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1. Geometry of the Unit Tangent Bundle

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1. Geometry of the Unit Tangent Bundle 1 1. Geometry of the unit tangent bundle The main reference for this section is [8]. In the following, we consi- der (M, g) an n-dimensional smooth manifold endowed with a Riemannian metric g. 1.1. Notations Given a local chart (U, φ) on M, we will denote by (xi)16i6n the local coordinates and we write in these coordinates n X i j g = gijdx dx , i,j=1 ∂ ∂ ij where gij(x) = g , . We denote by (g ) the coefficient of the ∂xi ∂xj inverse matrix. Given x ∈ M, the norm of v ∈ TxM is given by |v|x = 1/2 1/2 gx(v, v) , which will sometimes be denoted hv, vix . In the following, we will often drop the index x. Note that if we are given other coordinates (yj), then one can check that in these new coordinates : n X ∂xi ∂xj k l (1.1) g = gij dy dy ∂yk ∂yl i,j,k,l=1 We define the musical isomorphism at x ∈ M by ∗ TxM → Tx M [ : v 7→ v[ = g(v, ·) This is an isomorphism between the two vector bundles TM and T ∗M since they are equidimensional and g is symmetric definite positive, thus non-degenerate. Given an orthonormal basis (ei) of TxM, we will denote i ∗ by (e ) the dual basis of Tx M which is, in other words, the image of the basis (ei) by the musical isomorphism. If E is a vector bundle over M, then the projection will be denoted π : E → M. We denote by Γ(M, E) the set of smooth sections of E. Γ(M) de- m ∗ notes Γ(M, T M), the set of vector fields. We will denote by Γ(M, ⊗S T M) the set of smooth symmetric covariant m-tensors on M. On the cotangent bundle T ∗M, we denote by ω the canonical symplectic form, which we Pn i i write in coordinates ω = i=1 dp ∧ dx . We recall that it is obtained as the differential of the canonical 1-form λ ∈ Ω1(T ∗M), defined intrinsically as λ(x,p)(ξ) = p dπ(x,p)(ξ) , 2 ∗ ∗ ∗ for a point (x, p) ∈ T M, ξ ∈ T(x,p)T M and where π : T M → M denotes the projection. Given a point x ∈ M, if (ei) is an orthonormal basis of TxM, we define dvol = e1 ∧ ... ∧ en. In local coordinates, it is given by the formula : q 1 n (1.2) dvol = det(gij)dx ∧ ... ∧ dx We recall that the Laplacian in local coordinates is given by : n X 1 (1.3) ∆f = √ ∂ pdet ggij∂ f det g i j i,j=1 We denote by ∇ the Levi-Civita connection on TM. In local coordinates, the Christoffel symbols are defined such that : n ∂ X k ∂ (1.4) ∇ ∂ = Γij ∂xi ∂xj ∂xk k=1 They are given by the Koszul formula : n k 1 X kl ∂gil ∂gjl ∂gij (1.5) Γij = g + − 2 ∂xj ∂xi ∂xl l=1 We denote the torsion tensor T ∇, which is defined as : ∇ T (X, Y ) = ∇X Y − ∇Y X − [X, Y ] The curvature tensor is denoted by F ∇ and defined as : ∇ (1.6) F (X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z In particular, we clearly have F ∇(X, Y ) = −F ∇(Y, X). We recall that the Levi-Civita connection is the unique torsion-free and g-metric connection, namely it satisfies for any X, Y, Z ∈ Γ(M) : ∇ (1.7) T (X, Y ) = ∇X Y − ∇Y X − [X, Y ] = 0 (1.8) Z · (g(X, Y )) = g(∇Z X, Y ) + g(X, ∇Z Y ) The sectional curvature K is given at x by ∇ Kx(e1, e2) = hF (e1, e2)e1, e2i, where e1, e2 ∈ TxM are orthogonal. In particular, in the case of a surface, which is what we will be mostly interested in, the sectional curvature is a real number referred to as the Gaussian curvature (or simply the curvature if the context is not ambiguous). 