presented by Vertgeim L. B.

Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of
Sciences, 630090, Novosibirsk, Russia.

In the report integral geometry and reconstruction problems are studied for operators, acting in a finite-dimensional hermitian vector bundle over a compact riemannian manifold. A similiar reconstruction problem for a connection on a bundle by known parallel transport operators between boundary points is considered in the V. A. Sharafutdinov's paper [2]. One of the basic tool is some new more general tensor calculus, developed there.

1. In the integral geometry problem the unknown is an operator A, acting in a finite-dimensional hermitian vector bundle over a compact riemannian manifold M with the boundary. One has to reconstruct it by the known integrals with given weights along all the geodesics, joining pairs of the boundary points. Under certain restrictions on the weights and curvatures of the manifold and a connection on the bundle an uniqueness and stability theorem is proven.

2. In the nonlinear problem an unknown operator A is to be determined by the boundary values of the fundamental matrix of a system of differential equations, related to A, along geodesics, joining pairs of the boundary points. Under a priori smallness estimates for A an uniqueness and stability theorem is proven.

References

1. Vertgeim L. B.  Integral geometry with a matrix weight and a nonlinear problem of recovering matrices // Reports of the Sov. Acad. of Sciences. 1991. Vol. 319, N 3. P. 531-534. ( Engl. transl. Sov. Math. Dokl. 44, N 1, 132-135 (1992) ).

2. Sharafutdinov V. A.  On an inverse problem of determining a connection on a vector bundle // Max-Planck Institut fur Mathematik. Preprint Series. 1997 (109).