Robert J. Fisher and H. Turner Laquer

J. Korean Math. Soc. 36 (1999), No. 5, pp. 959{1008

SECOND ORDER TANGENT VECTORS IN RIEMANNIAN GEOMETRY Robert J. Fisher and H. Turner Laquer

Abstract. This paper considers foundational issues related to con-

nections in the tangent bundle of a manifold. The approach makes use of second order tangent vectors, i.e., vectors tangent to the tangent bundle. The resulting second order tangent bundle has certain properties, above and beyond those of a typical tangent bundle. In particular, it has a natural secondary vector bundle structure and a canonical involution that interchanges the two structures. The involution provides a nice way to understand the torsion of a connection. The latter parts of the paper deal with the Levi-Civita connection of a Riemannian manifold. The idea is to get at the connection by rst nding its spray. This is a second order vector eld that encodes the second order dierential equation for geodesics. The paper also develops some machinery involving lifts of vector elds from a manifold to its tangent bundle and uses a variational approach to produce the Riemannian spray.

0. Introduction The subject of dierential geometry continues to reveal that the concept of a connection is fundamental. In the case of vector bundles, connections are de ned in a variety of ways|covariant derivatives, forms on frame bundles, Christoel symbols, and horizontal distributions being the most common. An alternative approach is to encode the horizontal lift process as a function taking values in the tangent bundle of the vector bundle. This approach is not very well-known Received April 7, 1999. 1991 Mathematics Subject Classi cation: Primary 53B05; Secondary 53C05, 58A05, 58E10. Key words and phrases: connections in vector bundles, second order tangent vectors, canonical involution, torsion, geodesic spray, horizontal, canonical, and linear lifts, symmetric vector elds.

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R. J. Fisher and H. T. Laquer

but, when properly formulated, is an extremely useful description of a connection. The rst big idea needed to make the horizontal lift process precise is that the tangent bundle to a vector bundle has a natural secondary vector bundle structure. In section 1, we describe this secondary structure and give intrinsic descriptions of the secondary vector bundle operations in terms of curves representing tangent vectors. These operations are needed to express the horizontal lift process cleanly (see section 3). If the original vector bundle is the tangent bundle TM to a manifold M then the second order tangent bundle T (TM ) has two natural vector bundle structures as a bundle over TM: However, as is described in section 2, there is a canonical involution of T (TM ) that carries one structure into the other (theorem 1). Our understanding of the involution uses a natural submersion from a manifold of 2-jets, see lemma 2.1. The canonical involution returns in section 3 in the description of the torsion of a connection in the tangent bundle. By using second order tangent vectors, we develop a pointwise version of the classical torsion tensor, see (3.3) and theorem 2, and show that a connection is torsion-free if and only if the horizontal distribution is preserved by the canonical involution (corollary 3.1). One reason for using this approach to connections is to understand the spray of a connection in TM: The spray is a way of expressing the second order dierential equation for the geodesics of a connection. In this setting, it is natural to consider vector elds on TM that are also symmetric. Lemmas 4.1 and 4.2 characterize symmetric vector elds and sprays, respectively. The correspondence between sprays and torsion-free connections is given in lemma 4.3. Section 4 also introduces the canonical lift and, more generally, linear lifts. Section 5 deals with the spray of a Riemannian connection. The main goal of the section is to reconcile an algebraic approach for the spray with a variational approach, see propositions 5.1 and 5.2. Lemma 5.3 provides an important formula for the variation of energy in terms of a Lie derivative by a canonical lift. Finally, in section 6 we give a proof of the Fundamental Theorem of Riemannian Geometry from this point of view. In particular, lemma 6.1 gives a way to understand metric compatibility in terms of horizontal lifts.

Second order tangent vectors

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This paper is largely expository. Indeed, a number of its foundational themes have a rich history. However, certain aspects of the approach are not widely known|the central role of the canonical involution is not fully appreciated. In addition, our approach emphasizes the intrinsic description of ideas and coordinate free dierentiation, as compared to more classical local computations. Although [4] was the source of much of our inspiration, other authors who have dealt with some of these ideas include [1], [2], [3], and [7]. Related ideas involving second order frames have also been used to understand projective and conformal structures, see [6, chapter 4].

1. The Tangent Bundle To A Vector Bundle In [4, 16.15.7], Dieudonne proves the following result: Proposition 1.1. Let : E ;! M be a vector bundle. Then T : T E ;! TM is a vector bundle so that the manifold T E has two vector bundle structures, namely, its primary vector bundle structure as the tangent bundle of the smooth manifold E ; and a secondary structure with TM as the base manifold. Dieudonne's proof of proposition 1.1. By way of introduction, it is convenient to sketch the Dieudonne argument here. Since the question of proposition 1.1 is local with respect to the base manifold M; we assume that : E ;! M is a trivial vector bundle with a preferred trivialization E = M E: Also, M can be assumed to be a nite dimensional real vector space. In turn, the bundle projection is the rst projection

= pr1 : M E ;! M : Next, in the canonical way, identify TM to M M and T (M E ) to (M E ) (M E ): A point of T E = T (M E ) will be written in the form (m; x; h; v) and corresponds to the tangent vector at (m; x) represented by the curve t 7;! (m + th; x + tv) :

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The mapping T : T E ;! TM is realized by the linear function

T (m; x; h; v) = (m; h) : In turn, the secondary vector bundle operations in the ber (T E )(m;h) = (T);1(m; h) are de ned by: (1.1)

(m; x; h; v) +2 (m; y; h; w) = (m; x + y; h; v + w) 2 (m; x; h; v) = (m; x; h; v)

where (m; x; h; v); (m; y; h; w) 2 (T E )(m;h) and 2 R ; cf. (2.7). Next, it must be argued that the vector bundle structure is independent of the trivialization of E from which we started. Since : E ;! M is a vector bundle, two trivializations are related by a dieomorphism : M E ;! M E of the form (m; x) = (m; g(m) x) where g : M ;! GL(E ) is a smooth map. The tangent map of the dieomorphism T : T E ;! T E given by (1.2)

;

is

T (m; x; h; v) = (m; x) ; D (m; x) (h; v) ; = m; g(m) x ; h; Dg(m) h x + g(m) v ;

where D is used to denote the derivative of a mapping between vector spaces, cf. [4, 8.1]. Dieudonne completes the proof by observing that by (1.2) the function

T : (T E )(m;h) ;! (T E )(m;h) is linear, i.e., linear in the pair (x; v) for each (m; h) 2 TM: While the proof of proposition 1.1 establishes T : T E ;! TM as a vector bundle, the proof by its very nature does not reveal in intrinsic terms the secondary vector bundle operations. For example, given two lifts v; w 2 T E of a vector h 2 TM; how does one describe the secondary sum v +2 w without appealing to a local vector bundle chart? This issue is addressed in the following subsection.


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Intrinsic operations. Returning to the general setting of propo-

sition 1.1, let h 2 TM: Note, in the general setting, we let h denote a generic tangent vector to M and will write hm if it is necessary to indicate that the vector is based at m: Set (T E )h = T;1(h) |all \lifts" of the tangent vector h: Next, let +2 and 2 serve as the symbols for the secondary vector bundle operations. For any v; w 2 (T E )h ; it will be argued that the secondary sum v +2 w is described intrinsically as follows: given any curve m(t) that represents h; there are curves x(t) and y(t) representing v and w; respectively, so that (x(t)) = (y(t)) = m(t): The pointwise sum x(t)+ y(t) is then de ned in E and

v +2 w = [x(t) + y(t)] |the tangent vector represented by the curve x(t)+ y(t): In particular, v +2 w is a lift of h based at x(0)+ y(0): Similarly, the secondary scaling of v by a scalar is the tangent vector represented by the curve x(t) where x(t) is any curve representing v; i.e., (1.4) 2 v = [ x(t)] : Finally, let : M ;! E be the zero section. It is clear from (1.3) that for each h 2 TM; the zero vector of (T E )h is T h; the image of h under the tangent map of : Given two submersions i : E i ;! M (i = 1; 2); let E 1 M E 2 denote their bered product, that is, the closed submanifold of E 1 E 2 de ned by E 1 M E 2 = f(x; y ) j 1 (x) = 2 (y )g : When the i : E i ;! M are ber bundles, it can be shown by elementary means that the bered product E 1 M E 2 is naturally a ber bundle in three ways, namely over M; E 1 and E 2 : If the i : E i ;! M are vector bundles then the bered product is the familiar Whitney sum E 1 E 2 ; see [4, 16.16]. The following two general facts are needed in order to prove (1.3) and (1.4). Their proofs are elementary and are deferred until the end of this section.

(1.3)

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Lemma 1.1 (Lift Lemma). Let : E

;! M be a smooth ber bun-

dle. For each h 2 TM and for each curve m(t) that represents h; i.e., h = [m(t)] = dtd t=0 m(t); and for each v 2 T E such that T v = h; there exists a curve x(t) in E such that x(t) represents v and ( x)(t) = m(t): Lemma 1.2. Let i : E i ;! M; (i = 1; 2); be two ber bundles. Then the manifolds T (E 1 M E 2 ) and T E 1 TM T E 2 are canonically dieomorphic. The vector bundle addition in : E ;! M is a smooth mapping +E : E M E ;! E : Given a local vector bundle chart : E jU ;! U E; the mapping +E is described as follows: for each m 2 U and each x; y 2 E m ;

x +E y = ;1 m; pr2 ((x)) + pr2 ((y)) where pr2 : U E ;! E is the second projection map and the internal + in (1.5) is the vector space addition of the model ber E: The vector bundle addition from proposition 1.1 for T : T E ;! TM is a wellde ned smooth function +2 : T E TM T E ;! T E : A local vector bundle chart : E jU ;! U E for : E ;! M induces one for T : T E ;! TM; namely, it is the function T : T E jTU ;! T (U E ) = TU TE : Here T (U E ) is identi ed to TU TE in the canonical way and, via the canonical bijection between TE and E E; TE is regarded as a vector space. By analogy with (1.5), +2 is described as follows: for any h 2 TU and all v; w 2 (T E )h ; ; (1.6) v +2 w = T ;1 h ; Tpr2 (T (v)) + Tpr2 (T (w)) : Note that under the identi cation T (U E ) = TU TE; the second projection map pr2 : TU TE ;! TE and Tpr2 : T (U E ) ;! TE are the same. (1.5)


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Let : T E TM T E ;! T (E M E ) denote the canonical dieomorphism of lemma 1.2. By de nition, an element of T E T M T E is a pair of tangent vectors (v; w) such that T v = T w: The tangent vector (v; w) is represented by a curve (x(t); y(t)) 2 E M E where x(t) and y(t) are curves representing v and w such that (x(t)) = (y(t)): We will argue next that (1.7)

+2 = (T +E ) :

Given local vector bundle charts as above, let h 2 TU and let v; w 2 (T E )h : Take curves x(t); y(t) so that (v; w) = [x(t); y(t)]: Then m(t) = (x(t)) = (y(t)) represents h so that by the de nition of a tangent mapping, see [4, 16.5.3.1], and by (1.5) and (1.6) (T +E )((v; w)) = (T +E ) [x(t); y(t)] = [x(t) +E y(t)] ; = ;1 m(t) ; pr2 ((x(t))) + pr2 ((y(t))) (1.8) ; = T ;1 h ; Tpr2 (T (v)) + Tpr2 (T (w)) = v +2 w : Note that in the next to last line of (1.8), the internal addition is the canonical addition of the vector space TE: Equation (1.7) follows immediately. Also note that embedded in (1.8) is the intrinsic description of +2 : Let R : R ;! T R denote the zero section and let (R; id): R T E ;! T R T E be the canonical mapping. Next, let E : R E ;! E denote scalar multiplication in the bundle E : Then, in a manner more straightforward than for +2 ; it can be argued that

2 = (T E ) (R; id) : The intrinsic description (1.4) of 2 follows immediately.

