Geometric structures on loop and path spaces

arXiv:0801.3545v2 [math.SG] 15 Sep 2010 GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES VICENTE MUÑOZ AND FRANCISCO PRESAS Abstract. Is is known that the loop space associated to a Riemannian manifold admits a quasi-symplectic structure. This article shows that this structure is not likely to recover the underlying Riemannian metric by proving a result that is a strong indication of the “almost” independence of the quasi-symplectic structure with respect to the metric. Finally conditions to have contact structures on these spaces are studied. 1. Introduction The loop space L(M ) of a manifold M comes equipped with a natural section of its tangent bundle defined as α : L(M ) → T L(M ) γ → γ′. Whenever we fix a Riemannian metric g on M we can define an associated metric on the space of loops as Z 1 g(X(t), Y (t))dt, (gL )γ (X, Y ) = 0 1 ∗ Γ(S , γ T M ) where X, Y ∈ Tγ L(M ) ≃ are two tangent vectors. This metric gives us an isomorphism between T L(M ) and T ∗ L(M ). Therefore α and gL allow us to define a 1-form Z 1 1 (1) µ(X) = g(X(t), γ ′ (t))dt, 2 0 whose exterior differential ω = dµ happens to be quasi-symplectic. This means that the kernel of the form is finite dimensional, specifically ker(ω)γ = {X ∈ Γ(S 1 , γ ∗ T M ) ; ∇γ ′ X = 0}. (We assume throughout this article that the spaces considered are in the C ∞ category and have a natural Fréchet structure, unless something else is declared.) Date: November, 2007. 1991 Mathematics Subject Classification. Primary: 58B20. Secondary: 53D35, 55P35. Key words and phrases. Loop space, symplectic structures. First author partially supported through grant MEC (Spain) MTM2007-63582. 1 2 VICENTE MUÑOZ AND FRANCISCO PRESAS This quasi-symplectic structure can be enriched in many cases. This is well known in the case of loop groups (i.e., M is a Lie group). In this case it is possible to define an integrable complex structure making a finite codimensional closed manifold of a loop group into a Kähler manifold. Extending our space to the path space defined as P(M ) = {γ : [0, 1] → M }, we still have the canonical section of the tangent bundle given by α : P(M ) → T P(M ) γ → γ′. As in the loop space we will easily check that equation (1) is a 1-form whose differential is symplectic. Proposition 1.1. The 2-form ω = dµ in P(M ) induces a symplectic structure. Moreover (L(M ), ω) is a closed quasi-symplectic submanifold of P(M ). It could be thought that the symplectic structure makes life easier in comparison to the quasi-symplectic one. At least, in terms of the stability of the structure this is not the case. In particular, we prove Theorem 1.2. Fix a smooth closed manifold M of even dimension. Denote as ωg the quasi-symplectic form on L(M ) associated to a Riemannian metric g on M . Then given two Riemannian metrics g0 and g1 on M , there exists a smooth isotopy φǫ on L(M ) such that (φǫ )∗ ωg0 is ǫ-close to ωg1 in L2 -norm. The proof of this result cannot be generalized to the case of P(M ). Also, the theorem cannot be improved to obtain an isotopy that matches ωg0 and ωg1 on the nose. Certainly, allowing ǫ go to zero makes the norm of the isotopy go to infinity and so discontinuities are developed. So the result can be understood as a sort of “approximate uniqueness” of the quasi-symplectic structure. The proof of Theorem 1.2 is based on an adaptation of Moser’s trick to this setting. It is surprising that this type of argument works in an infinitedimensional non-compact setting, since Moser’s trick needs the compactness of the manifold. Somehow, the compactness of the underlying manifold makes the job in our case. This shows that the symplectic geometry of the loop space probably does not recover the Riemannian geometry of the underlying manifold in the even dimensional case. This statement will be clearer after the proof of Theorem 1.2, in which the geometric obstruction for the complete “uniqueness” of the quasi-symplectic structure is shown. Finally we discuss how to find contact hypersurfaces in loop (and path) spaces. The most natural construction is given by Theorem 1.3. Assume that the Riemannian manifold (M, g) admits a vector field X which satisfies LX g = g and is locally gradient-like, then the lift GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES 3 of X to L(M ) (respectively P(M )) is a Liouville vector field for the length function. This shows in particular that stabilizing the manifold M , i.e. considering M × R, we obtain contact hypersurfaces in the loop space. 2. Symplectic structure 2.1. Basic definitions. The quasi-symplectic structure in the space of loops of a Riemannian manifold is defined by taking the exterior differential of the 1-form µ given by equation (1). To do that we recall the formula (2) dα(X, Y ) = X(α(Y )) − Y (α(X)) − α([X, Y ]), which is valid for any 1-form α and it does not depend on the vector fields X, Y chosen to extend X(γ) and Y (γ) for a given point (loop) γ ∈ L(M ). In our case we start with two vectors U, V ∈ Γ(S 1 , γ ∗ T M ) ≃ Tγ L(M ). First define θ : (−ε, ε)2 × S 1 → M, satisfying: (i) θ(0, 0, t) = γ(t), ∂θ (0, 0, t) = U (t), (ii) ∂u ∂θ (iii) ∂v (0, 0, t) = V (t). And so define γ ′ = (3) ∂θ ∂t , Û = ∂θ ∂u and V̂ = ∂θ ∂v . They clearly satisfy [Û , V̂ ] = 0, since they are derivatives of the coordinates of a parametrization. This allows us to compute  Z  ∂θ   d 1 1 ∂θ Û (µ(V̂ )) = (u, 0, t), (u, 0, t) dt gθ(u,0,t) du 2 0 ∂v ∂t Z 1 1 = (4) (g (∇ V̂ , γ ′ (t)) + gθ(0,0,t) (V, ∇Û γ ′ ))dt. 2 0 θ(0,0,t) Û In the same way, we obtain Z 1 1 (g (∇ Û , γ ′ (t)) + gθ(0,0,t) (U, ∇V̂ γ ′ ))dt. V̂ (µ(Û )) = 2 0 θ(0,0,t) V̂ We are using the torsion-free Levi-Civita connection for the computations, so ∇γ ′ Û = ∇Û γ ′ and ∇γ ′ V̂ = ∇V̂ γ ′ . Also ∇Û V̂ − ∇V̂ Û = [Û , V̂ ] = 0. We shall use the notation ∇γ ′ U = ∂U ∂t . So we have, by applying the formula (2), that (5) ω(U, V ) = ω(Û , V̂ ) = dµ(Û , V̂ ) = Û (µ(V̂ )) − V̂ (µ(Û )) = Z ∂U
∂V
 1 1 g dt. V, − g = γ(t) γ(t) U, 2 0 ∂t ∂t 4 VICENTE MUÑOZ AND FRANCISCO PRESAS Moreover we have  Z 1 Z 1 d ∂U
∂V
