An $1$-differentiable cohomology induced by a vector field

TL;DR

This paper introduces a new cohomology theory induced by a vector field on manifolds, linking it to classical de Rham cohomology and exploring its properties in complex and holomorphic contexts, with applications to harmonic forms.

Contribution

It defines a novel 1-differentiable cohomology induced by vector fields and establishes its relationship with classical cohomologies, including applications to harmonicity.

Findings

Established a link between the new cohomology and de Rham cohomology.

Analyzed the cohomology in complex and holomorphic cases.

Applied the theory to study harmonicity of 1-differentiable forms.

Abstract

A new cohomology, induced by a vector field, is defined on pairs of differential forms ($1$--differentiable forms) in a manifold. It is proved a link with the classical de Rham cohomology and an $1$-differentable cohomology of Lichnerowicz type associated to an one form. Also, the case when the manifold is complex and the vector field is holomorphic is studied. Finally, an application of this theory to the harmonicity of $1$-differentiable forms is studied in a particular case.