(1.9)

Remark. It is an interesting exercise to show directly that (1.7) and (1.9) de ne vector space structures in the bers of T : T E ;! TM: One approach is to encode the vector space axioms in the bers of : E ;! M as a collection of commutative diagrams. The corresponding tangent-map diagrams, in which all spaces and mappings are

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replaced by tangent spaces and tangent mappings, will commute by the chain rule. T is, after all, a functor. Modulo certain canonical mappings, these tangent-map diagrams encode the berwise vector space axioms for T : T E ;! TM: Proof of lemma 1:1. Let (U; E; ) be a local ber bundle trivialization of E and let y(t) be any curve that represents v; i.e., v = [y(t)]: Then we have a smooth curve z : R ;! E such that ;

(y(t)) = (y(t)); z(t) : De ne

x(t) = ;1 (m(t); z(t)) : Then clearly ( x)(t) = m(t); and by a routine argument, [x(t)] = v: Proof of lemma 1:2. Since the Ti : T E i ;! TM are submersions, the bered product T E 1 TM T E 2 is a manifold. In particular, the question of whether T E 1 TM T E 2 is a ber bundle over TM is not relevant at this point. De ne a function

: T (E 1 M E 2 ) ;! T E 1 TM T E 2 as follows: let z 2 T (E 1 M E 2 ): By de nition, z is represented to rst order by a curve c(t) = (x(t); y(t)) in E 1 M E 2 so that (1.10)

(x(t)) = (y(t)) :

De ne

(z) = ([x(t)]; [y(t)]) |the pair of tangent vectors represented by the curves x(t) and y(t): By (1.10) and the chain rule, T [x(t)] = T [y(t)] so that (z) 2 T E 1 TM T E 2 : Next, it is clear that is injective and, by an obvious application of the Lift Lemma, is also surjective.


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Finally, to show that and its inverse are smooth, it suces to argue that there are atlases for the source and target in which the local description of is the identity mapping. The bundle projection 12 : E 1 M E 2 ;! M is de ned by 12 (x1; x2) = 1 (x1 ) = 2 (x2) : Local ber bundle charts i : E i jU ;! U Ei induce natural charts for E 1 M E 2 ; T (E 1 M E 2 ) and T E 1 TM T E 2 as follows: the function 12 : (E 1 M E 2 )U ;! U (E1 E2 ) de ned by ; 12 (x1 ; x2) = 12 (x1; x2) ; (pr2 1 )(x1); (pr2 2 )(x2) is a dieomorphism so that the function T 12 : T (E 1 M E 2 )TU ;! TU (TE1 TE2 ) de ned by ; T 12 z = T12 z ; T (pr2 1 pr1 ) z; T (pr2 2 pr2 ) z is also a dieomorphism. On the other hand, the local description 12 : (T E 1 TM T E 2 )TU ;! TU (TE1 TE2 ) is given by ; 12 (v1 ; v2) = T12 (v1; v2 ) ; T (pr2 1 ) v1 ; T (pr2 2 ) v2 : Thus (12 T ;121 ) (h; a1; y1; a2; y2) = (12 T ;121 ) [m(t); a1 + ty1 ; a2 + ty2 ] = (12 ) ;1 1 (m(t); a1 + ty1 ) ; ;2 1 (m(t); a2 + ty2) ; = 12 [;1 1 (m(t); a1 + ty1)] ; [;2 1 (m(t); a2 + ty2)] ; = 12 T ;1 1 (h; a1; y1) ; T ;2 1 (h; a2; y2) = (h; a1; y1; a2; y2) and the proof is now complete. The following result is a corollary of proposition 1.1. Its proof makes use of the intrinsic description of the secondary vector bundle operations.

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Corollary 1.1. Let i : E i

;! Mi (i = 1; 2) be a pair of vector

bundles and let (g; f ) be a vector bundle morphism from the rst to the second, which is represented by the commutative diagram (1.11)

E1 ? 1 ? y

g ;;;; !

E2 ? ? y 2

f M1 ;;;; ! M2: Then the pair of tangent mappings (Tg; Tf ) is a vector bundle morphism between the secondary vector bundle structures, i.e., the diagram below represents a vector bundle morphism.

(1.12)

Tg T E 1 ;;;; ! TE2 ?

T1 ? y

? ?T y 2

Tf TM1 ;;;; ! TM2 :

Proof. Diagram (1.12) commutes by the chain rule so that for any h 2 TM1 ; Tg maps (T E 1 )h into (T E 2 )Tf h: Next, let v; w 2 (T E 1 )h : By the Lift Lemma, there are smooth curves x(t) and y(t) with v = [x(t)]; w = [y(t)]; and 1 (x(t)) = 1 (y(t)): Then by two applications of (1.3) and by the berwise linearity of g;

Tg(v +2 w) = Tg [x(t) + y(t)] = [g(x(t)) + g(y(t))] = [g(x(t))] +2 [g(y(t))] = Tg(v) +2 Tg(w) : The argument that Tg( 2 v) = 2 Tg(v) is similar.

2. The Canonical Involution On T(TM)

Let M be a smooth manifold. By de nition a second order tangent vector to M is an ordinary tangent vector to TM: A second order tangent vector v 2 T (TM ) is said to be symmetric i


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TM v = TM (v); that is, v is a lift of the vector at which it is based. For h; x 2 TM with M (h) = M (x); set Tx (TM )h = fv 2 T (TM ) j TM (v) = x ; TM v = hg |all lifts of h that are based at x: By proposition 1.1, T (TM ) has two vector bundle structures as a bundle over TM; and to quote from Dieudonne, these two structures are \quite distinct". However, this is not entirely accurate because the two vector bundle structures are, in fact, canonically isomorphic. There is a canonical involution of T (TM ) that interchanges the two structures. This involution is known at a local level, cf. [4, 16.20, exercise 2]. Our purpose in this section is to give a more global proof of the following result. Theorem 1. For any smooth manifold M; there is a canonical involution I : T (TM ) ;! T (TM ) that is a TM -isomorphism of vector bundles in the following way: I ! T (TM ) T (TM ) ;;;; ?

TM ? y

? ?T y M

= TM ;;;; ! TM: Moreover, the symmetric second order tangent vectors are precisely the ones xed by I : Remark. The map I ; being an involution, equals its own inverse. Also, the inverse of a vector bundle isomorphism is again a vector bundle isomorphism. Thus, I is a TM -isomorphism from the secondary structure to the primary structure, as well. In outline form, the proof of the theorem has three steps: Step 1. In a natural way, the second order tangent bundle T (TM ) is a quotient of J02 (R 2 ; M ); the manifold of 2-jets of maps from R 2 to M with source 0: We call this quotient map Q : J02 (R2 ; M ) ;! T (TM ): Step 2. There is a canonical involution Ie : J02 (R 2 ; M ) ;! J02 (R 2 ; M ) that descends via Q to T (TM ) giving the involution I in theorem 1.

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Step 3. The involution I is a TM -isomorphism between the primary

and secondary vector bundle structures on T (TM ); its xed points are precisely the symmetric second order tangent vectors. Proof of step 1. Both the 2-jet manifold J02(R 2 ; M ) and the manifold T (TM ) have natural atlases that can be described as follows: let E be a nite dimensional real vector space. Then T (TE ) and J02 (R2 ; E ) are naturally regarded as nite dimensional real vector spaces. In the canonical way, set TE = E E and in turn set T (TE ) = TE TE: Elements of T (TE ) will be written as quartets of the form (m; x; h; v): Next, let L(R 2 ; E ) denote the vector space of linear functions A : R 2 ;! E and let S2 (R 2 ; E ) denote the vector space of all symmetric bilinear functions B : R 2 R2 ;! E: The function ; [] 7;! (0); D(0); D2(0) is the canonical isomorphism J02 (R2 ; E ) ;! E L(R2 ; E ) S2 (R2 ; E ) relative to which the two vector spaces are thought of interchangeably. Let ' : U ;! E be a local chart for M: Set J02 (R2 ; U ) = f[] 2 J02(R 2 ; M ) j (0) 2 U g : Then the function '^ : J02 (R 2 ; U ) ;! J02 (R2 ; E ) de ned by '^[] = [' ] ; = (' )(0); D(' )(0); D2(' )(0) is a local chart for J02 (R2 ; M ): Similarly, a local chart for T (TM ) is given by the second order tangent map of the local chart '; i.e., T 2 ' = T (T'): T (TU ) ;! T (TE ) : Next, let f : E ;! F be a smooth mapping between two nite dimensional vector spaces. By way of the canonical identi cation, Tf : TE ;! TF is described as Tf (m; x) = (f (m); Df (m) x)


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so that the second order tangent map of f is the function

T 2f = T (Tf ): T (TE ) ;! T (TF ) given by ;

T 2 f (m; x; h; v) = Tf (m; x) ; D(Tf )(m; x) (h; v) (2.1) ; = f (m); Df (m) x; Df (m) h; Df (m) v + D2 f (m) x h : We will also be using the second order chain rule for tangent mappings: given smooth functions g : E ;! F and f : F ;! G;

T 2(f g) = T 2 f T 2g : Let : R 2 ;! M be a smooth mapping. Then T 2 (0; e1; e2; 0) is the value of the second tangent map of on the second order tangent vector (0; e1; e2; 0) 2 T (T R2 ) where fe1 ; e2 g is the standard basis for R2 : Lemma 2.1. The function Q : J02 (R 2 ; M ) ;! T (TM ) de ned by

Q([]) = T 2 (0; e1; e2; 0) is a submersion so that T (TM ) is a quotient manifold of J02 (R2 ; M ): Therefore, a second order tangent vector to M is represented by a 1-parameter family of curves : R 2 ;! M: In particular, when interpreted as a curve of curves,

Q([]) = [t 7! [s 7! (s; t)]] ; where [s 7! (s; t)] is the tangent vector to the curve s 7! (s; t) at s = 0 and where [t 7! [s 7! (s; t)]] is the second order tangent vector represented by the given curve of rst order tangent vectors, cf. [1; 1:18].

