Tachibana operator

Background. The article considers the Hodge - De Rham laplacian and the Tachiban operator, functioning on differential forms of the compact Riemannian manifold. When the study of eigenvalues and properties, in general, of the first operator may be refered to as the classics of Riemannian geometry, the second operator has been introduced relatively recently by the first author. This operator is an elliptic one, and therefore on a compact manifold its kernel, consisting of conformal Killing forms, has a finite dimensionality, namd as the Tachibana number, similar to the Betti number, that equals ot the dimensionality of harmonic form space, forming the kernel of the Hodge - De Rham laplacian. Previously the authors have determined the properties of Tachibana numbers and relation thereof to Betti numbers of the compact Riemannian manifold. Particularly, the authors obtained “lower boundaries” for the first eigenvalues of the Hodge - De Rham laplacian and the Tachibana operator on the compact conformal plane Reimannian manifold of even dimensionality with fixed-sign scalar curvature. The research is aimed at acquisition of necessary and sufficient conditions that characterize harmonic, closed and coclosed conformal Killing forms using the Tachibana operator, as well as discovering the first eigenvalues of the Hodge - De Rham laplacian and the Tachibana operator on the Riemannian manifolds of constant curvature and determining the order thereof. Materials and methods. The object of research is an insufficiently studied elliptic differential operator of the second order, functioning on differential forms of the compact Riemannian manifold. The authors use the methods of classical tensor geometry and theory of differential operators on manifolds. Results. In the present article, using the Tachibana operator, the authors obtained the necessary and sufficient conditions that characterize harmonic, closed and coclosed conformal Killing forms, which generalize the already known characteristic thereof, obtained by K. Yano, as well as discovered the first eigenvalues of the Hodge - De Rham laplacian and the Tachibana operator on the Riemannian manifolds of constant curvature and determined the order thereof.