Vladimir A. Sharafutdinov
Some questions of integral geometry on Anosov manifolds
(56K, AMSTeX)

ABSTRACT.  A closed Riemannian manifold is called an Anosov manifold if its geodesic flow
is of Anosov type. If $f$ is a smooth function
on an Anosov manifold such that $f$ integrates to zero over every closed
geodesic, then $f$ itself must be zero. The corresponding theorem for one-forms
reads: If $f$ is a smooth one-form on an Anosov manifold which integrates to
zero around every closed geodesic, then $f$ is an exact form. For symmetric
tensor fields of degree $m\geq 2$, we obtain the weaker result: The subspace of
potential fields on an Anosov manifold $M$ has a finite codimension in the
space of symmetric tensor fields that integate to zero over every closed
geodesic. The latter statement is proved under the additional assumption that
the stable and unstable foliations belong to the class $W^1_p$ with some $p>2
dim M-1$.