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Proof. First of all, Q is a well de ned function|for suppose that [] = [ ] and that ' : U ;! E is a local chart with (0) = (0) 2 U: Then a routine derivative computation using (2.1) shows that

(T 2 ' T 2 ) (0; e1; e2 ; 0) = (T 2 ' T 2 ) (0; e1; e2; 0) and hence, because T 2 ' : T (TU ) ;! T (TE ) is a local chart for T (TM );

T 2 (0; e1; e2 ; 0) = T 2 (0; e1; e2 ; 0) : To complete the proof, it will be argued, using the natural atlases for J02 (R2 ; M ) and T (TM ); that the local form of Q is a surjective linear map. Let ' : U ;! E be a local chart of M that, without loss of generality, we assume maps onto the vector space E: Let (z; A; B ) 2 J02 (R2 ; E ); the corresponding 2-jet is the one represented by ^ x 7;! z + A x + 21 B x x :

Then (T 2' Q '^;1 ) (z; A; B ) = (T 2' Q) [';1 ^] ; = T 2' T 2(';1 ^) (0; e1; e2; 0) = T 2^ (0; e1; e2 ; 0) = (z; A e1 ; A e2 ; B e1 e2 ) (by (2.1)) : However, by elementary linear algebra, the function

Qb : J02 (R 2 ; E ) ;! T (TE ) given by

Qb(z; A; B ) = (z ; A e1 ; A e2 ; B e1 e2 ) : is a surjective linear map; its kernel is K = f(0; 0; B ) j B e1 e2 = 0g: It follows immediately that Q is a submersion so that by general principles, see [4, 16.7], Q is a quotient map and the proofs of the lemma and step 1 are complete.


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Proof of step 2. Let f : R 2 ;! R 2 denote the ip map f(s; t) = (t; s): De ne Ie : J02 (R2 ; M ) ;! J02 (R 2 ; M ) by Ie[] = [ f] : An elementary chain rule argument shows that Ie is a well de ned dieomorphism; in terms of the natural atlas for J02 (R2 ; M ); cf. the proof of step 1 just above, the local form of Ie is the linear involution of J02(R 2 ; E ) de ned by

(z; A; B ) 7;! (z; A f; B (f; f)) where (f; f) = (f pr1 ) (f pr2 ): R 2 R 2 ;! R 2 R 2 : We next argue that Ie maps bers of Q to bers of Q; that is, (2.2)

Q([]) = Q([ ])

=)

Q(Ie([])) = Q(Ie([ ])) :

Once established, (2.2) and step 1 imply by general principles, see [4, 16.7.7 (ii)], that Ie descends uniquely to a smooth function

I : T (TM ) ;! T (TM ) : Additionally, from the functional equation

Q Ie = I Q and the fact that Ie : J02 (R2 ; M ) ;! J02(R 2 ; M ) is an involution, it follows routinely that I is also an involution. Let ' : U ;! E be a local chart of M; and let '^ : J02(R 2 ; U ) ;! J02 (R2 ; E ) and T 2 ' : T (TU ) ;! T (TE ) be the induced charts on

J02 (R2 ; M ) and T (TM ); respectively. From the proof of lemma 2.1, the following functional equation holds

Qb '^ = T 2' Q ; so that if Q([]) = Q([ ]) then (2.3)

(Qb '^)[] = (Qb '^)[ ] :

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Next, [] 2 J02(R 2 ; U ) implies that Ie[] 2 J02 (R 2 ; U ): Thus (T 2 ' Q Ie)[] = (Qb '^ Ie)[] = Qb[ ' f ] ; = Qb (' )(0); D(' )(0) f; D2(' )(0) (f; f) (2.4) ; = (' )(0); D(' )(0) e2 ; D(' )(0) e1 ; D2(' )(0) e2 e1 : By equation (2.3) and the symmetry of the second derivative, it follows rst that ; (' )(0); D(' )(0) e2 ; D(' )(0) e1 ; D2(' )(0) e2 e1 ; = (' )(0); D(' )(0) e2 ; D(' )(0) e1 ; D2 (' )(0) e2 e1 so that after replacing [] with [ ] in (2.4), (2.5) (T 2 ' Q Ie)[] = (T 2 ' Q Ie)[ ] : On the other hand, because T 2 ' : T (TU ) ;! T (TE ) is injective, it follows immediately from (2.5) that (Q Ie)[] = (Q Ie)[ ] : The proof of step 2 is now complete. Proof of step 3. We begin by proving that for each h 2 TM; I ( Th (TM ) ) = T (TM )h : Let v 2 Th (TM ): Choose any [] 2 Q;1 (v): The pair (T 2 ; T) is a vector bundle morphism in two ways|from (T (T R2 ); T R2 ; T R2 ) to (T (TM ); TM; TM ) and, by corollary 1.1, from (T (T R2 ); T R2 ; TR2 ) to (T (TM ); TM; TM ): It follows that TM (I v) = (TM T 2 ) (0; e2; e1; 0) = (T TR2 ) (0; e2; e1; 0) = T (0; e1) = (T T R2 ) (0; e1; e2; 0) = (TM T 2 ) (0; e1; e2; 0) = h:


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Thus I (Th (TM )) T (TM )h: By symmetric reasoning I (T (TM )h) Th (TM ) so that I (Th (TM )) = T (TM )h follows. To complete the proof that I is a morphism of vector bundles, it remains to show that I : Th (TM ) ;! T (TM )h is a linear function. Let ' : U ;! E be a local chart for M: By corollary 1.1, (T 2 '; T') is the isomorphism of vector bundles that is expressed by the diagram T 2' T (TU ) ;;;; ! T (TE ) ?

TU ? y

? ?T y E

T' TU ;;;; ! TE: 2 On the other hand, the pair (T '; T') is by nature an isomorphism between the primary vector bundle structures as well. Thus, there is a natural atlas for the manifold T (TM ) that is simultaneously a vector bundle atlas for T (TM ) relative to the primary and secondary structures. For the natural vector bundle atlas of the previous paragraph, the local description of I is the function Ib = T 2 ' I (T 2');1 that is given by (2.6) Ib(m; x; h; v) = (m; h; x; v) : Next, the local description of the secondary operations are those of (1.1), while the primary operations are given by (m; x; h; v) +1 (m; x; k; w) = (m; x; h + k; v + w) (2.7) 1 (m; x; h; v) = (m; x; h; v) : It is now immediately veri ed from this information that ; Ib a 1 (m;x; h; v) +1 b 1 (m; y; h; w) = a 2 Ib(m; x; h; v) +2 b 2 Ib(m; y; h; w) : Finally, it also follows from the local form (2.6) of I that the xed points of I are the symmetric second order tangent vectors. This completes the proof of step 3 and theorem 1 as well.

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3. Torsion

Let M be a smooth manifold, let C 1 (M ) denote the ring of smooth real valued functions on M; and let X(M ) denote the C 1 (M )-module of vector elds on M: A Koszul connection or covariant derivative in the tangent bundle is any function r : X(M ) X(M ) ;! X(M ) that is C 1 (M )-linear in the rst factor and that satis es the product rule: ; rX (fY ) = (Xf )Y + f rX Y 8X; Y 2 X(M ); f 2 C 1 (M ) in the second factor. The torsion of a connection r is the TM -valued 2-tensor T that is de ned on pairs of vector elds X; Y by (3.1) T (X; Y ) = rX Y ; rY X ; [X; Y ] : The torsion is a tensor so that for each m 2 M; T de nes a skewsymmetric bilinear function T : Tm M Tm M ;! Tm M: However, in order to use (3.1) to compute T (h; k) for h; k 2 TmM; local vector elds extending h and k must be chosen. In this section, we use the canonical involution I from theorem 1 along with Dieudonne's de nition of a connection to give an equivalent formulation of the concept of torsion. From the outset, this description of torsion has two advantages over (3.1). Firstly, T (h; k) is computed directly without requiring local extensions of h and k: And secondly, the formulation shows that there is a natural symmetry in any torsion zero connection, see corollary 3.1 ahead. Also see [2, p. 106] for an alternate presentation of corollary 3.1. Connections. Following Dieudonne, cf. [4, 17.16.3], a connection in a vector bundle : E ;! M is a smooth function C : TM E ;! T E that is a vector bundle morphism in two ways: C ! TE C ! TE (C 1) TM E ;;;; (C 2) TM E ;;;; ?

? ? y E

pr2 ? y E

= ;;;; !

E

?

pr1 ? y

TM

? ? yT

= ;;;; ! TM:


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Both the primary and secondary vector bundle structures of T E are involved in the Dieudonne formulation of a connection. For each h 2 TM and x 2 E such that M (h) = (x); set Tx E h = Tx E \ (T E )h |the tangent vectors to E that are lifts of h and based at x: From (C1) and (C2), it is clear that for each (h; x) 2 TM E we have C (h; x) 2 Tx E h : Geometrically, C (h; x) is the \horizontal lift of h that is based at x". Let ;(E ) be the C 1 (M )-module of smooth sections of the vector bundle E : There is a natural correspondence between connections C and covariant derivatives r : X(M ) ;(E ) ;! ;(E ) that is described brie y as follows: for each h 2 Tm M and each section 2 ;(E ); de ne rh to be the element of E m such that (3.2) ((m); rh ) = T h ;1 C (h; (m)) where : E E ;! V T E is the canonical morphism to the vertical bundle, see lemma 3.2 ahead. For more details, see [4, 17.17] or [2, p. 87]. Connections in TM . Specializing to the case E = TM; let C : TM TM ;! T (TM ) be a connection. For each pair (h; x) 2 Tm M TmM; it follows from the basic properties of I ; see theorem 1, that both C (h; x) and I C (x; h) are lifts of h that are based at x; so that their primary dierence is a vertical tangent vector to TM that is based at x: De ne T C : TM TM ;! T (TM ) by (3.3) T C (h; x) = I C (x; h) ;1 C (h; x) : Then T C (h; x) 2 Tx (TM )0m so that by the canonical TM -isomorphism : TM TM ;! V; there is a unique vector T^(h; x) 2 Tm M such that (3.4) T C (h; x) = (x; T^(h; x)) : The following theorem relates T^ to the classical torsion.

