Tachibana, Killing and planarity numbers of compact Riemannian manifolds

TL;DR

This paper explores the definitions and properties of special differential forms on Riemannian manifolds, introduces associated numerical invariants, and investigates their relationships with classical topological invariants.

Contribution

It defines Tachibana, Killing, and planarity numbers as analogs of Betti numbers and establishes conditions and relationships among these invariants.

Findings

Defined Tachibana, Killing, and planarity numbers for Riemannian manifolds

Established conditions characterizing these numbers

Analyzed relationships between these numbers and Betti numbers

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.