign is the negative of the one in many more traditional approaches, since? parametrizes the horizontal bndle. Let Cf 1 (N; ) denote the space of all sections along f of, isomorphic to the space C 1 (f ) of sections of the pllback bndle. The covariant derivative may then be viewed as a bilinear mapping r : X(N) Cf 1(N; )! C1 f (N; ). It has the following properties which follow directly from the denitions: (3) r X s is C 1 (N; R)-linear in X 2 X(N). For x 2 N also we have r X (x)s = K:T s:x(x) = (r X s)(x) 2. (4) r X (h:s) = dh(x):s + h:r X s for h 2 C 1 (N; R). (5) For any manifold Q, smooth mapping g : Q! N, and Y y 2 T y Q we have r T g:yy s = r Yy (s g). If Y 2 X(Q) and X 2 X(N) are g-related, then we have r Y (s g) = (r X s) g. For vector elds X, Y 2 X(M) and a section s 2 C 1 () the crvatre R 2 2 (M; L(; )) of the connection is given by (6) R(X; Y )s = ([r X ; r Y ]? r [X;Y ])s Theorem. Let K : T! be the connector of a linear connection on a vector bndle (; p; M). If s : N! is a section along f := p s : N! M then we have for vector elds X, Y 2 X(N) (7) r X r Y s? r Y r X s? r [X;Y ]s = = (K T K? K T K) T T s T X Y = = R (T f X; T f Y )s : N! ; where R 2 2 (M; L(; )) is the crvatre. Proof. Let rst m t :! denote the scalar mltiplication. Then we t j 0 m t = vl where vl :! T is the vertical lift. We se then lemma 3 and some obvios commtation relations to get in trn: vl K t j 0 m t K t j 0 K m T t r X r Y s? r Y r X s? r [X;Y ]s = T t j 0 m T t = T K vl (T ; ;) : = K T (K T s Y ) X? K T (K T s X) Y? K T s [X; Y ] K T s [X; Y ] = K vl K T s [X; Y ] = K T K vl T T s [X; Y ] = K T K T T s vl T N [X; Y ] = K T K T T s ((T Y X? N T X Y ) (T?) 0 T N Y ) = K T K T T s T Y X? K T K T T s N T X Y? 0:

5 TH JACOBI FLOW 5 Now we sm p and se T T s N = T T s to get the rst reslt. If in particlar we choose f = Id M so that s is a section of! M and X; Y are vector elds on M, then we get the crvatre R. To see that in the general case (K T K? K T K) T T s T X Y coincides with R(T f X; T f Y )s one has to write ot (1) and (T T s T X Y )(x) 2 T T in canonical charts indced from vector bndle charts of. 5. Torsion. Let K : T T M! M be a linear connector on the tangent bndle, let X; Y 2 X(M). Then the torsion is given by Tor(X; Y ) = (K M? K) T X Y: If moreover f : N! M is smooth and U; V 2 X(N) then we get also Tor(T f:u; T f:v ) = r U (T f V )? r V (T f U)? T f [U; V ] Proof. (9) We have in trn = (K M? K) T T f T U V: Tor(X; Y ) = r X Y? r Y X? [X; Y ] = K T Y X? K T X Y? K vl T M [X; Y ] K vl T M [X; Y ] = K ((T Y X? M T X Y ) (T?) 0 T M Y ) = K T Y X? K M T X Y? 0: An analogos comptation works in the second case, and that (K M?K)T T f T U V = Tor(T f:u; T f:v ) can again be checked in local coordinates. 6. Sprays. Given a linear connector K : T T M! M on the tangent bndle with its horizontal lift mapping C : T M M T M! T T M, then S := C diag : T M! T M M T M! T T M is called the spray. This notion is de to [1]. The spray has the following properties: T M S = Id T M a vector eld on T M; T ( M ) S = Id T M S m T M t = T (m T t M ) m T t T M S `qadratic'; a second order dierential eqation; where m t is the scalar mltiplication by t on a vector bndle. From S one can reconstrct the torsion free part of C. The following reslt is well known: Lemma. For a spray S : T M! T T M on M, for X 2 T M geo S (X)(t) := M (Fl S t (X)) denes a geodesic strctre on M, where Fl S is the ow of the vector eld S. The abstract properties of a geodesic strctre are obvios: geo : T M R U! M geo(x)(0) = M (X); geo(tx)(s) = t j 0 geo(x)(t) = X geo(geo(x) 0 (t))(s) = geo(x)(t + s) From a geodesic strctre one can reconstrct the spray by dierentiation.