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Theorem 2. Let

C : TM TM ;! T (TM ) be a connection in the tangent bundle of a smooth manifold and let T denote the torsion of C: Let T^ : TM TM ;! TM denote the smooth function de ned by equation (3:4). Then T^ = T; that is, T^ is the torsion of the covariant derivative r corresponding to the connection C: The proof of theorem 2 breaks naturally into two steps. Step 1. T^ is a skew-symmetric TM -valued tensor of type (0,2). Step 2. T^ = T: Our proof of step 1 appeals to some basic facts that are applications of proposition 1.1. We state these results here and defer their proofs to the end of the section. Lemma 3.1. Let : E ;! M be a vector bundle and let M : M ;! E TM be the zero section of TM: The vertical bundle V = ker(T) ;! E is dieomorphic to the pullback of the secondary vector bundle T : T E ;! TM by M so that V is naturally a vector bundle over M: Lemma 3.2. Let : E ;! M be a vector bundle. De ne : E E ;! V by (x; y) = [x + ty] |the tangent vector represented by the curve x + ty: Then is both an E -isomorphism and an M -isomorphism of vector bundles. Lemma 3.3 (Intertwining Laws). Let : E ;! M be a vector bundle. Let x; y 2 E m ; h; k 2 Tm M: Then for all vectors u 2 Tx E h ; v 2 Ty E h ; w 2 Tx E k and z 2 Ty E k and all scalars ; ; the following laws hold: 1) (u +2 v) +1 (w +2 z) = (u +1 w) +2 (v +1 z) 2) 1 (u +2 v) = ( 1 u) +2 ( 1 v) 3) 2 (u +1 w) = ( 2 u) +1 ( 2 w) 4) 1 ( 2 v) = 2 ( 1 v):


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Proof of step 1. By a straightforward exercise using lemmas 3.1 and 3.3 in conjunction with the morphism properties of I that are established in theorem 1, T C has the following properties: 1) T C (h; ): Tm M ;! Vm is a linear map, where m = M (h); 2) T C ( ; x): TM (x) M ;! Tx (TM ) is a linear map. Next, by lemma 3.2, the canonical dieomorphism : TM TM ;! V is both a TM -morphism of vector bundles and an M -morphism of vector bundles. This fact together with the two properties of T C listed just above imply routinely that T^ is a TM -valued tensor of type (0; 2) on M: For example, the additivity of T^ in the second variable goes as follows: let h; x; y 2 Tm M: Then (x + y; T^(h; x + y)) = T C (h; x + y) = T C (h; x) +2 T C (h; y) = (x; T^(h; x)) +2 (y; T^(h; y)) = (x + y; T^(h; x) + T^(h; y)) so that T^(h; x + y) = T^(h; x) + T^(h; y) : The other requirements for the bilinearity of T^ : Tm M Tm M ;! ^ we appeal to Tm M are argued similarly. For the skew-symmetry of T; the fact the xed points of the canonical involution I are the symmetric second order tangent vectors. More precisely, given any x 2 Tm M; the de nition of C implies that C (x; x) 2 Tx (TM )x ; that is, C (x; x) is a symmetric second order tangent vector. Thus, (x; T^(x; x)) = T C (x; x) = 0x so that by lemma 3.2, T^(x; x) = 0m : It follows immediately that T^ is skew-symmetric and therefore the proof of step 1 is complete. Proof of step 2. The functions T^ and T are both TM -valued tensors of type (0,2) so that to argue their equality, it suces to show that for all vector elds X; Y of M; (Y; T (X; Y )) = (Y; T^(X; Y )) :

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However, the validity of this last equation can be shown locally as follows. Given a local chart ' : U ;! E for M and the corresponding vector bundle charts T' : TU ;! TE and T 2 ' : T (TU ) ;! T (TE ); the local representatives of and C are the functions : TE TE ;! V and C : TE TE ;! T (TE ) de ned by

(m; x; v) = (m; x; 0; v) and

C (m; h; x) = (m; x; h; ;;(m) h x) : Note that ; is the local Christoel symbol of C; that is, it is a smooth map of E into the vector space L2 (E ; E ) of all bilinear mappings E E ;! E; cf. [4, 17.16.4]. In order to agree with traditional Christoel symbols, we use ;; rather than ;: Similarly, the local representative of the canonical involution is the function I : T (TE ) ;! T (TE ) given by I (m; x; h; v) = (m; h; x; v) : Let X and Y be vector elds with local representatives x; y 2 1 C (E; E ) so that X (m) = (m; x(m)) and Y (m) = (m; y(m)): Then (Y; T^(X; Y )) = T C (X; Y ) = I C (Y; X ) ;1 C (X; Y ) (3.5) = (m; y; x; ;; y x) ;1 (m; y; x; ;; x y) = (m; y; 0; ; x y ; ; y x) where the argument m of the functions ;; x and y has been suppressed. Next, by an application of lemma 3.2, (Y;T (X; Y )) = (Y ; X + X ; rX Y ; rY X ; [X; Y ]) (3.6) = (Y; rX Y ) ;2 (X; rY X ) +2 (X; ;[X; Y ]) : From (3.2) in the case of E = TM;

(Y; rX Y ) = (m; y; 0; Dy x + ; x y) (X; rY X ) = (m; x; 0; Dx y + ; y x) :


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On the other hand, [x; y] = Dy x ; Dx y so that after substituting into (3.6) and simplifying we get (Y; T (X; Y )) = (m; y; 0; ; x y ; ; y x) : By (3.5), (Y; T (X; Y )) = (Y; T^(X; Y )) and hence T^ = T by lemma 3.2. The proof of step 2 and theorem 2 are now complete. Corollary 3.1. A connection C in the tangent bundle TM is torsion-free, that is, T = 0; if and only if T C = 0: Thus, a torsion-free connection C is described geometrically as follows: C is torsion-free if and only if for each pair of tangent vectors h and x based at the same point m; C (h; x) = I C (x; h) : Equivalently, the canonical involution I preserves the horizontal distribution H = f C (h; x) j (h; x) 2 TM TM g :

Proof. By theorem 2,

T C (h; x) = (x; T (h; x)) so that because is a TM -isomorphism of vector bundles, cf. lemma 3.2, T C (h; x) = 0x () (x; T (h; x)) = 0x () T (h; x) = 0m : Proof of lemma 3:1. By general principles, the vertical bundle V and the pullback M T E are vector bundles over E and TM; respectively. As such, they have smooth manifold structures. Moreover, generalities about restrictions of smooth mappings to submanifolds can be used to show that the natural mapping : M T E ;! V de ned by (m; v) = v is a smooth dieomorphism. By way of the dieomorphism ; V is a vector bundle over M: The ber of V at a point m is (T E )0m : The vector space operations are given by the vector bundle operations +2 and 2 of T : T E ;! TM; see (1.3) and (1.4).

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Proof of lemma 3:2. Clearly, is an injective E -morphism of the vector bundles pr1 : E E ;! E and E : V ;! E : But both vector bundles have the same rank so that is an E -isomorphism of vector bundles. By lemma 3.1, the vertical bundle V of E is a vector bundle over M: It is straightforward to argue that is also an M -morphism from E E ;! M to V ;! M: For example, to argue the additivity of means that (x + w; y + z) = (x; y) +2 (w; z) : However, this is clear because the curves x + ty and w + tz represent (x; y) and (w; z); respectively, and they are also lifts of the constant curve m so that the pointwise sum of these curves x + w + t(y + z) represents (x; y) +2 (w; z); cf. (1.3). But evidently, this same curve also represents (x + w; y + z): Because the rst paragraph of the proof establishes that is a dieomorphism, it follows that is also an M isomorphism of vector bundles. Proof of the intertwining laws. The above identities involve vectors that all lie in a single vector bundle chart so that the identities are \local" properties of the manifold T E : So to prove the lemma, it suces to argue in the case where M is a vector space and E = M E: For example, identity 2 is veri ed by the calculation

1 ( (m; x; h; u) +2 (m; y; h; v) ) = 1 (m; x + y; h; u + v) = (m; x + y; h; (u + v)) = (m; x; h; u) +2 (m; y; h; v) = 1 (m; x; h; u) +2 1 (m; y; h; v) ; cf. (1.1) and (2.7). Identities 1, 3 and 4 follow similarly.

4. Second Order Dierential Equations, Sprays, and Linear Lifts

Second order differential equations. Let M be a smooth

manifold. A vector eld S on the tangent bundle to M is said to be symmetric provided it is also a smooth section of the secondary vector


983

bundle TM : T (TM ) ;! TM: Formally, a symmetric vector eld is a smooth function S : TM ;! T (TM ) that satis es two conditions: 1) TM S = idTM ; 2) TM S = idTM : Equivalently, a symmetric vector eld is one whose values are symmetric second order tangent vectors, as in section 2. Let m : J ;! M be a smooth curve in M: By the prolongation of the curve m(t) we mean the curve m1 : J ;! TM de ned by m1 (t) = Tm (t; 1) where (t; 1) is the standard unit tangent to R at t: Classically, m1 is called the tangent vector eld to the curve m(t): Similarly, the second prolongation of the curve m(t) is the smooth function m2 : J ;! T (TM ) de ned by m2 (t) = Tm1 (t; 1) = T 2 m (t; 1; 1; 0) : Note, a second order tangent vector v is symmetric if and only if v = m2 (0) for some curve m(t) in M: Let S : TM ;! T (TM ) be a symmetric vector eld. In terms of the natural charts introduced in x2, S is locally a function of the form S : TE ;! T (TE ) given by (4.1) S (m; x) = (m; x; x; v(m; x)) where (m; x) 7;! v(m; x) is a smooth function. The local form of the ordinary dierential equation de ned by S is then (4.2) (m(t); x(t); m0(t); x0(t)) = S (m(t); x(t)) : Equation (4.2) is equivalent to the second order equation m00 (t) = v(m(t); m0 (t)) so that a solution to (4.2) is necessarily the prolongation of a curve m(t) in the model E: In global terms, it follows that the rst order dierential equation determined by S; namely, x1 (t) = S (x(t)) ; is equivalent to the second order equation (4.3) m2 (t) = S (m1 (t)) : Geometrically, (4.3) characterizes symmetric vector elds. This is summarized in the following lemma whose proof is left to the reader.

