Shape tensor and geometry of embedded manifolds

TL;DR

This paper reviews the shape tensor of embedded manifolds, highlighting its role in understanding intrinsic and extrinsic geometry, and introduces shape-minimizing curves that optimize the shape tensor magnitude.

Contribution

It provides a comprehensive review of the shape tensor concept and introduces the novel idea of shape-minimizing curves for geometric optimization.

Findings

Shape tensor unifies intrinsic and extrinsic geometry concepts.

Shape-minimizing curves extend classical geodesics by minimizing shape tensor magnitude.

The approach offers new insights into curvature and parallel transport.

Abstract

We review the notion of shape tensor of an embedded manifold, which efficiently combines intrinsic and extrinsic geometry, and allows for intuitive understanding of some basic concepts of classical differential geometry, such as parallel transport, covariant differentiation, and curvature. We introduce shape-minimizing curves, i.e., curves between given two points that minimize integrated value of the shape tensor magnitude.