Travelling times in scattering by obstacles in curved space
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 is connected. Suppose there is another codimension-0 submanifold S whose boundary is strictly convex, such that . We will call K an obstacle and S a bounding submanifold. A generalised geodesic in is any unit speed, piecewise smooth extremal of the length functional 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 , which reflects off the boundary ∂K symmetrically across the normal. More precisely still, there are discrete points where γ is not differentiable, and there we have with respect to any tangent v to ∂K. If γ is a geodesic between two points then we say that γ is an -geodesic. We define the set of travelling times of K to be the set of all triples where is the length of an -geodesic. We also call the travelling time of γ.
Let be the unit tangent bundle of and define the quotient by identifying angles with their reflection across the boundary of K, according to (1). Let γ be a generalised geodesic in generated by a point . We say that γ is non-trapped if there are distinct such that . Otherwise, we say that that γ is trapped. Denote the set of all which generate a trapped generalised geodesic by . Also let We will define a generalisation of the geodesic flow as follows. Let be the unique generalised geodesic in defined by the initial conditions Define for each the generalised geodesic flow, by This is the billiard flow as defined in [3] on general Riemannian manifolds. Note that if K is empty then is simply the geodesic flow (see e.g. [4]), which we denote by . Let K and L be two obstacles with the same bounding manifold . K and L are said to have conjugate flows if there exists a homeomorphism Such that and Moreover, let . We say that K and L have almost the same travelling times if for almost all .
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 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 .
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 that are disjoint unions of strictly convex bodies with smooth boundaries are also uniquely recoverable - this was proved in [10] for and in [11] for . 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 the generalised geodesic defined by is not tangent to anywhere.
Lemma 4 Fix . The set of pairs of distinct directions 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 from to . Then , where is the length of the geodesic γ. Proof Suppose is split into geodesic segments for Let be any variation of γ fixing the endpoint, . Reparameterise each so that for all h. Consider the derivative
Negative curvature
The following four results have long technical proofs. For submanifolds of () 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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