Curvature and Tachibana Numbers

TL;DR

This paper introduces the r-th Tachibana number as a measure of conformal Killing forms on Riemannian manifolds and explores its properties as an analogue to Betti numbers.

Contribution

It defines the r-th Tachibana number for Riemannian manifolds and investigates its properties, establishing an analogue to Betti numbers.

Findings

Defined the r-th Tachibana number for conformal Killing forms.

Established properties of Tachibana numbers analogous to Betti numbers.

Provided theoretical insights into conformal geometry of Riemannian manifolds.

Abstract

The purpose of this paper is to define the r-th Tachibana number t(r) of an n-dimentional closed and oriented Riemannian manifold (M,g) as the dimension of the space of all conformal Killing r-forms for r=1,2,...,n-1 and to formulate some properties of these numbers as an analogue to properties of the r-th Betty number b(r) of a closed and oriented Riemannian manifold.