On the synectic metric in the tangent bundle of a Riemannian manifold

TL;DR

This paper explores the properties of covariant derivatives of covector and Killing vector fields under the synectic lift in the tangent bundle of a Riemannian manifold, focusing on the Cg-complete and Va-vertical lifts.

Contribution

It introduces new insights into the behavior of covariant derivatives under the synectic lift in tangent bundles of Riemannian manifolds.

Findings

Analysis of covariant derivatives of covector fields under synectic lift

Investigation of Killing vector fields in tangent bundles

Characterization of the Cg-complete and Va-vertical lifts

Abstract

The purpose of this paper is to investigate applications the covariant derivatives of the covector fields and killing vector fields with respect to the synectic lift a in a the Riemannian manifold to its tangent bundle, where Cg-complete lift of the Riemannian metric, Va-vertical lift of the symmetric tens field of type (0,2) in Mn .