Travelling times in scattering by obstacles in curved space

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Abstract

We consider travelling times of billiard trajectories in the exterior of an obstacle K on a two-dimensional Riemannian manifold M. We prove that given two obstacles with almost the same travelling times, the generalised geodesic flows on the non-trapping parts of their respective phase-spaces will have a time-preserving conjugacy. Moreover, if M has non-positive sectional curvature, we prove that if K and L are two obstacles with strictly convex boundaries and almost the same travelling times then K and L are identical.

Introduction

Let M be a geodesically complete, 2-dimensional Riemannian manifold and let K be a smooth codimension-0 submanifold of M with boundary, such that M\K is connected. Suppose there is another codimension-0 submanifold S whose boundary is strictly convex, such that KS. We will call K an obstacle and S a bounding submanifold. A generalised geodesic γ:[a,b]SK in SK=S\K is any unit speed, piecewise smooth extremal of the length functionalab||γ˙||dt, with respect to variations of paths fixing the endpoints ([1], [2]). That is, γ is a piecewise smooth curve made up of smooth geodesic segments in SK, which reflects off the boundary ∂K symmetrically across the normal. More precisely still, there are discrete points t1,t2, where γ is not differentiable, and there we haveγ˙(ti),v=γ˙(ti+),v with respect to any tangent v to ∂K. If γ is a geodesic between two points x,yS then we say that γ is an (x,y)-geodesic. We define the set of travelling times TK of K to be the set of all triples (x,y,tγ) where tγ is the length of an (x,y)-geodesic. We also call tγ the travelling time of γ.

Let T1SK be the unit tangent bundle of SK and define the quotient T˚1SK by identifying angles with their reflection across the boundary of K, according to (1). Let γ be a generalised geodesic in SK generated by a point (x,ω)T1SK. We say that γ is non-trapped if there are distinct t0,t0R such that γ(t0),γ(t0)S. Otherwise, we say that that γ is trapped. Denote the set of all (x,ω)T1SK which generate a trapped generalised geodesic by Trap(SK). Also letTrap(SK)={(x,ω)Trap(SK):xS} We will define a generalisation of the geodesic flow as follows. Let γ(x,ω) be the unique generalised geodesic in SK defined by the initial conditionsγ(x,ω)(0)=xγ˙(x,ω)(0)=ω Define for each tR the generalised geodesic flow, Ft:T˚1SKT˚1SK byFt(x,ω)=(γ(x,ω)(t),γ˙(x,ω)(t)) This is the billiard flow as defined in [3] on general Riemannian manifolds. Note that if K is empty then Ft is simply the geodesic flow (see e.g. [4]), which we denote by ϕt:T1SKT1SK. Let K and L be two obstacles with the same bounding manifold SM. K and L are said to have conjugate flows if there exists a homeomorphismΦ:T˚1SK\Trap(SK)T˚1SL\Trap(SL) Such that Φ|TS=Id andFt(L)Φ=ΦFt(K) for all tR Moreover, let TK(x,y)={t[0,):(x,y,t)TK}. We say that K and L have almost the same travelling times if TK(x,y)=TL(x,y) for almost all (x,y)S×S.

We are now ready to state the two main results of this paper.

Theorem 1 Conjugacy Theorem

Two obstacles K and L with the same bounding manifold SM have conjugate flows if and only if they have almost the same travelling times.

Theorem 2 Uniqueness of Convex Obstacles

Suppose that M has non-positive sectional curvature. If K and L are two (disjoint unions of) strictly convex obstacles with almost the same travelling times in M, then K=L.