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Lemma 4.1. Let S : TM

;! T (TM ) be an ordinary vector eld on

TM: Then S is symmetric i the integral curves of S are prolongations of curves in M: In light of the lemma, a second order autonomous dierential equation on the manifold M is, by de nition, the choice of a symmetric vector eld on TM: Sprays. A spray on a manifold M is a symmetric vector eld S : TM ;! T (TM ) that satis es the following quadratic homogeneity condition: for all tangent vectors x 2 TM and all scalars 2 R ; (4.4)

S (x) = 1 2 S (x) :

In terms of the local description (4.1) of S; the spray condition is equivalent to the smooth function (m; x) 7;! v(m; x) being quadratic in x in the usual sense, i.e.,

v(m; x) = 2v(m; x) : Given a connection C : TM TM ;! T (TM ); the function S : TM ;! T (TM ) de ned by (4.5)

S (x) = C (x; x)

is a spray so that, by nature, sprays exist in abundance. Geometrically, sprays are characterized by the following well-known fact. Lemma 4.2. Let S : TM ;! T (TM ) be a symmetric vector eld and let ' : dom(') R TM ;! TM denote the ow of S: Then S is a spray if and only if for all nonzero

'(t; x) = '(t; x) so that the unique solution to the initial value problem

2(t) = S ( 1(t))

1(0) = x


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is (t) = m(t); t 2 (; ; ); where m(t); t 2 (;; ) is the solution to m2 (t) = S (m1 (t)) with initial condition m1 (0) = x: In particular, the image or \trajectory" of the curve (t) coincides with that of m(t): A proof of the lemma can be found in [4, 18.4.2]. The curve m(t) = M ('(t; x)) and collectively all such curves are called the geodesics of the spray. Moreover, what is classically known as the exponential map of the spray is given by exp(tx) = M ('(t; x)); cf. [4, 18.4.3] and [7, chapter 4]. The following basic fact about sprays will be used in section 6. Lemma 4.3. For each spray S : TM ;! T (TM ); there is a unique torsion-free connection C : TM TM ;! T (TM ) having spray S: Proof. For any x; h 2 Tm M and for any triple of second order tangent vectors a 2 Tx+h (TM )x+h; b 2 Th (TM )h; and c 2 Tx (TM )x ; there is a unique v 2 Tx (TM )h such that

a = (b +2 v) +1 (c +2 I (v)) : From this it follows immediately that there is a unique function

C : TM TM ;! T (TM ) such that (4.6)

;

;

S (x + h) = S (h) +2 C (h; x) +1 S (x) +2 I (C (h; x)) :

Relative to a standard chart on T (TM ); S has the form of (4.1) while C (h; x) has the form (m; x; h; w(m; h; x)): Equation (4.6) is then equivalent to

w(m; h; x) = 21 [ v(m; h + x) ; v(m; h) ; v(m; x) ] : Evidently, w(m; h; x) is a smooth function of (m; h; x) so that C is a smooth function. The local function m ;! v(m; ) is a smooth map of the model vector space E into Q(E; E ); the vector space of smooth quadratic functions v : E ;! E: A standard result is that a quadratic function

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v 2 Q(E; E ) has the property v(x) = 12 D2 v(0) x x so that the quadratic condition is equivalent to the parallelogram-identity: ;

v(x + y) + v(x ; y) = 2 v(x) + v(y) : From the parallelogram-identity, it is argued that

B (x; y) = 21 [ v(x + y) ; v(x) ; v(y) ] de nes the unique symmetric bilinear function B : E E ;! E such that v(x) = 12 B (x; x); cf. [5, p. 245]. Let S2 (E ; E ) denote the vector space of all symmetric bilinear functions B : E E ;! E: From the previous paragraph, m 7;! w(m; ; ) is a smooth function E ;! S2 (E ; E ) so that C satis es the two bundle morphism conditions for a connection, see (C1) and (C2) in section 3. Also, because m 7;! w(m; ; ) takes its values in S2(E ; E ); C is a torsion-free connection. In the standard notation, the local Christoel symbol for C is given by (4.7)

;(m) h x = ; 12 [ v(m; h + x) ; v(m; h) ; v(m; x) ]

from which it is immediate that the spray of C is S:

The canonical lift to TM of a vector field on M . Let

: E ;! M be a vector bundle and let X be a vector eld on M: A lift of X to E is any vector eld Xe : E ;! T E that is -related to X; that is, T Xe = X : The following lemma is a basic fact that is needed just ahead. It is stated in a somewhat restricted context although the result is valid for general -related vector elds. The proof of the lemma is left to the reader. Lemma 4.4. Let : E ;! M be a vector bundle and let Xe be a lift to E of a vector eld X on M: Let ' : dom(') R M ;! M and e respectively. 'e : dom('e) R E ;! E denote the ows of X and X; Let J (m) and Je(x) denote the domains of the maximal integral curves of X and Xe that start at m and x; respectively. Then ('e(t; x)) = '(t; (x))


987

for t 2 Je(x): In particular, Je(x) J ((x)): Given a vector eld X on M; the canonical lift of X is the vector eld X : TM ;! T (TM ) de ned by (4.8) X = I TX where TX : TM ;! T (TM ) is the tangent mapping of X : M ;! TM and where I : T (TM ) ;! T (TM ) is the canonical involution introduced in x2. Note that by virtue of I being a morphism between the primary and secondary vector bundle structures on T (TM ); the canonical lift is a linear lift, as will be described later in this section, cf. [4, 18.6, exercise 2]. For any X; the canonical lift X is never a symmetric vector eld. While an integral curve x(t) of X is necessarily a lift of an integral curve m(t) of X; it need not be the prolongation of m(t): Indeed, this occurs i x(0) = m1 (0): The following lemma gives a characterization of the canonical lift that shows it to agree with the concept of a \complete lift", cf. [3, p. 330]. Lemma 4.5. Let X be a smooth vector eld on a manifold M and let X denote its canonical lift. Let ' : dom(') R M ;! M and ' : dom(') R TM ;! TM denote the ows of X and X; respectively. Then the ow of X is determined by the ow of X as follows: 8 x 2 TM and s 2 J (M (x)); (4.9) '(s; x) = T's x where 's = '(s; ): Next, given any smooth curve m : [a; b] ;! M; de ne a 1-parameter family of curves v : (;; ) [a; b] ;! M by (4.10) v(s; t) = '(s; m(t)) : Then for each t; the curve s 7;! T v (s; t; 0; 1) is an integral curve of X; that is, (4.11) T v (s; t; 0; 1) = '(s; m1(t)) where m1 (t) = Tm(t; 1) and where (s; t; 0; 1) 2 R 4 denotes the tangent vector (0; 1) based at (s; t) 2 R 2 : Finally, T's maps symmetric second order tangent vectors to symmetric second order tangent vectors. Moreover, this condition is another characterization of X among the lifts of X:

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Proof. Let x = [m(t)] and (s; t) = '(s; m(t)): By lemma 2.1,

Q[] = [t 7! [s 7! '(s; m(t))]] = [t 7! X (m(t))] = TX x : Next, (4.12)

X (x) = I TX x = I Q[] = [ s 7! [t 7! '(s; m(t))] ] = [s 7! T's x] :

If s 2 J (m); then 's is de ned on an open neighborhood of m so that T's x is de ned for each x 2 Tm M: Consider the curve (s) = T's x: Then

1(s) = [u 7! (s + u)] = [u 7! T's+u x] = [u 7! T'u (T's x)] = X (T's x) = X ( (s)) where the fourth line follows from (4.12) after replacing x by T's x: We conclude that (s) is the integral curve of X starting at x; i.e., (4.9) holds. Next we use (4.9) to argue (4.11) as follows:

T v(s; t; 0; 1) = [u 7! '(s; m(t + u))] = T's m1 (t) = '(s; m1(t)) : Let w 2 Tx (TM )x; i.e., a symmetric second order tangent vector. Choose a curve m(t) such that m1 (0) = x and m2 (0) = w: As stated earlier, a second order tangent vector is symmetric if and only if it is the


989

second prolongation at t = 0 of a curve in M: So let n(t) = 's (m(t)): Then n1 (t) = T's m1 (t) so that

T's w = T's m2 (0) = [t 7! '(s; m1(t))] = [t 7! T's m1 (t)] = n2(0) ; and thus T's w is symmetric. Finally, let Xe be any lift of X with ow ': e Then, for each x 2 TM; e s 2 J (x); and v 2 Tx (TM )x ; T 'es v is a second order tangent vector that is based at 'es (x) and is a lift of T's x: Therefore, T 'es v is symmetric i T's x = 'es (x): By (4.9), Xe = X: Remark. Loosely, the geometric signi cance of (4.11) is as follows.

Starting with a curve m(t) in M; we can use the ow of X to create a new curve t 7;! '(s; m(t)) that has t 7;! Tv (s; t; 0; 1) as its prolongation. Alternatively, we can prolong the curve m(t) and then use the ow of X to produce a curve t 7;! '(s; m1(t)): The lemma states that these two methods yield the same curve in TM: Linear lifts. Let : E ;! M be a vector bundle and let X be a vector eld on M: A lift Xe of X is said to be linear if, in addition, the e X ) is a morphism from the vector bundle : E ;! M to the pair (X; bundle T : T E ;! TM: In the notation of x1, this means that for all m 2 M; x; y 2 E m ; and scalars a; b 2 R ;

Xe (ax + by) = a 2 Xe (x) +2 b 2 Xe (y) : There are two important examples of the linear lift concept, namely, the canonical lift of a vector eld to its tangent bundle, as described earlier in this section, and the horizontal lift (4.13) X C (x) = C (X ((x)); x) of a vector eld X on M as determined by a connection C : TM E ;! T E : It follows immediately from the de nition of a connection, see x3, that X C is a linear lift of X:

990


We say that the ow of Xe is berwise linear i for all m 2 M; x; y 2 E m ; a; b 2 R ; and t 2 Je(x) \ Je(y) \ Je(ax + by); 'e(t; ax + by) = a 'e(t; x) + b 'e(t; y) : Lemma 4.6. A lift Xe : E ;! T E of a vector eld X is linear if and only if its ow 'e is berwise linear. Proof. Suppose rst that Xe is a linear lift of X: For x; y 2 E m and a; b 2 R ; consider the curve (t) = a'e(t; x)+b'e(t; y) for t 2 Je(x)\Je(y): By lemma 4.4, the two curves 'e(t; x) and 'e(t; y) project to '(t; m) so that by the intrinsic description of the secondary vector bundle operations on T E ; cf. (1.3) and (1.4), it follows that

0(t) = a 2 Xe ('e(t; x)) +2 b 2 Xe ('e(t; y)) ; = Xe a'e(t; x) + b'e(t; y) : Thus (t) is an integral curve of Xe beginning at ax + by: By the uniqueness of integral curves, 'e(t; ax + by) = a'e(t; x) + b'e(t; y) for t 2 Je(x) \ Je(y): In particular, this shows Je(x) \ Je(y) Je(ax + by): Conversely, suppose that 'e is berwise linear. Then 'e(t; ax+by) and a'e(t; x) + b'e(t; y) are the same curve for t 2 Je(x) \ Je(y) \ Je(ax + by): Hence they represent the same tangent vector at t = 0 which, by the de nition of a ow and by (1.3) and (1.4) again, gives Xe (ax + by) = a 2 Xe (x) +2 b 2 Xe (y) : Maintaining the notation of lemma 4.6, let m 2 M and suppose that fx1 ; : : :; xng is a basis for E m : Let x = Pi ai xi 2 E m : Then by lemma 4.6 X 'e(t; x) = ai 'e(t; xi) i

for all t 2 \i Je(xi ) so that \i Je(xi ) Je(x): But \i Je(xi ) is an open interval about 0 and thus the following statement holds: (4.14) 8 m 2 M; 9 > 0 s.t. 8x 2 E m ; (;; ) Je(x) : With (4.14) in hand, we next prove the following fact.


