A classification of space-time structures

Abstract

The paper contains a review of various bundles which may be associated to the bundle of linear frames and used to describe properties of space relevant to physics. Restrictions, extensions, prolongations and reductions are defined in terms of morphisms of principal bundles. It is shown that the holonomic prolongation of a G-structure exist iff the corresponding structure function vanishes. G-connections are related to restrictions of the bundle of second-order frames. It is shown that these restrictions may be used to classify theories of space-time and gravitation. A distinction is made between a projective connection and a geodetic structure. In the framework of the Einstein-Cartan theory, the projective connection of a space-time is compatible with its metric tensor iff the spin density is bivector-valued. As an example, we mention a new theory of gravitation and electromagnetism based on the Weyl-Cartan structure of space-time and on the Yang quadratic Lagrangian.