Inverse problems related to metric rigidity have been studied for a very long time in Riemannian geometry - see e.g. [5] and the references there for more information. A different kind of problems studied extensively recently for various types of dynamical systems concern the so called Marked Length Spectrum, defined as the set of all lengths of periodic orbits in phase space together with their marking - see [6], [7] and the references there for more information. Various similar problems have been considered in scattering by obstacles in Euclidean spaces in the last 20 years. A natural and rather important problem in inverse scattering by obstacles in Rn is to get information about the obstacle K from its so-called scattering length spectrum, which is in a certain way related to travelling times of billiard (and more general) trajectories in the exterior of the obstacle - see [8] for details. An analogue of the Conjugacy Theorem (Theorem 1) above was first established in [8] in this context. For travelling times of trajectories in the exterior of obstacles in Rn, such a theorem was established in [9]. In both [8] and [9] the Conjugacy Theorem was used to recover geometric information about the obstacle from travelling times.

It turned out that some kinds of obstacles are uniquely recoverable from their travelling times (and also from their scattering length spectra), e.g. star-shaped obstacles are in this class, as shown in [8]. Obstacles in Rn that are disjoint unions of strictly convex bodies with smooth boundaries are also uniquely recoverable - this was proved in [10] for n3 and in [11] for n=2. In [12] a certain generalisation was established of the well-known Santalo's Formula in Riemannian geometry. As a consequence, it was shown in [12] that, assuming the set of trapped points has Lebesgue measure zero, one can recover for example the volume of the obstacle from travelling times.

It should be remarked that in general, the set of trapped points could be rather large. As an example of M. Livshits shows (see e.g. Figure 1 in [10] or in [12]), in some cases the set of trapped points contains a non-trivial open set, and then the obstacle cannot be recovered from travelling times. Theorem 2 above establishes a result similar to the one in [11], although the situation considered in this paper is significantly more complicated.

This paper is separated into a preliminary section, three main sections and an appendix. In Sections 3 and 4, we prove Theorem 1, Theorem 2 respectively. While in Section 5 we give proofs for three technical propositions which are of fundamental importance to proving Theorem 2. Throughout the paper we draw on arguments from [9] and [11], although in general we either adapt and extend the arguments to the more complicated case of Riemannian 2-manifolds or provide completely new proofs.

Section snippets

Preliminaries

We now state some results which will be useful in proving Theorem 1. The following result is well known, see [13] for a proof.

Lemma 3

For almost all (x,ω)TS\Trap(SK) the generalised geodesic defined by γ(t)=Ft(x,ω) is not tangent to SK anywhere.

Lemma 4

Fix x0S. The set of pairs of distinct directions ω1,ω2TSx0 which generate generalised geodesics with the same endpoint and the same travelling time is countable.

Proof

See Appendix A. 

The proof of the following fact would use the exact same argument as

The Conjugacy Theorem

In this section we first prove two useful results, Lemma 7 and Theorem 8. Then we finally give a proof of Theorem 1.

Lemma 7

Suppose that γ is a non-trapped generalised geodesic in SK from xS to yS. Then gradxT=γ˙(t0)/||γ˙(t0)||, where T(x,y) is the length of the geodesic γ.

Proof

Suppose γ:[a,b]SK is split into geodesic segments γ|[ti1,ti] fora=t0<t1<<tn1<tn=b. Let γh be any variation of γ fixing the endpoint, γh(tn)=γ(tn)=y. Reparameterise each γh so that tn(h)=b for all h. Consider the derivative

Negative curvature

The following four results have long technical proofs. For submanifolds of Rn (n2) an analogue of Proposition 9 was proved in [15] (see Lemma 5.2 there). However in the case of Riemannian 2-manifolds considered here the proof is more complicated. It should be stressed that this proposition is of fundamental importance for the proof of Theorem 2. For dispersing billiards in Euclidean spaces Proposition 10, Proposition 11 have been well-known and widely used for a very long time - see the

Proofs of Propositions 9, 10 and 11

The common difficulty shared by the proofs in this section is exhibiting the curvature of the convex front, whether it be from the influence of the manifolds intrinsic negative sectional curvature as in Proposition 10, Proposition 11, or simply by construction as in Proposition 9. Although Proposition 10 is done in a completely coordinate-free way, in both Proposition 9, Proposition 11 we use a Riemannian coordinate chart to allow us to use a Taylor expansion of the geodesics in question.

Proof of Proposition 9

Define

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