Lagrangian Multiplier and Riemannian Spaces

Abstract

The Riemannian geometries, which are derivable from a quadratic action principle, are generated by a new mathematical approach, based on the method of the Lagrangian multiplier. This changes the 10 differential equations of fourth order for the $g_{\mathrm{ik}}$ to 24 differential equations of second order for a new field quantity. This field quantity is a tensor of third order, antisymmetric in one pair of indices, in remarkable analogy to the fundamental $\Lambda$-tensor of Einstein's theory of distant parallelism. The field equations are tied together by 13 identities. The basic structure of the field equations for infinitesimal fields is investigated and their relations to Einstein's theory of gravity and Einstein's theory of distant parallelism discussed.