Specialized Orthonormal Frames and Embedding

TL;DR

This paper explores specialized orthonormal frames in flat frame bundles over indefinite geometries, analyzing their symmetries and embedding properties through differential systems and Cartan characters.

Contribution

It introduces specific linear constraints on connection forms that lead to new insights into isometric and torsion-free embeddings in various gravity theories.

Findings

Closed sets of linear constraints on connection forms identified.

Well-posedness of embedding equations established via Cartan analysis.

Symmetries of embedded geometries characterized through specialized frames.

Abstract

We discuss some specializations of the frames of flat orthonormal frame bundles over geometries of indefinite signature, and the resulting symmetries of families of embedded Riemannian or pseudo-Riemannian geometries. The specializations are closed sets of linear constraints on the connection 1-forms of the framing. The embeddings can be isometric, as in minimal surfaces or Regge-Teitelboim gravity, or torsion-free, as in Einstein vacuum gravity. Involutive exterior differential systems are given, and their Cartan character tables calculated to express the well-posedness of the underlying partial differential embedding and specialization equations.