Product and anti-Hermitian structures on the tangent space

TL;DR

This paper introduces a new 0-homogeneous lift of a Riemannian metric on a manifold's tangent bundle, leading to a pseudo-Riemannian structure with preserved homogeneity and new geometric properties.

Contribution

It proposes a novel 0-homogeneous lift of the metric, enabling the study of associated almost complex and product structures on the tangent bundle.

Findings

Defined a new pseudo-Riemannian metric on the slit tangent bundle.

Established properties of the natural almost complex and product structures.

Derived new geometric results related to homogeneity preservation.

Abstract

Noting that the complete lift of a Rimannian metric defined on a differentiable manifold is not 0-homogeneous on the fibers of the tangent bundle . In this paper we introduce a new lift which is 0-homogeneous. It determines on slit tangent bundle a pseudo-Riemannian metric, which depends only on the metric . We study some of the geometrical properties of this pseudo-Riemannian space and define the natural almost complex structure and natural almost product structure which preserve the property of homogeneity and find some new results.