Overview and comparative analysis of the properties of the Hodge-de Rham and Tachibana operators

Abstract

In the present paper we consider two natural, elliptic, self-adjoint second order dierential
operators acting on exterior dierential forms on Riemannian manifolds. These operators are the wellknown
Hodge-de Rham and little-known Tachibana operators. Basic properties of these operators are very
similar, or vice versa are dual with respect to each other. We review the results (partly obtained by the
authors) on the geometry of these operators and demonstrate the comparative analysis of their properties