27.02.2018

Geodesic flows: notes 7

Geodesic flows, notes 7: recap

The section numbers correspond to the numbering of previous blog entries. 

0. The goal

The goal is to show that a function on a suitable Riemannian manifold (M,g)(M,g) is uniquely determined by its integrals over all maximal geodesics. To this end, we must understand the geometry of geodesics in great detail. The geodesic X-ray transform takes a function f:MRf:M→R to a function If:ΓRIf:Γ→R, where ΓΓ is the set of all geodesics on MM. This II is a linear integral transform, and it is known as the geodesic X-ray transform. The question is whether this transform is injective.

1. Symplectic manifolds and Hamilton flows

Definition. A symplectic manifold is a pair (N,σ)(N,σ) where NN is an 2n2n-dimensional smooth manifold, and σσ is symplectic form, that is, a closed 22-form on NN which is nondegenerate in the sense that for any ρNρ∈N, the map Iρ:TρNTρN,Iρ(s)=σ(s,)Iρ:TρN→Tρ∗N,Iρ(s)=σ(s,⋅) is bijective.

Example 1. The space R2nR2n has a standard symplectic structure given by the 22-form σ=dx1dxn+1++dxndx2nσ=dx1∧dxn+1+…+dxn∧dx2n.

Example 2. More generally, if MM is an nn-dimensional CC∞ manifold, then N=TMN=T∗M becomes a symplectic manifold as follows: if π:TMMπ:T∗M→M is the natural projection, there is a 11-form λλ on NN (called the Liouville form) defined by λρ=πρ,ρTMλρ=π∗ρ,ρ∈T∗M. Then σ=dλσ=dλis a closed 22-form. If xx are local coordinates on MM, and if (x,ξ)(x,ξ) are associated local coordinates (called canonical coordinates) on TMT∗M, then in these local coordinates

λσ=ξjdxj,=dξjdxj.λ=ξjdxj,σ=dξj∧dxj.

It follows that σσ is nondegenerate and hence a symplectic form.

A Riemannian metric is an isomorphism TMTMTM→T∗M, so it can be used to give a natural symplectic structure on the tangent bundle of a Riemannian manifold.

Definition. Let (N,σ)(N,σ) be a symplectic manifold. Given any function fC(N)f∈C∞(N), the Hamilton vector field of ff is the vector field HfHf on NNdefined by

Hf=I1(df)Hf=I−1(df)

where dfdf is the exterior derivative of ff (a 11-form on NN), and II is the isomorphism TNTNTN→T∗N given by the nondegenerate 22-form σσ.

Example 2. In R2nR2n one has I(s,t)=(t,s)I(s,t)=(t,−s)s,tRns,t∈Rn, and

Hf=ξfxxfξ.Hf=∇ξf⋅∇x−∇xf⋅∇ξ.

Definition. Let (N,σ)(N,σ) be a symplectic manifold, and let fC(N)f∈C∞(N). Denote by φtφt the flow on NN induced by HfHf, that is,

φt:ρ(0)ρ(t) where ρ˙(t)=Hf(ρ(t)).φt:ρ(0)↦ρ(t) where ρ˙(t)=Hf(ρ(t)).

Any Hamilton flow map is symplectic ((φt)σ=σ(φt)∗σ=σ) and consequently volume-preserving.

2. The geodesic flow

The geodesic flow on a Riemannian manifold (M,g)(M,g) is a dynamical system on TMT∗M (or TMTM, the two are naturally isomorphic via the Riemann metric). A geodesic is uniquely determined by its initial position and velocity. The tangent bundle TMT∗M is a symplectic manifold, and the geodesic flow can be realized as a Hamilton flow. It is given by the Hamilton function

f:TMR,  f(x,ξ)=12|ξ|2g1=12gjk(x)ξjξk.f:T∗M→R,  f(x,ξ)=12|ξ|g−12=12gjk(x)ξjξk.

