(Pseudo)Generalized Kaluza-Klein G-Spaces and Einstein Equations

TL;DR

This paper develops a generalized geometric framework for Kaluza-Klein spaces using Lie algebroids, deriving connections, curvature, and Einstein equations in this new setting.

Contribution

It introduces a Lie algebroid-based generalization of Kaluza-Klein geometry, including new connections, curvature identities, and Einstein equations.

Findings

Formulated the Lie algebroid generalized tangent bundle for Kaluza-Klein spaces.

Derived Cartan and Bianchi identities in this framework.

Established Einstein equations for (pseudo) generalized Kaluza-Klein G-spaces.

Abstract

Introducing the Lie algebroid generalized tangent bundle of a Kaluza-Klein bundle, we develop the theory of general distinguished linear connections for this space. In particular, using the Lie algebroid generalized tangent bundle of the Kaluza-Klein vector bundle, we present the $\left( g,h\right) $-lift of a curve on the base $M$ and we characterize the horizontal and vertical parallelism of the $\left( g,h\right) $-lift of accelerations with respect to a distinguished linear $\left( \rho ,\eta \right) $-connection. Moreover, we study the torsion, curvature and Ricci tensor field associated to a distinguished linear $\left( \rho ,\eta \right) $-connection and we obtain the identities of Cartan and Bianchi type in the general framework of the Lie algebroid generalized tangent bundle of a Kaluza-Klein bundle. Finally, we introduce the theory of (pseudo) generalized Kaluza-Klein G-spaces and we develop the Einstein equations in this general framework.