On the triviality of the generalized tangent bundle

TL;DR

This paper investigates the conditions under which the generalized tangent bundle of a manifold is trivial, establishing that it is trivial for parallelizable manifolds but not necessarily conversely, and explores related geometric structures.

Contribution

It proves the triviality of the generalized tangent bundle for parallelizable manifolds and demonstrates that the converse does not hold, with applications to various geometric structures.

Findings

Generalized tangent bundle of a parallelizable manifold is trivial.

The converse implication (triviality implies parallelizability) does not hold.

Connections between triviality and generalized geometric structures are established.

Abstract

We study the relations between the triviality of the tangent bundle $TM$ and the generalized tangent bundle $\mathbb{T}M = TM\oplus T^*M$ of a manifold. We show that the generalized tangent bundle of a paralellizable manifold is trivial. We also prove that the converse implication does not hold, by studying the cases of the M\"obius strip, spheres and projective spaces. Finally, we relate the triviality of the generalized tangent bundle to generalized geometric structures.