The modified horizontal derivative and some of its applications

Abstract

In [1] R. G. Mukhometov published a proof of the following result which remains the strongest among those known in integral geometry: a simple Riemannian metric on a compact manifold with boundary is uniquely determined in a prescribed conformal class by the distances between boundary points (a metric is called simple, if a unique geodesic joints each pair of points). The proof is based on some differential identity for the kinetic equation on a Riemannian manifold. Using the identity, R. G. Mukhometov “in passing” obtained the following two strong results: he proved uniqueness for a solution to the linear problem of integral geometry and found a formula that expresses the volume of a Riemannian manifold in terms of the distances between boundary points. Unfortunately, the formula is given in [1] in regretfully noninvariant form. Closed results were obtained by I. M. Bernstĕin and M. L. Gerver [2]. Mukhometov’s identity has a cumbersome formulation and very difficult proof. The present article was originally intended as purely methodological: to find an exposition of Mukhometov’s results appropriate for their inclusion into the author’s lecture course on integral geometry. As it often happens in such matters, the research goes beyond the preliminary framework, leading to the results of interest in their own right. First, the stability estimate that we obtained in the nonlinear problem of determining a metric differs from the similar estimate by Mukhometov; the differences can occur essential for some questions. Second, our formula for the volume of a Riemannian manifold has an invariant form and depends linearly on the boundary distances; this circumstance turned out to be unexpected for the author. In the present article, it is Mukhometov’s results on the linear problem of integral geometry that undergo the broadest generalization. To clarify the nature of these generalizations, we recall that the integral geometry problem is equivalent to the inverse problem of determining a source in the stationary kinetic equation. The latter equation has a simple physical meaning: it describes the distribution of particles (or a radiation) moving along certain trajectories (in our case along the geodesics of a Riemannian metric) and not interacting with each other and with a medium. If we wish to take account of interaction of particles with the medium, then we have to insert extra summands into the equation. The simplest of such summands describes attenuation of particles by the medium. From the standpoint of integral geometry, the taking of attenuation into account leads to the fact that, in the main integral operator, there appears a weight factor depending exponentially