991

Lemma 4.7 (Domain of de nition for the ow of a linear lift). For all m 2 E and all x 2 E m ; Je(x) = J (m):

Proof. Fix m 2 M and let [a; b] J (m) be a closed interval containing 0 in its interior. We will argue that [a; b] Je(x) for each x 2 E m : Set m(t) = '(t; m); t 2 J (m): For each t0 2 [a; b]; choose 0 > 0 so that (4.14) holds at m(t0 ): Let > 0 be a Lebesgue number for the open cover f(t0 ; 0 ; t0 + 0 ) j t0 2 [a; b]g of [a; b]: We claim that the following holds:

(4.15)

8 u 2 [a; b]; 8 y 2 E m(u) ;

; 3 ; 3 Je(y) :

So let u 2 [a; b]: Then by the de ning property of a Lebesgue number u ; 3 ; u + 3 (t0 ; 0 ; t0 + 0 ) for some t0 2 [a; b] so that (4.16)

u ; t0 2 (;0 ; 0 ) Je(x)

for all x 2 E m(t0 ) : The rst relevant point en route to proving (4.15) is that (4.17)

'e(u ; t0 ; ): E m(t0 ) ;! E m(u) is a linear isomorphism.

Note, if the function in (4.17) is meaningful, then it is linear by lemma 4.6. Also by (4.16), 'e(u ; t0 ; x) is de ned for all x 2 E m(t0 ) : Next the range of (4.17) is correct by lemma 4.4 and the concatenation property of the ow of X; cf. [4, 18.2.3.2]. On the other hand, by [4, 18.2.3.1], (4.18)

Je(x) ; (u ; t0 ) = Je('e(u ; t0 ; x)) ;

so that (t0 ; u) 2 Je('e(u ; t0 ; x)): Concatenation of ows now applies to give 'e(t0 ; u; 'e(u ; t0 ; x)) = x for each x 2 E m(t0 ) : Therefore, the linear function of (4.17) is injective, hence an isomorphism. Next, to complete the proof of (4.15), let y 2 E m(u) and let t 2 ; 3 ; 3 : By (4.17), y = 'e(u ; t0 ; x) for a unique x 2 E m(t0 ) : Then

t + u 2 u ; 3 ; u + 3 (t0 ; 0 ; t0 + 0 )

992


so that t + u ; t0 2 (;0 ; 0 ) Je(x): By (4.18) we have t 2 Je('e(u ; t0 ; x)) = Je(y): P Finally, each t 2 [a; b] is expressible as a nite sum t = pi=1 ti where all the ti have the same sign and jti j 3 : Clearly, the partial sums satisfy t1 + + ti 2 [a; b]: Then by (4.15), concatenation of ows, and induction we have

ti 2 Je('e(t1 + + ti;1 ; x)) for all i: Note that the rst step in the induction uses 0 2 [a; b]: Taking i = p; we get

tp 2 Je('e(t1 + + tp;1 ; x)) = Je(x) ; (t1 + + tp;1 ) P

so that t = i ti 2 Je(x):

Remark. The following example shows that the conclusion of lemma 4.7 can fail for nonlinear lifts. Let E = R R ; X (m) = (m; 1); and Xe (m; x) = (m; x; 1; x2): Then J (m) = R while

Je(m; x) =

8 > < > :

(;1; x1 )

R

( x1 ; 1)

x>0 x=0 x < 0:

Parallel translation. Let : E ;! M be a vector bundle and let C : TM E ;! T E be a connection in E : Let N M be an embedded curve without reference to a parametrization. The well-known concept of parallel translation of tangent vectors along N (relative to C ) states that for any pair of points p; q 2 N; there is a unique linear isomorphism p;q : E p ;! E q that is referred to as parallel translation along N from p to q: It depends only on N and the connection C; and is de ned as follows: let X be any vector eld of M that is tangent to and nowhere vanishing along N: Let X C denote its horizontal lift, see (4.13). Let ' : dom(')

Second order tangent vectors R

993

M ;! M and 'C : dom('C ) R E ;! E denote the ows of

X and X C : Then

p;q (x) = 'C (t; x) where t is de ned uniquely by '(t; p) = q: Lemmas 4.4, 4.6 and 4.7 can be used to show that p;q is a linear isomorphism. On the other hand, it follows from the basic theory of ODE's that p;q is also independent of the vector eld X: Veri cation of this point is left to the reader. Additionally, it follows from the concatenation property of a ow that (4.19) p;r = q;r p;q for all p; q; r 2 N: Finally, x a base point m0 2 N and let m(t) = '(t; m0 ); the parametrization of N induced by X and m0 : Then it is customary to write t : E m(u) ;! E m(t+u) in place of p;q leaving the starting point p = m(u) unspeci ed. In view of (4.19), the notational advantage is that, in context, the equation s t = s+t holds.

5. The Spray of a Riemannian Metric

Let (M; g) be a Riemannian manifold. There is a natural spray S : TM ;! T (TM ) associated to the Riemannian metric. Three common approaches to producing S are as follows. 1) Classically, the covariant derivative r : X(M ) X(M ) ;! X(M ); referred to as the Levi-Civita connection of the metric, is the unique covariant derivative that has torsion zero and is compatible with the metric. The proof of its existence does not require knowing its spray. By way of equation (3.2), r determines a unique connection C : TM TM ;! T (TM ) in the sense of x3. With C in hand, S is then de ned by (4.5). 2) An algebraic approach to S is to use the natural Poisson structure on the manifold TM: The spray then arises as the Hamiltonian vector eld of the quadratic function associated to the Riemannian metric, cf. (5.5) below and [2, p. 113]. 3) In variational calculus, the geodesics of a Riemannian metric are de ned as the critical points of the energy functional associated to the metric. The spray S emerges naturally out of this problem.

994


Of the three approaches, the third is the most geometric while the spray appears somewhat serendipitously in the second approach. Our objective in this section is to reconcile the second and third approaches by using canonical lifts of vector elds, as presented in x4. Via such lifts, one is lead naturally to a description of the energy functional, or more precisely its rst variation, that directly involves both the canonical symplectic form on T M and the dierential of the quadratic function of the given Riemannian metric. The canonical symplectic form and its spray. Let : T M ;! M be the cotangent bundle. The canonical 1-form on T M is the smooth function : T (T M ) ;! R de ned by (5.1) (v) = hT M (v); T vi |the pairing of the base point of v with the projection of v; cf. [4, 16.20.6]. Note that in writing (v); the base point x = T M (v) is tacitly understood. This convention departs from the standard notation of (x)(v) which emphasizes as a smooth section of T (T M ): Similarly, the Riemannian metric is expressed as a function g : TM TM ;! R that is berwise an inner product. Next, g induces an M -isomorphism of vector bundles g^ : TM ;! T M by letting g^(x) = g(x; ): In particular, (5.2) (^g)(v) = g(TM (v); TM v) (8v 2 T (TM )) : The canonical symplectic form on T M is by de nition ! = ;d : The canonical symplectic structure !g on the tangent bundle of (M; g) is de ned by (5.3) !g = g^! = ;d(^g) : The quadratic function q : TM ;! R of the Riemannian metric g is given by (5.4) q(x) = 21 g(x; x) : Because !g is nondegenerate, there is a unique vector eld S : TM ;! T (TM ) such that (5.5) (S ) !g = dq :


995

Lemma 5.1. The vector eld S is a spray.

Proof. The local form for a metric on M is a function of the form g(m; x; y) = g(m)(x; y) where m 7;! g(m) is a smooth mapping of the model E into S2+ (E; R); the space of inner products on E: Next, in a natural vector bundle chart, the a priori local form of S is S (m; x) = (m; x; h(m; x); v(m; x)) : The local form of (5.5) is then (Dg(m) h1 )(x; h(m; x)) + g(m)(v1; h(m; x)) ; (Dg(m) h(m; x))(x; h1) ; g(m)(v(m; x); h1) (5.6) = 21 (Dg(m) h1 )(x; x) + g(m)(x; v1) for all (m; x; h1; v1) 2 T (TE ): Taking h1 = 0 forces h(m; x) = x and therefore S is a symmetric vector eld. Substituting h(m; x) = x into (5.6) and solving for v(m; x) yields ; v(m; x) = pr2 g^;1 m ; 12 (Dg(m) ( ))(x; x) ; (Dg(m) x)(x; ) : Clearly then v(m; x) = 2 v(m; x) so that S must also be a spray.

