Parametric Manifolds II: Intrinsic Approach

TL;DR

This paper introduces a new geometric framework for parametric manifolds, extending classical differential geometry concepts to include a parameter-dependent structure and a novel object called deficiency.

Contribution

It develops an intrinsic approach to parametric manifolds, defining new differentiation operators and introducing the deficiency as a measure of manifold properties.

Findings

Defined generalized Lie, exterior, and covariant derivatives for parametric manifolds.

Introduced the deficiency as a new geometric object analogous to torsion.

Provided conditions under which parametric manifolds can be viewed as families of orthogonal hypersurfaces.

Abstract

A parametric manifold is a manifold on which all tensor fields depend on an additional parameter, such as time, together with a parametric structure, namely a given (parametric) 1-form field. Such a manifold admits natural generalizations of Lie differentiation, exterior differentiation, and covariant differentiation, all based on a nonstandard action of vector fields on functions. There is a new geometric object, called the deficiency, which behaves much like torsion, and which measures whether a parametric manifold can be viewed as a 1-parameter family of orthogonal hypersurfaces.