3 1.2. Structure of SM 1.2.1. The horizontal distribution Let us first state some very general results. We consider a vector bundle π : E → M of rank r with a connection ∇. Given a path γ : I → M such −1 that γ(0) = x, γ˙ (0) = X ∈ TxM and an initial value s0 ∈ Ex = π ({x}), there exists a unique lift of γ to a path s : I → E — it is the parallel transport of s0 along γ — such that s(0) = s0, π(s) = γ and (1.9) ∇γ˙ s = 0 Now, we can associate to X the vector ˜ d (1.10) X = s(t) ∈ Ts0 E dt t=0 One can check that X˜ is well defined and only depends on the choice of (γ(0), γ˙ (0)). It is a linear application TxM → Ts0 E which we denote by i ∂ j θ : X 7→ X˜. In local coordinates, if we write X = X , s0 = s ej, then ∂xi the parallel transport equation (1.9) yield to : i ∂ k i j ∂ (1.11) θ(X) = X − ΓijX s ∂xi ∂sk We define the horizontal distribution on (E, ∇) at s0 ∈ E as : (1.12) ∇ = Span(θ(X),X ∈ T (s )) ⊂ T E Hs0 π 0 s0 In the following, we will drop the notation ∇ for Hs0 , but we insist on the fact that Hs0 is entirely determined by the choice of ∇. Actually, one can check that choosing a connection ∇ is strictly equivalent to choosing a smooth distribution of vector spaces H ⊂ TE. Note that we have the following theorem : Theorem 1.1. — The following assertions are equivalent : — The distribution s0 7→ Hs0 is integrable. — The connection ∇ is flat, i.e. T ∇ vanishes. — The holonomy i.e. the parallel transport of a section along a closed loop is invariant by homotopy of this loop. Since θ is clearly injective by definition, it defines an isomorphism bet- ∼ ween Tπ(s0) −→ Hs0 . We define the vertical distribution on (E, ∇) at s0 as Vs0 = ker dπs0 = Eπ(s0) ⊂ Ts0 E. Thanks to (1.9), it is easy to check that θ is exactly the inverse of the differential dπs0 restricted to the subspace Hs0 of Ts0 E. We therefore have the splitting : (1.13) Ts0 E = Hs0 ⊕ Vs0 4 Given a vector ξ ∈ Ts0 E, which we write in local coordinates ξ = ∂ ∂ Xi + Y k , (1.11) shows that it is in ∇ if and only if Hs0 ∂xi ∂sk k k i j (1.14) Y + ΓijX s = 0, i for all 1 6 k 6 r, and it is in Vs0 if and only if X = 0 for all 1 6 i 6 n. 1.2.2. The Sasaki metric We now apply the previous formalism to the particular case when E = TM and construct a canonical metric hh·, ·ii on TM, called the Sasaki metric. If ∇ is the Levi-Civita connection on TM, then it automatically determines a splitting like in (1.13). In this case, there also exists a canonical application K : T(x,v)TM → TxM whose kernel is exactly H(x,v). Somehow, it can be seen as a com- plementary application to dπ. Given ξ ∈ T(x,v)TM, we can consider a local curve c in TM such that c(0) = (x, v) and c˙(0) = ξ. We can write c(t) = (γ(t),Z(t)), where Z is a vector field along the curve γ. We define : (1.15) K(x,v)(ξ) = ∇γ˙ Z(0) One can easily check that the application K now defines an isomorphism between V(x,v) and TxM and its kernel is precisely H(x,v), the horizontal distribution. Definition 1.2. — Given a point (x, v) ∈ TM and ξ, η ∈ T(x,v)TM, we set : — If ξ, η are vertical, namely ξ, η ∈ V(x,v) = TxM, then hhξ, ηii := hK(ξ), K(η)i, — If ξ, η are horizontal, then hhξ, ηii := hdπ(x,v)(ξ), dπ(x,v)ηi, — V(x,v) and H(x,v) are orthogonal for hh·, ·ii. Note that the last line induces in particular that the decomposition (1.13) is orthogonal for the Sasaki metric. In the rest of this paragraph, we explain how to recover the Sasaki metric from the symplectic point of view. We recall that λ is the canonical 1-form defined on T ∗M and introduced in the previous section. It can be pulled 5 back via the musical isomorphism to get α = [∗λ ∈ Ω1(TM). Then α(x,v)(ξ) = λ(x,v[)(d [(x,v)(ξ)) [ = v dπ(x,v[) ◦ d [(x,v)(ξ) [ = v d (π ◦ [)(x,v) (ξ) = hv, dπ(x,v)(ξ)i = hhZ, ξii, for some vector field Z, according to Riesz representation theorem. We claim that Z is actually the geodesic vector field X. Indeed, first observe that X is a horizontal vector field (i.e. X(x, v) ∈ H(x,v)), which is an immediate consequence of the expressions in local coordinates (??) and (1.14). In particular, in local coordinates, expression (1.11) yield to : i ∂ (1.16) dπ(x,v)(X(x, v)) = v ∂xi In order to prove that Z = X, we just have to check that the expressions hv, dπ(x,v)(ξ)i and hhX, ξii agree when ξ runs through a basis of T(x,v)TM. If ξ is in V(x,v) (that is ξ is vertical), this is immediate since dπ(ξ) = 0 and ∂ k j ∂ ∂ X is horizontal. If ξ = ξi = − Γijv , then dπ(ξi) = by (1.11) ∂xi ∂vk ∂xi and hv, dπ(x,v)(ξ)i = vi.
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