The symplectic form !g induces a natural Poisson structure on TM: Given a pair of smooth scalar valued functions F; H on TM; their Poisson bracket is de ned by fF; H g = !g (XF ; XH ) where a function F determines the vector eld XF by the equation dF = (XF )!g ; cf. [8]. In the language of this subject, S is the Hamiltonian vector eld of q: An elementary relation between the function q and its Hamiltonian vector eld S is that q is constant along any integral curve of S: This fact can be argued directly as follows: let ' : dom(') R TM ;! TM denote the ow of S: Then for any x 2 TM; ; dq S ('(t; x)) = !g S ('(t; x)) ; S ('(t; x)) =0 so that the curve t 7;! q('(t; x)) is constant. On the other hand, because S is a spray, the integral curves of S are the prolongations of the geodesics of S; cf. lemma 4.1. Thus, the geodesics of S have constant speed, i.e., q is constant along m1 (t) for any geodesic m(t):

996


The energy of a smooth curve. A variation of a smooth curve

m : [a; b] ;! M is a smooth function

v : (;; ) [a; b] ;! M such that v(0; t) = m(t): The variation is said to keep the endpoints of m(t) xed if v(s; a) = m(a) and v(s; b) = m(b): Next, two variations v; w of the curve m(t) are said to agree to rst order along m(t) if and only if T v (0; t; 1; 0) = T w (0; t; 1; 0) : If M = E is a real vector space, then agreement of v and w to rst order along m(t) is expressed as follows:

@ v (0; t) = @ w (0; t) : @s @s By (4.10), a vector eld X determines a variation of a curve m(t): We refer to this type of variation as a ow variation because the trace curve m(t) is being moved by the ow of X: If m(t) is an embedded curve, then to rst order along m(t); any variation of w of m(t) is a

ow variation. The point is that because m(t) is embedded in M; there is a vector eld X on M such that

X (m(t)) = T w (0; t; 1; 0) : Then v(s; t) = '(s; m(t)); where ' denotes the ow of X; agrees with w to rst order along m(t): The energy of a smooth curve m : [a; b] ;! M relative to a Riemannian metric g is de ned by

E (m) =

b

Z

a

g(m1(t); m1(t)) dt :

The following elementary fact will be used in the discussion ahead. Its proof is left to the reader.


997

Lemma 5.2. Let (M; g ) be a smooth Riemannian manifold. Let

m : [a; b] ;! M be a smooth curve. If v and w are smooth variations of m(t) that agree to rst order along m(t); then d E (v(s; )) = d E (w(s; )) : ds s=0 ds s=0 That is, if two variations agree to rst order along m(t); then the energies of the variations also agree to rst order. Classically, a geodesic of the metric is de ned to be a critical point of E ; that is, it is a curve m(t) such that for any variation v(s; t) keeping the endpoints of m(t) xed, d ds s=0 E (v(s; )) = 0 : To test whether an embedded curve is a critical point of E ; it suces by the previous lemma and the above discussion to consider dsd s=0 E (v(s; )) for ow variations. With this in mind, we have the following result. Lemma 5.3. Let (M; g ) be a smooth Riemannian manifold. Let m : [a; b] ;! M be a smooth embedded curve and let X be a vector eld on M: Next, let v(s; t) denote the ow variation of m(t) described by (4:10). Then (5.7)

Z

d ds s=0 E (v(s; )) = a

b

LX (^g) m2 (t) dt :

where X is the canonical lift of X; is the canonical 1-form on T M; cf. (5:1), and L denotes Lie derivative. Moreover, (5.8) LX (^g) m2 (t) = 2 dq X (m1 (t)) where q is the quadratic function of g; cf. (5:4). Consequently, if X vanishes at m(a) and m(b); then d E (v(s; )) ds s=0 Z b (5.9) ; =2 !g m2 (t);X (m1 (t)) ; dq X (m1 (t)) dt a

where !g is the canonical symplectic 2-form on TM; cf. (5:3).

998


Proof. Let v 2 Tx (TM ): By the notational convention set up in conjunction with (5.1), LX (^g) v means the value of the 1-form LX (^g) at the point x applied to the second order tangent vector v: From the de nition of the Lie derivative and (5.2), LX (^g) m2 (t) = @s@ s=0 's (^g) m2 (t) ; = @s@ s=0 g TM (T's m2 (t)) ; TM T's m2 (t) :

Next by (4.11) and the nal statement of lemma 4.5, T's m2 (t) is a symmetric second order tangent vector that is based at T v (s; t; 0; 1): Thus ; (5.10) LX (^g) m2 (t) = @s@ s=0 g T v (s; t; 0; 1) ; T v (s; t; 0; 1) so that because

E (v(s; )) =

Z

a

b

;

g T v (s; t; 0; 1) ; T v (s; t; 0; 1) dt ;

equation (5.7) follows. Next we prove equation (5.8). By (4.11), the curve s 7;! T v (s; t; 0; 1) represents the second order tangent vector X (m1 (t)) so that by the de nition of the exterior derivative of a function, it follows that ; 2dq X (m1 (t)) = 2 @s@ s=0 q T v (s; t; 0; 1) ; = @s@ s=0 g T v (s; t; 0; 1); T v (s; t; 0; 1) : By (5.10), equation (5.8) follows immediately. Finally, we prove (5.9). By Cartan's form of the Lie derivative, ; LX (^g) = (X )d(^g) + d (X )(^g) : Because g is berwise bilinear and X is assumed to vanish at the endpoints of m(t); it follows that b

Z

a

b

; d (X )(^g) m2 (t) dt = g m1 (t); X (m(t)) ;

=0

a


999

and thus b

Z

(5.11)

a

LX

(^g) m2 (t) dt = =

Z

a

Z

b b

a

(X ) d(^g) m2 (t) dt ;

!g m2 (t);X (m1 (t)) dt :

Therefore, by (5.7), (5.8), and (5.11) Z b d E (v(s; )) = LX (^g) m2 (t) dt ds s=0 a Z b = 2LX (^g) m2 (t) ; LX (^g) m2 (t) dt a Z b ;

=2

a

!g m2 (t);X (m1 (t)) ; dq X (m1 (t)) dt :

and the proof of the lemma is complete.

Lemma 5.4 (Energy Principle). Let m : [a; b] ;! M be an embed-

ded curve. Then m(t) is a critical point of the energy if and only if for each [c; d] [a; b]; the restricted curve m : [c; d] ;! M is a critical point of the energy. Proof. The reverse direction is obvious. We argue the forward direction as follows: let [c; d] [a; b] with a < c < d < b: Let Y be a vector eld that vanishes at m(c) and m(d): Let denote the ow of Y and let w : (;; ) [c; d] ;! M be de ned by w(s; t) = (s; m(t)): To argue that m : [c; d] ;! M is a critical point of the energy, it suces, by lemma 5.2, to show that dsd s=0 E (w(s; )) = 0 for all ow variations w of m(t): Given > 0; let f : M ;! R be a smooth function such that 0 f (m) 1 for all m 2 M and

f (m(t)) =

1 0

t 2 [c; d] t 2 [a; b] n (c ; ; d + ):

De ne X = fY and for (s; t) 2 (;; ) [a; b]; let v(s; t) = '(s; m(t)) where ' denotes the ow of X: Independently of m : [a; b] ;! M being

1000


a critical point of the energy, we claim that there exists K > 0; such that for all > 0 and all f as above, d d E (w(s; )) < K p : (5.12) ds s=0 E (v(s; )) ; ds s=0

First of all, X (m(t)) = Y (m(t)) for t 2 [c; d] implies that X (m1 (t)) = Y (m1 (t)) for t 2 [c; d]: Next, for t 2 [a; b] n (c ; ; d + ); (X m)(t) = (M m)(t) where M denotes the zero vector eld. Dierentiate this equation to get TX m1 (t) = TM m1 (t): Next, TM m1 (t) is the zero vector of T (TM )m1(t) ; see the sentence following (1.4), and I is a morphism of vector bundles, see theorem 2, so that X (m1 (t)) = I (TX m1 (t)) = 0m1(t) |the zero vector of Tm1(t) (TM ): Hence ; !g m2 (t);X (m1 (t)) ; dq X (m1 (t)) = 0 for all t 2 [a; b] n (c ; ; d + ): Thus, by two applications of (5.9) and the additivity of the integral Z b ; 2 1 d E (v(s; )) = 2 ( m ( t )) ! m ( t ) ;X g ds s=0 a d = ds s=0 E (w(s; )) Z c ;

+2

+2 so that

Z

c; d+ d

; dq X (m1 (t)) dt

!g m2 (t);X (m1 (t)) ; dq X (m1 (t)) dt

;

!g m2 (t);X (m1 (t)) ; dq X (m1 (t)) dt

d d ds s=0 E (v(s; )) ; ds s=0 E (w(s; )) Z c ;

2

+2

c ;

!g m2 (t);X (m1 (t)) ; dq X (m1 (t))

Z d+ ; 2 1 ! m ( t ) ;X ( m ( t )) g d

dt

; dq X (m1 (t))

dt

The following fact will be used just ahead in the discussion.

:


Lemma 5.5. For all x

metric v;

1001

2 TM and all v 2 Tx (TM )x; i.e., all sym

!g (v;X (x)) ; dq X (x) = f (M (x)) !g (v; Y (x)) ; dq Y (x) : by

Proof. In a natural vector bundle chart, the local form of X is given ;

X (m; x) = m; x; X(m); DX(m) x ; = m; x; f (m)Y(m); (Df (m) x)Y(m) + f (m)DY(m) x ; where the local forms of X and Y are X (m) = (m; X(m)) and Y (m) = (m; Y(m)): The local descriptions of !g and dq are such that 8 (m; x; x; v1); (m; x; h2; v2) 2 T (TE ); ;

!g (m; x) (x; v1) ; (h2 ; v2) ; dq(m; x) (h2 ; v2) = 21 (Dg(m) h2 )(x; x) ; (Dg(m) x)(x; h2) ; g(m)(v1; h2 ) : Taking (h2 ; v2) = (f (m)Y(m); (Df (m) x)Y(m) + f (m)DY(m) x) ; we get ;

!g (m; x) (x; v1) ; (h2; v2 ) ; dq(m; x) (h2 ; v2 ) = f (m) [ 12 (Dg(m) Y(m))(x; x) ; (Dg(m) x)(x; Y(m)) ; g(m)(v1; Y(m)) ] :

The statement of the lemma is now immediate.

