Metrical Distortion, Exterior Differential and Gauss's Lemma

TL;DR

This paper revises Gauss's Lemma by introducing the concept of metrical distortion, explores exterior differentials via covariant gradient transport, and exemplifies the theory with the geometry of the 2-sphere.

Contribution

It introduces the concept of metrical distortion as a non-identity point set association and develops a concrete exterior differential framework using covariant gradient transport.

Findings

Metrical distortion is determined by geodesically radial volume preservation.

Exterior differential is concretely defined through covariant gradient transport.

The theory is exemplified by the exterior geometry of the 2-sphere.

Abstract

Gauss's Lemma is revised by showing that the point set association of the double tangential space with the tangential space of a Riemannian manifold is not the identity. The latter point set association is called a metrical distortion, an isometry that actually induces the geometry. The defnition of the exterior differential, which linearizes a mapping that points on the top of a Riemannian manifold, is worked out concretely by covariant gradient transport, what includes a differential slip in contrast to inner differentials. A differential slip is a scalar gauge theory that considers the reparametrization of different length scales. The metrical distortion is determined by geodesically radial volume preservation, whereas the Riemannian exponential mapping is determined by geodesically radial length preservation. The theory is exemplifed by the exterior geometry of the 2-sphere.