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We define the Dirichlet to Neumann operator on exterior differential forms for a compact Riemannian manifold with boundary and prove that the real additive cohomology structure of the manifold is determined by the DN operator. In particular, an explicit formula is obtained which expresses Betti numbers of the manifold through the DN operator. We express also the Hilbert transform through the DN map. The Hilbert transform connects boundary traces of conjugate co-closed forms.

Abstract. A vector field on a Riemannian manifold is called conformal Killing if it generates one-parameter group of conformal transformation. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric and physical problems. We prove the statement: A trace-free conformal Killing tensor field is identically zero if it vanishes on some hypersurface. This statement is a basis of the theorem on decomposition of a symmetric tensor field on a compact manifold with boundary to a sum of three fields of special types. We also establish triviality of the space of trace-free conformal Killing tensor fields on some closed manifolds.

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