Differential geometry, Palatini gravity and reduction

TL;DR

This paper formulates vacuum Palatini gravity as a general variational principle using geometric tools, relating it to Einstein-Hilbert and other gravity variational problems through a generalized reduction scheme.

Contribution

It introduces a geometric formulation of Palatini gravity as a variational principle and extends Lagrange-Poincaré reduction to connect it with Einstein-Hilbert gravity.

Findings

Relates Palatini gravity to Einstein-Hilbert variational problem

Uses geometric tools from bundle of connections

Generalizes reduction scheme for variational problems

Abstract

The present article deals with a formulation of the so called (vacuum) Palatini gravity as a general variational principle. In order to accomplish this goal, some geometrical tools related to the geometry of the bundle of connections of the frame bundle $LM$ are used. A generalization of Lagrange-Poincar\'e reduction scheme to these types of variational problems allows us to relate it with the Einstein-Hilbert variational problem. Relations with some other variational problems for gravity found in the literature are discussed.