Parametric Manifolds I: Extrinsic Approach

TL;DR

This paper generalizes the Gauss-Codazzi formalism for parametric manifolds by introducing the concept of deficiency to handle non-hypersurface orthogonal foliations, with applications in general relativity.

Contribution

It introduces a new formalism for parametric manifolds that extends classical methods to non-hypersurface orthogonal cases using the notion of deficiency.

Findings

Generalization of Gauss-Codazzi formalism to non-orthogonal foliations

Introduction of deficiency as a measure of non-surface-forming tangent spaces

Application to initial value problems in general relativity

Abstract

A parametric manifold can be viewed as the manifold of orbits of a (regular) foliation of a manifold by means of a family of curves. If the foliation is hypersurface orthogonal, the parametric manifold is equivalent to the 1-parameter family of hypersurfaces orthogonal to the curves, each of which inherits a metric and connection from the original manifold via orthogonal projections; this is the well-known Gauss-Codazzi formalism. We generalize this formalism to the case where the foliation is not hypersurface orthogonal. Crucial to this generalization is the notion of deficiency, which measures the failure of the orthogonal tangent spaces to be surface-forming, and which behaves very much like torsion. Some applications to initial value problems in general relativity will be briefly discussed.