 (6) 0= g g(U, V ) dt = , V + g U, dt, dt ∂t ∂t 0 0 which implies (7) ω(U, V ) = Z 1 g 0 ∂U
, V dt. ∂t Now the kernel of this 2-form at a point γ is given by the parallel vector fields along γ. Therefore dim ker(γ) ≤ n. There are several ways of removing the kernel. The simplest one is to fix a point p ∈ M and to define Lp (M ) = {γ ∈ L(M ) ; γ(0) = p}. This forces the tangent vectors to satisfy X ∈ Tγ Lp (M ) ⇒ X ∈ Γ(S 1 , γ ∗ T M ), X(0) = 0. Therefore any parallel vector field is null. So the manifold Lp (M ) is symplectic. Extend our space to P(M ) where it is still possible to repeat all the previous computations. We highlight the differences. The equation (4) is exactly the same as it is symmetric. The equation (5) remains also without changes. We just need to rewrite equation (6) which is not true anymore and so the final expression for the exterior differential of µ becomes Z 1  ∂U
1 d ,V − g(U, V ) dt g dµ(U, V ) = ω(U, V ) = ∂t 2 dt 0 Z 1 g(U (1), V (1)) − g(U (0), V (0)) ∂U
, V dt − . g = ∂t 2 0 Is is obviously a closed (being exact) form. Let us compute its kernel. Assume that X ∈ ker(ω)γ . Considering ω(X, V ) = 0 for all vectors V ∈ Tγ P(M ) with V (0) = V (1) = 0, we obtain that ∂X = 0. ∂t Now by choosing all V ∈ Tγ P(M ) with V (0) 6= 0 and V (1) = 0, we conclude that X(0) = 0. By parallel transport, X = 0 and so the kernel of ω is trivial. Hence this form is symplectic. This proves Proposition 1.1. 2.2. Almost complex structures. There is a canonical almost-complex structure compatible with ω in Lp M . Let us construct it. Given a curve γ : [0, 1] → M , denote Pst the parallel transport isometry along γ. There is an isometric isomorphism between γ ∗ T M and the trivial Tγ(0) M bundle over I with constant metric gγ(0) . This allows to translate any section U (t) ∈ γ ∗ T M to a section Pt0 (U (t)) = Û (t) ∈ Tγ(0) M . This gives rise to a “développement” map Tγ Lp ∼ = L0 (Tγ(0) M ). GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES 5 Note that if we apply this map to γ ′ (t) itself, we get x(t) ∈ Tγ(0) M . Now we define Z t x(s)ds, a(t) = 0 which is known as the “développement de Cartan” of the curve γ in the tangent space Tγ(0) M . As the covariant derivative along γ becomes the ordinary derivative in Tγ(0) M , we have that γ is a geodesic just when its développement de Cartan is a line. Define the almost complex structure Jˆ in Tγ Lp (M ) as follows: take any vector field U ∈ Γ(S 1 , γ ∗ T M ) and compute its “développement” Û . Recall that Û (0) = Û (1) = 0 since U (t) ∈ Tγ Lp (M ). Fixing an isomorphism Tγ(0) M ∼ = Rn , we have Û (t) ∈ C ∞ (S 1 , Rn ). Take its Fourier series expansion, Û (t) = ∞ X ak e2πikt , k=−∞ where ak ∈ (8) Cn and a−k = āk . Then define X X ˜ Û )(t) = J( (−iak ) e2πikt + a0 + iak e2πikt , k<0 k>0 We substract the constant vector J(Û )(0) to get the almost-complex structure. So we have ˆ Û )(t) = J˜(Û )(t) − J( ˜ Û )(0) ∈ Tγ Lp (M ). J( To check that it is an almost complex structure we compute ˆ Û ) = J( ˆ J( ˜ Û ) − J( ˜ Û )(0)) = JˆJ( ˜ Û ) − J( ˜ Û )(0) − J˜J( ˜ Û )(0) + J( ˜ Û )(0) = = J˜J( ˜ Û ) − J˜J( ˜ Û )(0) = = J˜J( = (−Û + 2a0 ) − (−Û (0) + 2a0 ) = −Û . So we obtain an almost complex structure on L0 (Tγ(0) M ), then by using the “développement” we have an almost complex structure in Tγ Lp (M ), that is, in Lp (M ). To check that Jˆ is compatible with the symplectic form ω we just compute the value of ω when trivialized in the “développement”, to obtain ! X X X ap e2πipt , bq e2πiqt = (9) ω(Û , V̂ ) = ω 2πik Rehak , bk i, p q k where ak , bk ∈ Cn , and h , i is the standard Hermitian product in Cn . The associated metric X g(Û , V̂ ) = ω(Û , J V̂ ) = 2πk Re(hak , bk i + ha−k , b−k i) k>0 6 VICENTE MUÑOZ AND FRANCISCO PRESAS is clearly Riemannian. Moreover, Jˆ is smooth. To check it, recall that the map F : {f ∈ C ∞ (S 1 , Rn ) ; f (0) = 0} → SS ⊂ (Cn )∞ f 7→ (a1 , a2 , a3 , . . .), where SS is the Schwartz space of sequences of vectors in Cn with decay faster than polynomial, and {ak } are the Fourier coefficients of f , is a topological isomorphism (we take in SS the Fréchet structure given by the norms P t ||(ak )||t = k |ak |). (Note that the Fourier coefficients satisfy a−k = āk P and a0 = − k6=0 ak .) The map Jˆ is conjugated under F to the map J : SS → SS, (a1 , a2 , . . .) 7→ (ia1 , ia2 , . . .), which is smooth (actually an isometry). So Jˆ is smooth. This corrects the folklore statement saying that this almost complex structure is not smooth in general (see [Wu95, pag. 355]). In the case in which M is a Lie group, there is an alternative way of defining an almost complex structure for the space of loops based at the neutral element e ∈ G. To do it we just use the left multiplication to take the tangent space Tγ Le (G) to Γ(S 1 , Te G), so we obtain an isomorphism Tγ Le (G) ≃ Γ(S 1 , Rn )/Rn , preserving the metric by construction (the quotient is by the constant maps). So every particular vector field X ∈ Tγ Le (G) is transformed via the isomorphism to a loop in Rn . Recall that the isomorphism does not coincide with the one induced by the “développement” unless the group is flat (an abelian group). Once we have set up the previous identification, the formula (8) provides again an almost complex structure. Again we remark that it does not coincide with the previous one in the cases when both are defined. 2.3. Uniqueness. One may ask how canonical the symplectic structures on the loop spaces are. Such symplectic structure ω is associated to a metric g on M . Recall that the space of Riemannian metrics on a manifold is connected. Therefore, the Morse trick may help to prove that the associated loop spaces are all symplectomorphic. Recall that the Moser’s trick for exact symplectic forms works as follows. Assume that the 1-parametric family of forms ωt = dµt are symplectic on a manifold N . We want to find φt : N → N , such that φ∗t ωt = ω0 . Let Yt be the vector field generating φt . Then dµt
