98-412 Vladimir A. Sharafutdinov

Some questions of integral geometry on Anosov manifolds (56K, AMSTeX) Jun 3, 98

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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 $p>2 dim M-1$.

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