School of Mathematical Sciences
Monday, April 16, 2012
Schreiber 006, 12:15
Tel Aviv University
Inverse kinematic problem and rigidity of Riemannian metrics
We are given a compact domain with boundary (e. g. a ball) with an
unknown conformal Euclidean metric. The problem is to reconstruct the metric
from knowledge of lengths of all closed geodesics in the domain. This problem
was formulated more than hundred years ago in geophysics, where the length of a
geodesic is called travel-time. An explicit reconstruction of an isotropic velocity
field from travel-times was found by G. Herglotz for the spheric Earth model
with velocity depending only on depth.
Much later, in the seventies progress in theoretical geophysics (Novosibirsk
group, Bernstein, Gerver, Beylkin) gave rise a problem of Riemannian geometry:
in which extent a Riemannian metric on a manifold with boundary can be
determined from its travel-time function? Finally an answer was found for
metrics within a given conformal class.
If the conformal class is not known then a unique determination of a metric is
not possible because of shortage of information. A natural question is whether a
Riemannian metric in a manifold is rigid with respect to lengths of all closed
geodesics (Michel, Gromov, Pestov, Uhlmann,...). The rigidity problem is still
far from a complete solution.
More tags: Santalo's volume formula, "Goursat theorem" for a geodesic
geometry, integral geometry.
Coffee will be served at 12:00 before the lecture
at Schreiber building 006