The Hamiltonian equation of motion becomes exactly the geodesic equation. One can also see the geodesic flow from the Langrangian point of view, or geometrically via local length minimization.

4. The Sasaki metric

The tangent bundle of a smooth manifold is a smooth manifold (of double dimension). There is a canonical Riemannian metric on the tangent bundle of a Riemannian manifold. This is the Sasaki metric.

The tangent bundle describes possible directions of motion on MM; each point on TMTM contains a point xMx∈M and a vector vTxMv∈TxM. Similarly, TTMTTM describes the directions of motion on TMTM. It is natural to split motion in two components: motion within a fiber (vertically) or motion of the base point only (horizontally). This division is most clear when M=RnM=Rn; then TM=R2nTM=R2n and TTM=R4nTTM=R4n.

For any θ=(x,v)TMθ=(x,v)∈TM, we split TθTM=H(θ)V(θ)TθTM=H(θ)⊕V(θ). Note that if dim(M)=ndim⁡(M)=n, then dim(H(θ))=dim(V(θ))=dim(TxM)=ndim⁡(H(θ))=dim⁡(V(θ))=dim⁡(TxM)=n. It turns out that there are natural isomorphisms H(θ)TxMH(θ)→TxM and V(θ)TxMV(θ)→TxM.

Let π:TMMπ:TM→M be the canonical projection. The vertical fiber is then V(θ)=ker(dθπ)V(θ)=ker⁡(dθπ) — there is no movement in the base.

There is a connection map K:TTMTMK:TTM→TM. A point θTTMθ∈TTM describes (to first order) a curve on TMTM. This is a curve on MM and a vector field along it. The covariant derivative of this vector field along this curve is K(θ)TxMK(θ)∈TxM. The horizontal fiber is then H(θ)=ker(Kθ)H(θ)=ker⁡(Kθ) — there is no movement in the fiber (parallel transport).

The natural isomorphisms are dθπ|H(θ):H(θ)TxMdθπ|H(θ):H(θ)→TxM and Kθ|V(θ):V(θ)TxMKθ|V(θ):V(θ)→TxM. The Sasaki metric is obtained by declaring these to be isometries (inherit the metric from TxMTxM) and H(θ)V(θ)H(θ)⊥V(θ). We can split any vector TTMη=(ηh,ηv)TTM∋η=(ηh,ηv). In this notation

η,ξSasaki=ηh,ξh+ηv,ξv.⟨η,ξ⟩Sasaki=⟨ηh,ξh⟩+⟨ηv,ξv⟩.

6. Coordinate representations of the Sasaki metric

For any coordinates on MM, there are corresponding coordinates on TMTM given by the coordinate functions and their differentials. If xx denotes the coordinates on MM, let (x,y)(x,y) be the corresponding coordinates on TMTM. Let also (x,y,X,Y)(x,y,X,Y) be the corresponding coordinates on TTMTTM.

The vectors {δxj=xjΓljkykyl}nj=1{δxj=∂xj−Γjklyk∂yl}j=1n are a basis for the subspace H(θ)H(θ), where θ=(x0,y0)θ=(x0,y0). The vectors {yk}nk=1{∂yk}k=1n are a basis for the subspace V(θ)V(θ)θ=(x0,y0)θ=(x0,y0).

The operators dπ and KK can be described in these coordinates:

Kθ(δxj)dθπ(δxj)dθπ(yk)Kθ(yk)=0,=xj,=0,=xk.Kθ(δxj)=0,dθπ(δxj)=∂xj,dθπ(∂yk)=0,Kθ(∂yk)=∂xk.

Given vectors ξ,ηTθTMξ,η∈TθTM and writing them in the basis given by δxjδxj and xj∂xj, i.e.

ξ=Xiδxi+Ykyk,η=X~iδxi+Y~kyk,ξ=Xiδxi+Yk∂yk,η=X~iδxi+Y~k∂yk,

we get that

ξ,ηSasaki=Xixi,X~ixi+Ykxk,Y~kxk=gjkXjX~k+gjkYjY~k.
 

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