6 6 PTR W. MICHOR 7. Theorem. Let S : T M! T T M be a spray on a manifold M. Then T M T S : T T M! T T T M is a vector eld. Consider a ow line of this eld. Then we have: T Y (t) = Fl MT S t (Y (0)) c := M T M Y is a geodesic on M. _c = T M Y is the velocity eld of c. J := T ( M ) Y is a Jacobi eld along c. J _ = M Y is the velocity eld of J. J = K M Y is the covariant derivative of J. The Jacobi eqation is given by: 0 = J + R(J; _c) _c + Tor(J; _c) = K T K T S Y: This implies that in a canonical chart indced from a chart on M the crve Y (t) is given by (c(t); c 0 (t); J(t); J 0 (t)): Proof. Consider a crve s 7! X(s) in T M. Then each t 7! M (Fl S t (X(s))) is a geodesic in M, and in the variable s it is a variation throgh geodesics. Ths J(t) s j 0 M (Fl S t (X(s))) is a Jacobi eld along the geodesic c(t) := M (Fl S t (X(0))), and each Jacobi eld is of this form, for a sitable crve X(s). We consider now the crve Y (t) s j 0 Fl S t (X(s)) in T T M. Then by 2.(6) we t Y (t) s j 0 Fl S t (X(s)) = T s j t Fl S t (X(s)) = T s j 0 S(Fl S t (X(s))) = ( T M T S)(@ s j 0 Fl S t (X(s))) = ( T M T S)(Y (t)); so that Y (t) is a ow line of the vector eld T M T S : T T M! T T T M. Moreover sing the properties of from section 2 and of S from section 6 we get T M :Y (t) = T M :@ s j 0 Fl S t (X(s)) s j 0 M (Fl S t (X(s))) = J(t); M T M Y (t) = c(t); the t J(t) t T M :@ s j 0 Fl S t (X(s)) s j 0 M (Fl S t (X(s))); = s j t M (Fl S t (X(s))) = s j t M (Fl S t (X(s))) = s j 0 T M :@ t Fl S t (X(s)) = s j 0 (T M S) Fl S t (X(s)) = s j 0 Fl S t (X(s)) = M Y (t); J = t J = K M Y: Finally let s express the well known Jacobi expression, where we pt (t; s) := M (Fl S t (X(s))) for short and se most of the expressions from above: J + R(J; _c) _c + Tor(J; _c) = = :T :@ s + R(T :@ s ; T :@ t )T :@ t + Tor(T :@ s ; T :@ t ) = K:T (K:T (T :@ s ):@ t ):@ t + (K:T K: T M? K:T K):T T (T :@ t s :@ t + K:T ((K: M? K):T T s :@ t ):@ t

7 TH JACOBI FLOW 7 Note that for example for the term in the second smmand we have T T T :T t s :@ t = T (T (@ t ):@ s ):@ t t t : M :@ t :@ s = T M :@ t :@ t :@ s which at s = 0 eqals T M J. Using this we get for the Jacobi expression at s = 0: J + R(J; _c) _c + Tor(J; _c) = = (K:T K + K:T K: T M :T M? K:T K:T M + K:T K:T M? K:T K):@ t J = = K:T K: T M :T M :@ t J = K:T K: T M :@ t Y = K:T K:T S:Y; where we t J t ( M :Y ) = T t Y = T M : T M :T S:Y. Finally the validity of the Jacobi eqation 0 = K:T K:T S:Y follows trivially from K S = 0 T M. References 1. Ambrose, W; Palais, R.S., Singer I.M., Sprays, Acad. Brasileira de Ciencias 32 (1960), 163{ Gromoll, D.; Klingenberg, W.; Mayer, W., Riemannsche Geometrie im Groen, Lectre Notes in Math. 55, Springer-Verlag, Berlin, Heidelberg, Kolar, Ivan; Michor, Peter W.; Slovak, Jan, Natral operations in dierential geometry, Springer-Verlag, Berlin, Heidelberg, New York, Losik, M.V., On innitesimal connections in tangential stratiable spaces, Izv. Vyssh. Uchebn. Zaved., Mat. 5(42) (1964), 54{60. (Rssian) 5. Michor, Peter W., Riemannsche Dierentialgeometrie, Lectre corse at the Universitat Wien, 1988/ Urbanski, P., Doble bndles, These proceedings?? (??),??. Institt fr Mathematik, Universitat Wien, Strdlhofgasse 4, A-1090 Wien, Astria -mail address: Peter.Michor@esi.ac.at