LYt ω t = d dt is satisfied. This is true if dµ , (10) iYt ωt = dt which has a unique solution since ωt non-degenerate. In our case we are given two different Riemannian metrics g0 and g1 in M , so there is a path GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES 7 of metrics gt that joins them. Therefore there is a path of exact symplectic forms ωt in Lp (M ). Now we substitute into equation (10) in our case to obtain Z 1  Z  ∇Yt (γ)(s) d 1 gt (11) , X(s) ds = gt (X(s), γ ′ (s))ds, ds dt 0 0 for all X ∈ Γ(S 1 , γ ∗ T M ). (Note that Yt (γ) ∈ Tγ Lp (M ).) This equation does not have continuous solutions in general. This is because we can compute ∇Yt (γ)(s) in a continuous way and we may assume that Yt (γ)(0) = 0, but ds then there is no reason to expect that Yt (γ)(1) = 0. So, it turns out that we do not get a uniqueness result for the symplectic structure. We will see a way of partially avoiding this obstruction. For this we are forced to change our point of view and work over the space L(M ), where the forms ω are quasi-symplectic. Proof of Theorem 1.2. We try to apply Moser’s trick in the space L(M ). Recall that ω is not symplectic in this case, since it has a finite dimensional kernel. Denote by G(M ) the space of Riemannian metrics over M . Given a point p ∈ M , define so0 (p) = {a ∈ so(Tp M ) ; det(a) = 0}. Recall that so0 (p) 6= so(Tp M ) if dim(M ) is even. In that case it is a codimension 1 (singular) submanifold. This defines a fibration so0 (M ) → M, which is a (non-linear) subbundle of the bundle so(T M ). For a metric g ∈ G(M ), the curvature Rg associated to the metric is a section of the bundle so(T M ) ⊗ Ω2 (M ), therefore it defines a bundle map ^ 2 Rg : (T M ) → so(T M ) . Note that these bundles have the same rank. We say that a metric g is R-generic if for each p ∈ M there are two vectors u, v ∈ Tp M such that det(Rg (u, v)) 6= 0, in the case dim M ≥ 4, or if Rg vanishes only at a discrete set of points, in the case dim M = 2. Define G0 (M ) = {g ∈ G(M ) : g is R-generic}. Given two metrics g0′ and g1′ , there are two metrics g0 and g1 such that gi is as close as desired to gi′ and gi is R-generic. Moreover we may do the same for paths gt of R-generic metrics. This is clear in the case dim M = 2. If n = dim M ≥ 4, we work as follows: for each g ∈ G and p ∈ M , there exists a small perturbation of g which of p. This is V V is R-generic in a neighborhood true since the subspace Hom( 2 (Tp M ), so0 (p)) ⊂ Hom( 2 (Tp M ), so(Tp M )) 8 VICENTE MUÑOZ AND FRANCISCO PRESAS
is of codimension n2 , which is bigger than n. The result follows from a Sard-Smale lemma applied to the functional ^ G(M ) × M → Hom( 2 (T M ), so(T M )) (g, p) → Rg (p). Now define the set SO0 (Tp M ) = {A ∈ SO(Tp M ) ; A − Id is not invertible}. This is a codimension 1 stratified submanifold of SO(Tp M ) and defines a bundle SO0 (T M ) → M. Let us define Ls (M ) = {γ ∈ L(M ) ; P01 − Id is not invertible}. There is a map θ : L(M ) → SO(T M ) γ → (P01 )γ ∈ SO(Tγ(0) M ). Clearly Ls (M ) = θ −1 (SO0 (T M )). Therefore if θ is generic in a suitable sense, Ls (M ) will be a stratified codimension 1 submanifold. We claim that if g ∈ G0 (M ), this is the case. To check this, pick γ ∈ Ls (M ). Being g an R-generic metric at Tγ(0) M , there exist two vectors u, v ∈ Tγ(0) M such that det(Rg (u, v)) 6= 0. Recall [Be02, Subsection 15.4.1] that if we extend u, v to a neighborhood of γ(0) in such a way that they define a local pair of coordinates (x, y) where ∂ = u, ∂x (0,0) ∂ = v, ∂y (0,0) s as (in the coordinates (x, y)): and we define the path γu,v s (t) = (4st, 0), t ∈ [0, 1/4], • γu,v s • γu,v (t) = (s, 4s(t − 1/4)), t ∈ [1/4, 1/2], s (t) = (4s(3/4 − t), s), t ∈ [1/2, 3/4], • γu,v s • γu,v (t) = (0, 4s(1 − t)), t ∈ [3/4, 1], we obtain s (P01 )γu,v lim = Rg (u, v). s→0 s s , Take the path which is the juxtaposition of γ with γu,v s βs = γ ∗ γu,v . This family of paths determines a tangent vector in Tγ L(M ). We will show that it is transverse to the submanifold Ls (M ). The holonomy of βs is (12) (P01 )βs = P01 (Id +s Rg (u, v)) + O(s2 ), GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES 9 where P01 is the holonomy of γ. Now embed gl(n, R) ⊂ gl(n, C) by complexifying. Then P01 ∈ gl(n, C) admits a Jordan canonical form J = BP01 B −1 , where B ∈ GL(n, C). Multiply on the left and on the right by B and B −1 the expresion (12) to obtain (13) B(P01 )βs B −1 = J + s BP01 Rg (u, v)B −1 + O(s2 ). Now, it is easy to check that the eigenvalues of the right hand side of (13), for s small enough, are far away from zero or grow faster than ǫs, for some fixed ǫ > 0. Since B is just a change of coordinates matrix for (P01 )βs on the left hand side of (13), the eigenvalues of (P01 )βs are the same than those of the right hand side of (13). This implies that the tangent vector determined by βs is transverse to Ls (M ). In the case dim M = 2, we work as before when γ(0) does not coincide with either of the points p ∈ M with Rg (p) = 0. We can define Ls (M ) to be the same set as before together with the paths starting at a point p with Rg (p) = 0. This latter set is of codimension n, so Ls (M ) is still of codimension 1, as needed, in this case. Recall that it is fundamental for this argument to work that the dimension of M is even. If the dimension is odd, then Ls (M ) = L(M ). All the computations done at the beginning of this subsection remain valid for the family {gt } and so we can follow Moser’s trick to end up with equation (10) again. Now we recall that the solution Yt is not unique, basically because we are working with L(M ), and there is a space of parallel vector fields along γ, which are in the kernel of the quasi-symplectic form ω. In general we obtain Ytw = Yt + w as a valid solution where Yt is a particular solution and w is a parallel vector field along γ (here we may have w(0) 6= w(1) – actually, this will be the case). Again in general Ytw (1) 6= Ytw (0). However a careful choice of w may help. We fix the equation Yt (0) + w(0) = Yt (1) + w(1), that if solved for some w(0) ∈ Rn , leads to a smooth solution of equation (10). The previous equation leads to (14) P01 (w(0)) − w(0) = Yt (1) − Yt (0), that clearly has a unique solution whenever γ 6∈ Ls (M ). So equation (10) has a unique continuous solution outside a set of positive codimension in L(M ). We are aiming to construct an “approximate” solution to (14). To get this, we can perturb equation (14) to (15) (λP01 − Id)(wλ (0)) = Yt (1) − Yt (0), which always admits a solution for |λ| < 1 since P01 ∈ SO(n). Now we assume that λ is a smooth map λ : L(M ) × [0, 1] → [1 − ǫ, 1] satisfying (i) λ(γ, t) = 1 − ǫ if γ ∈ Ls (M ) for the metric gt . 