Sergey Stepanov

BETTI AND TACHIBANA NUMBERS

The Tachibana numbers tr(M), the Killing numbers kr(M), and the planarity numbers pr(M) are considered as the dimensions of the vector spaces of, respectively, all, coclosed, and closed conformal Killing r-forms with 1 ≤ r ≤ n − 1 “globally” defined on a compact Riemannian n-manifold (M,g), n >- 2. Their relationship with the Betti numbers br(M) is investigated. In particular, it is proved that if br(M) = 0, then the corresponding Tachibana number has the form tr(M) = kr(M) + pr(M) for tr(M) > kr(M) > 0. In the special case where b1(M) = 0 and t1(M) > k1(M) > 0, the manifold (M,g) is conformally diffeomorphic to the Euclidean sphere.

Harmonic transforms of complete Riemannian manifolds

Vanishing theorems for harmonic and infinitesimal harmonic transformations of complete Riemannian manifolds are proved. The proof uses well-known Liouville theorems on subharmonic functions on noncompact complete Riemannian manifolds.

On Dimensions of Vector Spaces of Conformal Killing Forms

In this article there are found precise upper bounds of dimension of vector spaces of conformal Killing forms, closed and coclosed conformal Killing (r)-forms ((1,{le }, r,{le }, n {-} 1)) on an (n)-dimensional manifold. It is proved that, in the case of (n)-dimensional closed Riemannian manifold, the vector space of conformal Killing (r)-forms is an orthogonal sum of the subspace of Killing forms and of the subspace of exact conformal Killing (r)-forms. This is a generalization of related local result of Tachibana and Kashiwada on pointwise decomposition of conformal Killing (r)-forms on a Riemannian manifold with constant curvature. It is shown that the following well known proposition may be derived as a consequence of our result: any closed Riemannian manifold having zero Betti number and admitting group of conformal mappings, which is non equal to the group of motions, is conformal equivalent to a hypersphere of Euclidean space.

On Space-like Hypersurfaces in a Space-time

In the present paper we study the global geometry of convex, totally umbilical and maximal space-like hypersurfaces in space-times and, in particular, in de Sitter space-times.

Seven invariant classes of the Einstein equations and projective mappings

By using the representation theory of groups, we define seven classes of the Einstein equations of the General Relativity Theory. Then we use this result for a more detailed study of the Einstein equations.