Space-time geometry and the bundle of biframes

Summary

When one considers bivectors on space-time as the basic geometrical objects in place of vectors one is led to the principal bundle of biframesL ² M in place of the linear frame bundleLM.L ² M is thus the natural arena for studying the geometry of bivectors on space-time. In this paper we develop the geometry onL ² M including the concepts of connections, soldering form, curvature, bitorsion, generalized Bianchi identities, and associated bundles. In particular we show that the soldering form onL ² M is a two-form, rather than a one-form as is the soldering form onLM. The corresponding torsion of a given connection one-form onL ² M, or bitorsion, is thus a three-form. Furthermore, we show that the structure equation for the bitorsion of a given connection onL ² M can be recast in a form which is analogous to the field equations which occur in typical non-Abelian (Yang-Mills type) gauge theories. In this analogy, the bitorsion itself plays the role of an external geometrical source for the field equations, while conservation of the total geometrical source is guaranteed by the bitorsion Bianchi identity. Finally, we show that the introduction of a difference form onL ² M, which is the difference between a general connection one-form onL ² M and a connection one-form onL ² M which is induced by a Riemannian connection one-form onLM, provides a fundamental relation between the biframe Ricci tensor and the Riemannian Einstein tensor.