10 VICENTE MUÑOZ AND FRANCISCO PRESAS (ii) λ(γ, t) = 1 on a small neighborhood of Ls (M ). This defines a family of flows {Ytλ } (for the given constant ǫ > 0). The integrals at time 1 of these flows are generating the family φǫ required in the statement of the theorem. The flow exists and is unique. This is due to the fact that it can be understood as a parametric (smoothly dependent) family of flows in M and there we have existence and uniqueness (for all times). The smooth dependency in the parameters gives that the flow is Fréchet smooth. We also need to prove that the flow is by diffeomorphisms since, being C ∞ (S 1 , M ) a Fréchet manifold, this is not automatic. But this follows from φ−ǫ ◦ φǫ = I , and so the flow maps admit inverses. It is a routine to check that φǫ takes the quasi-symplectic form associated to g0 to a form ǫ-close in L2 -norm to the quasi-symplectic form associated to g1 . Moreover, since gi and gi′ are as close as needed we can also claim that their associated quasi-symplectic structures are close to each other in L2 -norm.  Finally, it is remarkable to note that the possibility of addition of parallel vector fields has been the key to find a continuously varying family of functions which are solutions to (10) almost everywhere. This is the geometric reason for which we cannot extend the computation to the case of the (symplectic) space Lp (M ). 3. Loop spaces as contact manifolds. We want to check whether the symplectic manifold Lp (M ) has hypersurfaces of contact type on it. We prove now Theorem 1.3. Proof of Theorem 1.3. Let X be a vector field on M satisfying LX g = g. Then ∇X ∈ End(T M ), and its symmetrization is 21 Id. This follows since, for Y , Z vector fields on M , we have g(∇Z X, Y )+ g(∇Y X, Z) = = g(∇X Z, Y ) + g(∇X Y, Z) − g(LX Z, Y ) − g(LX Y, Z) = = X(g(Y, Z)) + (LX g)(Y, Z) − LX (g(Y, Z)) = = g(Y, Z), where we have used that LX Z = ∇X Z − ∇Z X on the second line. The antisymmetrization of ∇X is A(∇X) = A(∇X # )# = (dX # )# , where X # is the 1-form associated to X (“raising the index”), and the (·)# means “lowering the index” with the metric. Recall that a vector field is ”locally gradientlike” in a neighborhood U for a metric g if it is g-dual of some exact 1-form df , where f is a function f : U → R. Thus, if X is locally gradient-like, then X # is a locally exact, i.e. closed, 1-form and so A(∇X) = 0. Then ∇X = 21 Id. GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES 11 Associated to X there is an induced vector field X̂ on L(M ). It is defined as follows: for γ ∈ L(M ), X̂γ ∈ Tγ L(M ) is given by X̂γ (t) = X(γ(t)). We want to check that LX̂ µ = µ. For Y ∈ Tγ L(M ), we have α(Y ) = iX̂ ω(Y ) = ω(X̂, Y ) = Z 1 Z 1 ∂X
g(∇γ ′ X, Y )dt = g = , Y dt = ∂t 0 0 Z 1 1 = g(γ ′ , Y )dt = µ(Y ). 2 0 So α = µ. Then LX̂ µ = diX̂ µ + iX̂ dµ = diX̂ iX̂ µ + iX̂ ω = 0 + α = µ. From this, it follows that LX̂ ω = LX̂ dµ = dLX̂ µ = dµ = ω, as required.  Remark 3.1. The manifolds to which the previous result applies, that is, those satisfying LX g = g with X locally gradient-like, are locally of the form ∂ . This follows by (N × R, et (g + dt2 )) with expanding vector field X = ∂t writing X = grad f , with f > 0 and putting t = log(f ). Two examples are relevant: • M = N × R, with (N, g) a compact Riemannian manifold. Give M the metric et (g + dt2 ). • Let (N, g) be an open Riemannian manifold with a diffeomorphism ϕ : N → N such that ϕ∗ (g) = eλ g, λ > 0. Then take M = (N × [0, λ])/ ∼, where (x, 0) ∼ (ϕ(x), λ) and M has the metric induced by et (g + dt2 ). To finish, let us check that the familiar finite dimensional picture translates to this case. Proposition 3.2. Let (M, g) be a Riemannian manifold which has a locally gradient-like vector field X satisfying LX g = g. Then the hypersurface Lp,1 (M ) = {γ ∈ Lp (M ) ; length(γ) = 1} is a contact hypersurface of Lp (M ). Proof. We need to check that α = iX̂ ω is a contact form on Lp,1 (M ). First we claim that α is nowhere zero on that submanifold. If this were not the case, then we would have that (16) (iX̂ ω)|Lp,1 (M ) = 0, and we know that iX̂ ω(X̂) = 0. Note that X̂ is transversal to Lp,1 (M ) since the flow increases the length of the loop. Therefore we have that iX̂ ω = 0 and ω would not be symplectic, which is a contradiction. So α is nowhere zero. 12 VICENTE MUÑOZ AND FRANCISCO PRESAS Now we have the distribution (ker α, ω = dα) on Lp,1 (M ). To finish we check that it is symplectic. Assume that Y ∈ Tγ Lp,1 (M ) satisfies that ω(Y, Z) = 0, for all Z ∈ Tγ Lp,1 (M ). Moreover we have that ω(Y, X̂) = −α(Y ) = 0. Hence iY ω = 0, and we get a contradiction. Corollary 3.3. Given a Riemannian manifold (M, g), the manifold (M × R, eλ (g+dλ2 )) has an associated space of loops of length one with a canonical contact form. For a loop γ and Y vector field along γ, we denote γ = (γ1 , γ2 ) and Y = (Y1 , Y2 ) according to the decomposition M × R. Then the contact form is given by Z
1 1 γ2 (t) e g(γ1′ (t), Y1 (t)) + γ2′ (t)Y2 (t) dt. α(Y ) = µ(Y ) = 2 0 3.1. Reeb vector fields. We compute the Reeb vector field associated to the contact form. We need to do it in L1 (M ), that is a quasi-contact space (instead of contact). This is necessary in order to obtain a smooth Reeb vector field. Lemma 3.4. The Reeb vector field associated to (L1 (M ), α) is the vector R= γ′ · h(γ), ||γ ′ || where h : L1 (M ) → R is a Diff(S 1 )-invariant strictly positive function on the loop space. Proof. The condition for a vector V ∈ Tγ L(M ) to belong to Tγ L1 (M ) is R1 ′ ||γ + s dV dt ||dt = 0. lim 0 s→0 s So we are asking the vector to satisfy Z 1 ∂V ′
