Betti and Tachibana numbers of compact Riemannian manifolds

Abstract

We present definitions and properties of conformal Killing, Killing and planarity forms on a Riemannian manifold and determine Tachibana, Killing and planarity numbers as an analog of the well known Betti numbers. We state some set of conditions to characterize these numbers. Moreover, we formulate the main results on the relationship between the Betti, Tachibana, Killing and planarity numbers.

On the Betti and Tachibana Numbers of Compact Einstein Manifolds

“…The vector space of conformal Killing p-forms on a compact Riemannian manifold (M, g) has finite dimension t p (M ) named the Tachibana number (see e.g. [17,18,19]). Tachibana numbers t 1 (M ), .…”

Comparative Analysis of Spectral Properties of the Hodge–De Rham and Tachibana Operators

“…Moreover, Tachibana numbers possess a duality property similar to the Poincaré duality for Betti numbers. In [39,40], various properties of Tachibana numbers are found. For example, in [40], "lower bounds" for first eigenvalues of the Hodge-de Rham Laplacian and the Tachibana operator on a compact, conformally flat, even-dimensional Riemannian manifold with sign-definite scalar curvature are obtained.…”

“…Later, in [38][39][40], properties of the operator D * D were studied. In particular, it was proved that the kernel of the operator D * D on a compact manifold (M, g) consists of conformal Killing r-forms that constitute a finite-dimensional vector space.…”

“…On the other hand, for an arbitrary conformal Killing r-form ω on the Euclidean sphere S 2r of constant curvature δ > 0, we obtain from (3.3) the relation Δω = r(r + 1)δω. In addition (see [4,40,47]), t r (M ) ≤ t r (S n ) = (n + 2)! (r + 1)!…”

“…In this case, the r-form ω is called an eigenform of the operator corresponding to the eigenvalue μ r (see [40]). …”