First-order invariant Einstein-Cartan variational structures

Abstract

A first-order invariant Einstein-Cartan structure is a Lagrangian structure on a differential manifold defined by a generally invariant Lagrangian depending on a metric field, a connection field, and the first derivatives of these fields. Moreover, it is assumed that the metric and connection fields satisfy the so-called compatibility condition. In this paper the problem of finding all such invariant Einstein-Cartan structures is discussed. It is shown that each Lagrangian of these structures depends only on certain tensors constructed from the metric and the connection fields, which means that all the Lagrangians can be described within the framework of the classical theory of invariants. The maximal number of functionally independent Lagrangians is determined as a function of the dimension of the underlying manifold.