1 , γ (t) dt = 0. g ∂t ||γ ′ || 0 Thus the previous equation can be rewritten as  γ′  = 0. ω V, ′ ||γ || Imposing this condition for every non-zero vector V ∈ Tγ L1 (M ), we get that ′ the Reeb vector field is a positive multiple of ||γγ ′ || , so proving the statement. It remains to be checked that h(γ) is invariant by changes of parametrization preserving the origin. This is a consequence of the invariance under changes of coordinates of the defining equation  γ′  α(R) = iX̂ ω h(γ) = 1. ||γ ′ || GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES 13  There are more solutions to the Reeb vector field equation since for any parallel vector field w along γ, we can add it to R, so that R + w defines another solution to the equation. However those solutions are not continuous (as functionals on L(M )) in general. This is the reason that prevents us to define the Reeb field in the space Lp,1 (M ). Again the flexibility of the quasi-contact structure is the key to find the solution. We have the following Lemma 3.5. All the Reeb orbits of α are closed. For a loop γ : S1 → M t → γ(t), the Reeb orbit passing through it has period equal to the length of γ divided by h(γ). Proof. Take γ : S 1 → M . The arc-length parametrization of γ(S 1 ) is denoted as γp , where p is the point of γ(S 1 ) in which the arc-length parameter starts. So we define θ(s, t) = γγ(t) (h(γ)s) which is clearly the Reeb orbit starting at γ. It is periodic with period length(γ)/h(γ).  References [Be02] M. Berger, A panoramic view of Riemannian geometry. Springer-Verlag, Berlin, 2003. [GP88] M. A. Guest, A. N. Pressley, Holomorphic curves in loop groups, Commun. Math. Phys. 118 (1988), 511–527. [PS86] A. Pressley, G. Segal, Loop groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1986. [Wu95] T. Wurzbacher, Symplectic geometry of the loop space of a symplectic manifold, J. of Geometry and Physics 16 (1995), 345–384. Departamento de Matemáticas, Consejo Superior de Investigaciones Cientı́ficas, 28006 Madrid, Spain Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain E-mail address: [email protected] Departamento de Matemáticas, Consejo Superior de Investigaciones Cientı́ficas, 28006 Madrid, Spain Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain E-mail address: [email protected] arXiv:0801.3545v2 [math.SG] 15 Sep 2010 GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES VICENTE MUÑOZ AND FRANCISCO PRESAS Abstract. The loop space associated to a Riemannian manifold admits a quasi-symplectic structure (that is, a closed 2-form which is nondegenerate up to a finite-dimensional kernel). We show how to construct a compatible almost-complex structure. Finally conditions to have contact structures on loop spaces are studied. 1. Introduction Let M be a smooth manifold, and consider the loop space L(M ) consisting of C ∞ loops in M . This is naturally a Fréchet manifold. The tangent space to L(M ) at a loop γ is Tγ L(M ) ∼ = Γ(S 1 , γ ∗ T M ). The loop space L(M ) is equipped with a natural section of its tangent bundle defined as ξ : L(M ) → T L(M ) γ 7→ γ ′ . Whenever we fix a Riemannian metric g on M , we can define an associated weak metric on the space of loops as Z 1 g(X(t), Y (t))dt, (gL )γ (X, Y ) = 0 for X, Y ∈ Tγ L(M ). Recall that a weak metric is a symmetric non-degenerate bilinear form which makes the space a pre-Hilbert space, and such that it extends to the completion giving rise to a (topological) isomorphism between the space and its dual. In our case, gL gives the L2 -norm on Tγ L(M ). The section α and gL allow us to define a 1-form Z 1 1 g(X(t), γ ′ (t))dt, (1) µ(X) = 2 0 whose exterior differential will be denoted as ω = dµ. The forms µ and ω are known as the Atiyah forms of the loop space L(M ) (see [At84]). Definition 1.1. A 2-form β on a pre-Hilbert space is weakly symplectic if it is non-degenerate and closed (but note that it may not produce an isomorphism between the Hilbert space obtained after completion and its dual). Date: February, 2010. 1991 Mathematics Subject Classification. Primary: 58B20. Secondary: 53D35, 55P35. Key words and phrases. Loop space, symplectic structures, contact structures. First author partially supported through grant MEC (Spain) MTM2007-63582. 1 2 VICENTE MUÑOZ AND FRANCISCO PRESAS A 2-form is quasi-symplectic if it has a finite-dimensional kernel and it is weakly symplectic on the orthogonal of the kernel. In our case, we shall see that ω is quasi-symplectic with kernel (2) ker(ωγ ) = {X ∈ Γ(S 1 , γ ∗ T M ) ; ∇γ ′ X = 0}. This quasi-symplectic structure can be enriched in many cases. This is well known in the case of based loop groups (i.e., M is a Lie group). In this case it is possible to define an integrable complex structure making a finite codimensional closed manifold of a loop group into a Kähler manifold. Now consider the path space P(M ) consisting of C ∞ -maps γ : [0, 1] → M . We again have the canonical section of the tangent bundle given by ξ : P(M ) → T P(M ) γ 7→ γ ′ . As in the case of the loop space, we will easily check that the equation (1) yields a 1-form whose differential is symplectic. Therefore: Proposition 1.2. The 2-form ω = dµ in P(M ) induces a weakly symplectic structure. Moreover L(M ) is a closed quasi-symplectic submanifold of (P(M ), ω). Consider now Lp (M ) = {γ ∈ L(M ) ; γ(0) = p} . Then (Lp (M ), ω) is a symplectic submanifold of P(M ). We shall show how to construct a weak almost complex structure J on Lp (M ). This is a map from Tγ Lp (M ) into its Hilbert completion, such that J 2 = − Id. Proposition 1.3. Lp (M ) has a weak almost complex structure J compatible with ω, that is ω(·, J·) is a weak metric on Lp (M ). Finally we discuss how to find contact hypersurfaces in loop spaces. We define Definition 1.4. A non-vanishing 1-form δ on a pre-Hilbert space is weakly contact if its exterior differential dδ is non-degenerate when restricted to the kernel of δ (note that it may not produce an isomorphism between the Hilbert space obtained after completion of ker δ and its dual). A 1-form δ is quasi-contact if the restriction of its differential dδ to the kernel ker δ has finite dimensional kernel. We recall that a vector field X in a