Returning to the main argument of lemma 5.4, it follows jointly from lemma 5.5, the Cauchy-Schwarz inequality, and f being bounded by 1 on M; that Z c !g (m2(t);X (m1 (t))) c;

; dq X (m1 (t))

dt

k f m k[2c;;c] k !g (m2 ; Y (m1 )) ; dq Y (m1 ) k[2c;;c] p k !g (m2 ; Y (m1 )) ; dq Y (m1) k[2a;b]

1002


where, generically, the notation k k[2p;q] means the L2-norm over the interval [p; q]: There is a similar inequality for the integral d+

Z

d

;

!g m2 (t);X (m1 (t)) ; dq X (m1 (t)) dt :

Taking K = 4 k !g (m2 ; Y (m1 )) ; dq Y (m1 ) k[2a;b]; (5.12) follows directly. By a similar argument, (5.12) holds if c = a or d = b: Finally, suppose that m : [a; b] ;! M is a critical point of the energy. Then by (5.12) d p ds s=0 E (w(s; )) < K for arbitrary > 0 so that dsd s=0 E (w(s; )) = 0 and the proof is complete. Proposition 5.1. Let (M; g ) be a smooth Riemannian manifold

and let m : [a; b] ;! M be a smooth embedded curve. Then m(t) is a critical point of the energy E if and only if (5.13)

!g (m2 (t); v) = dq v

(8t 2 [a; b]; 8v 2 Tm1 (t) (TM ))

where !g and q are de ned in (5:3) and (5:4), respectively. Proof. Suppose rst that (5.13) holds. Let X be a vector eld that vanishes at m(a) and m(b): Then (5.9) vanishes so that the embedded curve m(t) is a critical point of the energy. Conversely, suppose that the embedded curve m(t) is a critical point of the energy. Then by (5.9) b

Z

a

;

!g m2 (t);X (m1 (t)) ; dq X (m1 (t)) dt = 0

for all vector elds X vanishing at the endpoints of m(t): By the energy principle, m(t) is a critical point of the energy along any [c; d] [a; b] where [c; d] can be taken small enough so that, for any vector eld Y vanishing at m(c) and m(d); the computation d

Z

c

;

!g m2 (t); Y (m1 (t)) ; dq Y (m1(t)) dt = 0


1003

can be done in a single local chart. In a natural chart, the above integrand is of the form ; !g m2 (t); Y (m1 (t)) ; dq Y (m1 (t)) = 12 (Dg(m) Y(m))(m0; m0 ) ; (Dg(m) m0 )(m0; Y(m)) ; g(m)(m00 ; Y(m)) = h(t) ; Y(m(t))i where Y is the notation for the local description of a vector eld Y; where h ; i : E E ;! R denotes the dual pairing, and where (t) = 12 (Dg(m(t)) ( ))(m0 (t); m0(t)) ; (Dg(m(t)) m0 (t))(m0(t); ( )) ; g(m(t))(m00 (t); ( )) : The map gives a smooth curve into E ; the dual vector space to the local model E: On the other hand, because m : [a; b] ;! M is an embedding, so too is the local description of m: It follows that every smooth curve x : [c; d] ;! E is of the form x(t) = Y(m(t)) for some vector eld Y on E: The proof of the following elementary fact is left to the reader. Lemma 5.6. Let E be a nite dimensional real vector space. Let : [c; d] ;! E be a smooth curve such that for all smooth curves x : [c; d] ;! E vanishing at c and d d

Z

c

h(t); x(t)i dt = 0

where h ; i : E E ;! R denotes the dual pairing. Then (t) = 0 for all t 2 [c; d]: As a consequence of the lemma, it follows immediately that for all t 2 [c; d] and all v 2 Tm1 (t) (TM ); !g (m2 (t); v) = dq v : Finally, because of the arbitrary nature of [c; d] [a; b]; equation (5.13) holds for all t in the interval [a; b]: Formally, a critical point of the energy functional and a geodesic of the spray S are distinct concepts. We complete the section by using proposition 5.1 to prove the equivalence of the two concepts.

1004


Proposition 5.2. A nonconstant smooth curve m : J

;! M is a

geodesic of the spray S if and only if m(t) is a critical point of the energy functional on any subinterval [a; b] J on which m(t) is an embedding. Proof. We prove the reverse direction rst. Let J1 be a maximal open subinterval of J on which m1 (t) 6= 0m(t) : Then J1 is a union of closed intervals [a; b] such that m : [a; b] ;! M is an embedding. By assumption, m : [a; b] ;! M is a critical point of the energy so that proposition 5.1 applies. Therefore, for each t 2 J1 and each v 2 Tm1 (t) (TM ) !g (m2 (t); v) = dq v : By the nondegeneracy of !g ; it follows immediately that

m2 (t) = S (m1 (t)) for all t 2 J1 so that m : J1 ;! M is a geodesic of S: But as observed earlier in this section, m : J 1 ;! M must then have constant speed. If J1 6= J then at least one endpoint of J1 ; say c; belongs to J by connectivity. Since the speed function is continuous on J and constant on J1 ; it follows that m1 (c) 6= 0m(c): This contradicts the nature of J1 so that J = J1 and m : J ;! M is a geodesic of S: Conversely, let m : J ;! M be a geodesic of the spray S: Then by de nition m2 (t) = S (m1 (t)) and thus equation (5.13) holds for all t 2 J: Proposition 5.1 then implies that m(t) is a critical point of E on any subinterval [a; b] J on which m(t) is an embedding.

6. The Fundamental Theorem of Riemannian Geometry Let (M; g) be a Riemannian manifold and let C : TM TM ;! T (TM ) be a connection in the tangent bundle of M: We say that C is compatible with g i for any embedded curve N M and any p; q 2 N; the parallel translation p;q : Tp M ;! Tq M is an isometry, see the end of x4. The Fundamental Theorem of Riemannian Geometry


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states that there is a unique connection in the tangent bundle of a Riemannian manifold that has zero torsion and is compatible with the metric. Classically, this connection is called the Levi-Civita or Riemannian connection of (M; g): In this section, we give a proof of this foundational result. By way of the canonical isomorphism : T (TM ) 2 T (TM ) ;! T (TM TM ) in lemma 1.2, the tangent map of g : TM TM ;! R gives rise to the berwise bilinear function f : T (TM ) T (TM ) ;! R Tg 2

de ned by f (v; w ) = (pr Tg )(v; w ) = d Tg 2 dt t=0 g (x(t); y (t)) where v = [x(t)]; w = [y(t)] and M (x(t)) = M (y(t)): Note, we use 2 to emphasize that this is the Whitney sum of the secondary structure T : T (TM ) ;! TM with itself. Lemma 6.1. A connection C is compatible with the metric g if and only if ;

f C (h; x); C (h; y ) = 0 (6.1) Tg

( 8m 2 M; 8h; x; y 2 Tm M ) :

Proof. Let h = [m(t)] and let t : Tm(0) M ;! Tm(t) M denote parallel translation to time t along the curve m(t): Then for any x; y 2 Tm(0) M; C (h; x) = [t (x)] and C (h; y) = [t (y)] so that ;

f C (h; x); C (h; y ) = d Tg dt t=0 g (t (x); t (y )) : Assuming that t is an isometry, g(t (x); t (y)) = g(x; y) so that (6.2) vanishes. Conversely, suppose that (6.1) holds. Let N be an embedded curve in M and let p 2 N: Choose a parameterization m : J ;! M of N with m(0) = p: We will argue that the parallel translations t : Tp M ;! Tm(t) M are isometries by proving that for x; y 2 Tp M; the curve t 7;!

(6.2)

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g(t (x); t (y)) is constant. Let [m(t)]t denote the element of Tm(t) M represented by the curve s 7;! m(t + s): Then ; f C ([m(t)]t ; (x)); C ([m(t)]t; (y )) 0 = Tg t t = dsd s=0 g(s t (x); s t (y)) = dsd s=0 g(s+t (x); s+t (y)) = dtd t g(t (x); t (y)) so that the curve t 7;! g(t (x); t (y)) has derivative zero everywhere. Let C : TM TM ;! T (TM ) be the torsion-free connection determined by the spray S of the Riemannian metric g; cf. lemma 4.3 and section 5. From the local form of the spray S; we will verify that (6.1) holds so that C is compatible with g and thus a Riemannian connecf ((m; x; h; v ); (m; y ; h; w )) is tion exists. Using our standard charts, Tg the derivative at time 0 of the curve t 7;! g(m + th)(x + tv; y + tw) so that ; f (m;x; h; v ); (m; y ; h; w ) Tg (6.3) = (Dg(m) h)(x; y) + g(m)(v; y) + g(m)(x; w) : The local form of C is C (m; h; x) = (m; x; h; ;;(m) h x) and so ; f C (m; h; x); C (m; h; y ) Tg ; (6.4) = (Dg(m) h)(x; y) ; g(m) ;(m) h x; y ; ; g(m) x; ;(m) h y : On the other hand, from lemma 5.1 the local form of S is such that v(m; x) satis es the equation ; (6.5) g(m) v(m; x); y = 21 (Dg(m) y)(x; x) ; (Dg(m) x)(x; y) for all y: The local Christoel symbol of C is given by (4.7) so that by (6.5) and the symmetric bilinearity of g(m); ; ;g(m) ;(m) h x ; y = 21 (Dg(m) y)(h; x) ; (Dg(m) h)(x; y) (6.6) ; (Dg(m) x)(h; y)


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holds for all h; x; y: Interchanging x and y; the analogous expression is obtained for ;g(m)(;(m) h y; x): Substituting the two expressions into (6.4) and simplifying in the obvious manner, we get ;

f C (m; h; x); C (m; h; y ) = 0 : Tg

Last of all, we address the uniqueness of the Riemannian connection, to wit, let C be any connection that has torsion zero and is g-compatible. We will argue that the spray of C equals the spray of g so that by lemma 4.3, the uniqueness of the Riemannian connection follows immediately. First of all, C is g-compatible so that by lemma 6.1 and (6.4) ;

(6.7)

0 = (Dg(m) h)(x; y) ; g(m) ;(m) h x ; y ; ; g(m) x ; ;(m) h y :

Cyclically permute h; x; y in (6.7). This leads to three equations that by adding and subtracting appropriately and also by using the symmetry of ;(m) in its two arguments, yields equation (6.6) so that the connection of the previous paragraph and C have the same local Christoel symbol or equivalently the same spray.

References [1] A. L. Besse, Manifolds all of whose Geodesics are Closed, Springer-Verlag, New York, 1978. [2] E. Binz, J. Sniatycki, and H. Fischer, Geometry of Classical Fields, Elsevier Science Publishers B.V., Amsterdam, 1988. [3] M. Crampin and F. A. E. Pirani, Applicable Dierential Geometry, Cambridge University Press, Cambridge, 1986. [4] J. Dieudonne, Treatise on Analysis I, III, IV, Academic Press, Inc., New York, 1969, 1972, 1974. [5] W. H. Greub, Linear Algebra, Springer-Verlag, New York, 1967. [6] S. Kobayashi, Transformation Groups in Dierential Geometry, Springer-Verlag, New York, 1995. [7] S. Lang, Dierential Manifolds, Springer-Verlag, New York, 1985. [8] D. McDu and D. Salamon, Introduction to Symplectic Topology, Oxford University Press, 1995.

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Department of Mathematics Idaho State University Pocatello, ID 83209-8085 USA E-mail : [email protected] [email protected]