Riemannian manifold (M, g) is locally gradient-like if the 1-form X # is closed. We mean by the symbol # the operation of raising the index by means of the metric. This is equivalent to be the gradient for a local function in the neighborhood of any point. The most natural construction is given by GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES 3 Theorem 1.5. Assume that the Riemannian manifold (M, g) admits a vector field X which satisfies LX g = g and is locally gradient-like, then the lift X̂ of X to L(M ) is a Liouville vector field for the weakly symplectic form ω. Moreover the lift X̂ is transverse to the level sets of the length functional. A vector field is called Liouville for a symplectic form ω if it satisfies LX ω = ω. As in the finite dimensional case, the existence of a Liouville vector field transverse to a hypersurface provides a contact form. Define L1 (M ) = {γ ∈ L(M ) ; length(γ) = 1}. and Lp,1 (M ) = {γ ∈ Lp (M ) ; length(γ) = 1}. They are smooth hypersurfaces of the manifolds L(M ) and Lp (M ) respectively. We have Corollary 1.6. The form α = iX̂ ω is a quasi-contact form in L1 (M ). Respectively the restriction of the form α to Lp,1 is a weakly contact form in that space. We will show in particular that stabilizing the manifold M , i.e. considering M × R, we obtain contact hypersurfaces in the loop space. 2. Symplectic structure In this section we shall prove Theorem 1.2. The quasi-symplectic structure in the space of loops of a Riemannian manifold is defined by taking the exterior differential of the 1-form µ given by equation (1). To do that we recall the formula (3) dα(X, Y ) = X(α(Y )) − Y (α(X)) − α([X, Y ]), which is valid for any 1-form α and it does not depend on the vector fields X, Y chosen to extend X(γ) and Y (γ) for a given point (loop) γ ∈ L(M ). In our case we start with two vectors U, V ∈ Γ(S 1 , γ ∗ T M ) ≃ Tγ L(M ). First define θ : (−ε, ε)2 × S 1 → M, satisfying: (i) θ(0, 0, t) = γ(t), ∂θ (0, 0, t) = U (t), (ii) ∂u ∂θ (iii) ∂v (0, 0, t) = V (t). And so define γ ′ = (4) ∂θ ∂t , Û = ∂θ ∂u and V̂ = ∂θ ∂v . [Û , V̂ ] = 0, They clearly satisfy 4 VICENTE MUÑOZ AND FRANCISCO PRESAS since they are derivatives of the coordinates of a parametrization. This allows us to compute  Z  ∂θ   d 1 1 ∂θ Û (µ(V̂ )) = (u, 0, t), (u, 0, t) dt g du 2 0 θ(u,0,t) ∂v ∂t Z 1 1 (5) (g (∇ V̂ , γ ′ (t)) + gθ(0,0,t) (V, ∇Û γ ′ ))dt. = 2 0 θ(0,0,t) Û In the same way, we obtain Z 1 1 V̂ (µ(Û )) = (g (∇ Û , γ ′ (t)) + gθ(0,0,t) (U, ∇V̂ γ ′ ))dt. 2 0 θ(0,0,t) V̂ We are using the torsion-free Levi-Civita connection for the computations, so ∇γ ′ Û = ∇Û γ ′ and ∇γ ′ V̂ = ∇V̂ γ ′ . Also ∇Û V̂ − ∇V̂ Û = [Û , V̂ ] = 0. We shall use the notation ∇γ ′ U = ∂U ∂t . So we have, by applying the formula (3), that (6) ω(U, V ) = ω(Û , V̂ ) = dµ(Û , V̂ ) = Û (µ(V̂ )) − V̂ (µ(Û )) = Z 1 1 ∂U
∂V
 = gγ(t) V, dt. − gγ(t) U, 2 0 ∂t ∂t Moreover we have  Z 1 Z 1 ∂U
d ∂V
 g g(U, V ) dt = , V + g U, (7) 0= dt, dt ∂t ∂t 0 0 which implies (8) ω(U, V ) = Z 1 g 0 ∂U
, V dt. ∂t Now the kernel of this 2-form at a point γ is given by the parallel vector fields along γ. Therefore dim ker(ωγ ) ≤ n. There are several ways of removing the kernel of ω. The simplest one is to fix a point p ∈ M and to define Lp (M ) = {γ ∈ L(M ) ; γ(0) = p}. This forces the tangent vectors to satisfy X ∈ Tγ Lp (M ) ⇒ X ∈ Γ(S 1 , γ ∗ T M ), X(0) = 0. Therefore any parallel vector field is null. So the manifold Lp (M ) is (weakly) symplectic. We shall take a second route. Extend our space to P(M ) = {γ : [0, 1] → M }, where it is still possible to repeat all the previous computations. We highlight the differences. The equation (5) is exactly the same as it is symmetric. The equation (6) remains also without changes. We just need to rewrite GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES 5 equation (7) which is not true anymore and so the final expression for the exterior differential of µ becomes Z 1  ∂U
1 d g dµ(U, V ) = ω(U, V ) = ,V − g(U, V ) dt ∂t 2 dt 0 Z 1
g(U (1), V (1)) − g(U (0), V (0)) ∂U , V dt − . g = ∂t 2 0 It is obviously a closed (being exact) form. Let us compute its kernel. Assume that X ∈ ker(ωγ ). Considering ω(X, V ) = 0 for all vectors V ∈ Tγ P(M ) with V (0) = V (1) = 0, we obtain that ∂X = 0. ∂t Now by choosing all V ∈ Tγ P(M ) with V (0) 6= 0 and V (1) = 0, we conclude that X(0) = 0. By parallel transport, X = 0 and so the kernel of ω is trivial. Hence this form is (weakly) symplectic. This proves Proposition 1.2. Remark 2.1. The Riemannian metric determines the symplectic form ω. But conversely, the form ω determines the Riemannian metric of M as follows: consider M embedded in L(M ) as the space of constant loops. Given p ∈ M , consider the constant loop γ(t) = p, for all t ∈ S 1 . Given v ∈ Tp M and f (t), a smooth real-valued function on S 1 with f (0) = 0, write Xv,f ∈ Tγ L(M ) for the vector field defined as Xv,f (t) = f (t)v. Then for v1 , v2 ∈ Tp M , one has Z 1 f1′ (t)f2 (t)dt . ωγ (Xv1 ,f1 , Xv2 ,f2 ) = hv1 , v2 i 0 Therefore we can recover the metric of M out of ω. 3. Almost complex structures There is a canonical almost-complex structure compatible with ω in Lp (M ). Let us construct it. Given a curve γ : [0, 1] → M , denote Pst the parallel transport isometry along γ. There is an isometric isomorphism between γ ∗ T M and the trivial Tγ(0) M bundle over I with constant metric gγ(0) . This allows to translate any section U (t) ∈ γ ∗ T M to a section Pt0 (U (t)) = Û (t) ∈ Tγ(0) M . This gives rise to a “développement” map Tγ Lp (M ) → L0 (Tγ(0) M ), where L0 (Tγ(0) M ) is the space of loops based at 0 in the tangent space Tγ (0)M such that they are C ∞ in (0, 1) and continuous at the origin. Realize that the derivatives, though bounded, are not continuous in general at the point 0. This is a continuous injective linear map. Moreover, realize that the map extends to an isomorphism between the completions of the spaces Tγ Lp (M ) and L0 (Tγ(0) M ). 6 VICENTE MUÑOZ AND FRANCISCO PRESAS Note that if we apply this map to γ ′ (t) itself, we get a curve x(t) ∈ Tγ(0) M . Now we define Z t x(s)ds, a(t) = 0 which is known as the “développement de Cartan” of the curve γ in the tangent space Tγ(0) M . As the covariant derivative along γ becomes the ordinary derivative in Tγ(0) M , we have that γ is a geodesic just when its développement de Cartan is a line. Define a weak almost complex structure Jˆ in Tγ Lp (M ) as follows: take any vector field U ∈ Γ0 (S 1 , γ ∗ T M ) and compute its “développement” Û , which obviously satisfies Û (0) = Û (1) = 0, since U (t) ∈ Tγ Lp (M ). Fixing an isomorphism Tγ(0) M ∼ = Rn , we have Û (t) : [0, 1] → Rn , which is C ∞ in (0, 1), continuous in [0, 1] and Û (0) = Û (1) = 0. Take its Fourier series expansion, Û (t) = ∞ X ak e2πikt , k=−∞ where ak ∈ (9) Cn and a−k = āk . Then define X X ˜ Û )(t) = J( (−iak ) e2πikt + a0 + iak e2πikt , k<0 k>0 We substract the constant vector J˜(Û )(0) to get the almost-complex structure. So we have ˆ Û )(t) = J( ˜ Û )(t) − J( ˜ Û )(0) ∈ L0 (Tγ(0) M ). J( To check that it is an almost complex structure we compute ˆ Û ) = J( ˆ J( ˜ Û ) − J( ˜ Û )(0)) = JˆJ( ˜ Û ) − J( ˜ Û )(0) − J˜J( ˜ Û )(0) + J( ˜ Û )(0) = = J˜J( ˜ Û ) − J˜J( ˜ Û )(0) = = J˜J( = (−Û + 2a0 ) − (−Û (0) + 2a0 ) = −Û . It is important to realize that the map is well defined wit image in the completion of L0 (Tγ(0) M ), and in fact it does not preserve the space without compactifying it. So we obtain a weak almost complex structure on L0 (Tγ(0) M ). This means a map from a pre-Hilbert space into its completion, which is a linear and bounded map satisfying that the square is minus the identity (note that J does not extend to the Hilbert completion of L0 (Tγ(0) M )). Finally, by using the “développement” we have a weak almost complex structure in Tγ Lp (M ), that is, in Lp (M ). GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES 7 To check that Jˆ is compatible with the symplectic form ω we just compute the value of ω when trivialized in the “développement”, to obtain ! X X X 2πipt 2πiqt = ap e , bq e (10) ω(Û , V̂ ) = ω 2πikhak , bk i, p q k Cn , where ak , bk ∈ and h , i is the standard Hermitian product in Cn . The associated metric X g(Û , V̂ ) = ω(Û , J V̂ ) = 2πk Re(hak , bk i + ha−k , b−k i) k>0 is clearly Riemannian. Moreover, Jˆ is smooth, meaning that at each tangent space Tγ Lp (M ), the map Jˆ between the respective completions is a linear bounded map. Moreover, Jˆ also depends smoothly on γ. Fix γ ∈ Lp (M ) and consider an open neighbourhood as follows: Let B ⊂ Γ0 ([0, 1], γ ∗ T M ) a small ball (in the Fréchet topology), and parametrize a neighbourhood U of γ by means of the exponential map: X(t) 7→ γX (t) = expγ(t) (X(t)) . Now we trivialize the tangent bundle over U by means of the “développement”: TγX Lp (M ) ∼ = U × L0 (Tp M ). With = L0 (Tp M ), which produces a chart T U ∼ respect to this chart, Jˆ is a constant operator, hence smooth. This corrects the folklore statement saying that this almost complex structure is not smooth in general (see [Wu95, pag. 355]). The case of Lie groups. In the case in which G is a Lie group, there is an alternative way of defining an almost complex structure for the space of loops based at the neutral element e ∈ G. To do it we just use the left multiplication to take the tangent space Tγ Le (G) to Γ0 (S 1 , Te G), so we obtain an isomorphism Tγ Le (G) ∼ = Γ0 (S 1 , Rn ) = {f ∈ C ∞ (S 1 , Rn ) ; f (0) = 0}, preserving the metric by construction. So every particular vector field X ∈ Tγ Le (G) is transformed via the isomorphism to a loop in Rn . Recall that the isomorphism does not coincide with the one induced by the “développement” unless the group is flat (an abelian group). Once we have set up the previous identification, the formula (9) provides again an almost complex structure. We remark that it does not coincide with the previous one in the cases when both are defined. This almost complex structure is compatible with the metric. Moreover, it is smooth. To check it, recall that the map F : Γ0 (S 1 , Rn ) → SS ⊂ (Cn )∞ f 7→ (a1 , a2 , a3 , . . .), 8 VICENTE MUÑOZ AND FRANCISCO PRESAS where SS is the Schwartz space of sequences of vectors in Cn with decay faster than polynomial, and {ak } are the Fourier coefficients of f , is a topological isomorphism (we take in SS the Fréchet structure given by the norms P t ||(ak )||t = k |a |). (Note that the Fourier coefficients satisfy a−k = āk P k and a0 = − k6=0 ak .) The map Jˆ is conjugated under F to the map J : SS → SS, (a1 , a2 , . . .) 7→ (ia1 , ia2 , . . .), which is smooth. So Jˆ is smooth. 4. Contact structures. We want to check whether the symplectic manifold Lp (M ) has hypersurfaces of contact type on it. We prove now Theorem 1.5. Proof of Theorem 1.5. Let X be a vector field on M satisfying LX g = g. Then ∇X ∈ Γ(M, End(T M )), and its symmetrization is 12 Id. This follows since, for Y , Z vector fields on M , we have g(∇Z X, Y )+ g(∇Y X, Z) = = g(∇X Z, Y ) + g(∇X Y, Z) − g(LX Z, Y ) − g(LX Y, Z) = = X(g(Y, Z)) + (LX g)(Y, Z) − LX (g(Y, Z)) = = g(Y, Z), where we have used that LX Z = ∇X Z − ∇Z X on the second line. The antisymmetrization of ∇X is A(∇X) = A(∇X # )# = (dX # )# , where X # is the 1-form associated to X (“raising the index”), and the (·)# means “lowering the index” with the metric. Recall that a vector field is “locally gradientlike” in a neighborhood U for a metric g if it is g-dual of some exact 1-form df , where f is a function f : U → R. Thus, if X is locally gradient-like, then X # is a locally exact, i.e. closed, 1-form and so A(∇X) = 0. Then ∇X = 21 Id. Associated to X there is an induced vector field X̂ on L(M ). It is defined as follows: for γ ∈ L(M ), X̂γ ∈ Tγ L(M ) is given by X̂γ (t) = X(γ(t)). We want to check that LX̂ µ = µ. For Y ∈ Tγ L(M ), we have α(Y ) = iX̂ ω(Y ) = ω(X̂, Y ) = Z 1 Z 1 ∂ X̂γ
g = g(∇γ ′ X, Y )dt = , Y dt = ∂t 0 0 Z 1 1 g(γ ′ , Y )dt = µ(Y ). = 2 0 So α = µ. Then LX̂ µ = diX̂ µ + iX̂ dµ = diX̂ iX̂ ω + iX̂ ω = 0 + α = µ. GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES 9 From this, it follows that LX̂ ω = LX̂ dµ = dLX̂ µ = dµ = ω, as required.  Remark 4.1. The manifolds to which the previous result applies, that is, those satisfying LX g = g with X locally gradient-like, are locally of the form ∂ . This follows by (N × R, et (g + dt2 )) with expanding vector field X = ∂t writing X = grad f , with f > 0 and putting t = log(f ). Two examples are relevant: • M = N × R, with (N, g) a compact Riemannian manifold. Give M the metric et (g + dt2 ). • Let (N, g) be an open Riemannian manifold with a diffeomorphism ϕ : N → N such that ϕ∗ (g) = eλ g, λ > 0. Then take M = (N × [0, λ])/ ∼, where (x, 0) ∼ (ϕ(x), λ) and M has the metric induced by et (g + dt2 ). To finish, let us check that the familiar finite dimensional picture translates to this case. Proposition 4.2. Let (M, g) be a Riemannian manifold which has a locally gradient-like vector field X satisfying LX g = g. Then the hypersurface L1 (M ) = {γ ∈ L(M ) ; length(γ) = 1} is a quasi-contact hypersurface of L(M ). Proof. We define a family of loops as follows γs (t) = ψs (γ(t)), where ψs : M → M is the flow associated to the vector field X. Let us compute the following derivative Z 1 d d length(γs ) g(γs′ (t), γs′ (t))1/2 dt |s=0 = |s=0 ds ds 0 Z 1 d g(dψs (γ ′ (t)), dψs (γ ′ (t)))1/2 dt |s=0 = ds 0 Z 1 d ((ψs∗ g)(γ ′ (t), γ ′ (t)))1/2 dt |s=0 = ds 0 Z 1 d = ((es g)(γ ′ (t), γ ′ (t)))1/2 dt |s=0 ds 0  Z 1  d s ′ ′ 1/2 |s=0 e (g(γ (t), γ (t))) dt = ds 0 = length(γ). Therefore the vector field X̂ is tranverse to the level sets of the functional length in L(M ). 10 VICENTE MUÑOZ AND FRANCISCO PRESAS Now, we need to check that α = iX̂ ω is a weak contact form on L1 (M ). First we claim that the form α is nowhere zero on that submanifold. If this were not the case, then we would have that (11) (iX̂ ω)γ0 |Tγ L1 (M ) = 0, for certain γ0 ∈ L1 (M ) , and we know that iX̂ ω(X̂) = 0. Therefore we have that iX̂ ω = 0 and then the vector field X̂ belongs to ker ω. Thus it satisfies the equation (2). Since the Levi-Civita connection is torsion-free we have that ∇γs′ X = ∇X γs′ . To set up the previous formula, we are extending γ0 to the family of loops γs in order to correctly define the commutators. We finish by computing ∇X g(γ0′ , γ0′ ) = g(∇X γ0′ , γ0′ ) + g(γ0′ , ∇X γ0′ ) = 0, because of the compatibility condition of the connection. So, we get that d length(γs ) |s=0 = 0, that is a contradiction. So α is nowhere zero. ds Now we have the distribution (ker α, ω = dα) on L1 (M ). To finish we check that it is quasi-symplectic. Assume that Y ∈ ker α ⊂ Tγ L1 (M ) satisfies that ω(Y, Z) = 0, for all Z ∈ ker α ⊂ Tγ L1 (M ). Moreover we have that ω(Y, X̂) = −α(Y ) = 0. Hence iY ω = 0, and we get that Y ∈ ker ω that is finite dimensional.  Corollary 4.3. The submanifold of Lp (M ) defined as Lp,1 (M ) = {γ ∈ Lp (M ) ; length(γ) = 1} is a weak contact submanifold. Proof. We have checked that L1 (M ) is quasi-contact. By choosing the submanifold Lp,1 (M ) we are removing the kernel as in the symplectic case.  Corollary 4.4. Given a Riemannian manifold (M, g), the manifold (M × R, eλ (g+dλ2 )) has an associated space of loops of length one with a canonical contact form. For a loop γ and Y vector field along γ, we denote γ = (γ1 , γ2 ) and Y = (Y1 , Y2 ) according to the decomposition M × R. Then the contact form is given by Z
1 1 γ2 (t) e g(γ1′ (t), Y1 (t)) + γ2′ (t)Y2 (t) dt. α(Y ) = µ(Y ) = 2 0 Reeb vector fields. Since the manifold L1 (M ) = {γ ∈ L(M ) ; length(γ) = 1} is quasi-contact, there is not a unique Reeb vector field, i.e. a vector field R satisfying iR α = 1, iR dα = 0. GEOMETRIC STRUCTURES ON LOOP AND PATH SPACES 11 However, this pair of equations has always solution as we now show. Lemma 4.5. A Reeb vector field associated to (L1 (M ), α) is the vector (12) R=2 γ′ . ||γ ′ || Proof. The condition for a vector V ∈ Tγ L(M ) to belong to Tγ L1 (M ) is that there s is a curve variation γs with γ0 = γ, dγ ds |s=0 = V and length(γs ) = 1. At first order, this is equivalent to: Z 1 1 ′ lim ||γ + s∇γ ′ V ||dt = 0. s→0 s 0 So we are asking the vector to satisfy Z 1 ∂V ′
1 , γ (t) dt = 0. g ∂t ||γ ′ || 0 Thus the previous equation can be rewritten as   γ′  γ′  ω V, ′ = dα V, ′ = 0. ||γ || ||γ || Imposing this condition for every non-zero vector V ∈ Tγ L1 (M ), we get that ′ the Reeb vector field is a positive multiple of ||γγ ′ || . Now we compute Z Z  γ′   γ′  γ′ 1 1 1 1 ′ γ′ 1 ′ = iX̂ ω = ω( ′ , X̂γ ) = g( ′ , γ ) = ||γ || = . α ′ ′ ||γ || ||γ || ||γ || 2 0 ||γ || 2 0 2 Therefore R=2 γ′ ||γ ′ || is a Reeb vector field.  There are more solutions to the Reeb vector field equation since for any parallel vector field w along γ, we can add it to R, so that R + w defines another solution to the equation. However those solutions are not continuous (as vector fields on L1 (M )) in general). We have the following Lemma 4.6. All the Reeb orbits for (12) in L1 (M ) are closed of period 1/2. Proof. Take γ : S 1 → M . The arc-length parametrization of γ(S 1 ) is denoted as γp , where p is the point of γ(S 1 ) in which the arc-length parameter starts. So we define θ(s, t) = γγ(t) (2s) which is clearly the Reeb orbit starting at γ. It is periodic with period 1/2.  12 VICENTE MUÑOZ AND FRANCISCO PRESAS References [At84] M. F. Atiyah, Circular symmetry and stationary phase approximation, in: Colloque en honneur de Laurent Schwartz, Vol. 1, Astérisque 131 (1984), 43–59. [Be02] M. Berger, A panoramic view of Riemannian geometry. Springer-Verlag, Berlin, 2003. [GP88] M. A. Guest, A. N. Pressley, Holomorphic curves in loop groups, Commun. Math. Phys. 118 (1988), 511–527. [PS86] A. Pressley, G. Segal, Loop groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1986. [Se88] G. Segal, Elliptic cohomology, in: Séminaire Bourbaki 40 (1987-88), no. 695, Astérisque 161-162 (1988), 187-201. [Wu95] T. Wurzbacher, Symplectic geometry of the loop space of a symplectic manifold, J. of Geometry and Physics 16 (1995), 345–384. Departamento de Geometrı́a y Topologı́a, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain E-mail address: [email protected] Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Serrano 113bis, 28006 Madrid, Spain Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain E-mail address: [email protected]