Extrinsic Characterisations of Immersions

TL;DR

This paper explores the history and mathematical framework of immersions of Riemannian manifolds, focusing on extrinsic characterizations and the relationships between intrinsic and extrinsic geometric quantities, with implications for understanding space deformation.

Contribution

It reviews historical developments and discusses new perspectives on extrinsic characterizations of immersions, highlighting the potential to describe submanifolds primarily through extrinsic quantities.

Findings

Analysis of the role of Nash's theorem in space immersion

Identification of technical challenges in relating intrinsic and extrinsic invariants

Discussion of the potential for extrinsic quantities to characterize submanifolds

Abstract

We outline the history of the idea of deformation of space, which lead to the concept of curvature invariants, as we understand them today, including contributions of E. Bacaloglu and F. Casorati, among others. We pursue the following question: what is the best way to quantify the deformation of space? This important question could be viewed in a new paradigm after 1956, when John F. Nash, Jr. proved that a Riemannian manifold can be immersed isometrically into an Euclidean ambient space of dimension sufficiently large. This important theorem allowed to view the representation of space from its exterior, from an outside perspective. In 1968, S.-S. Chern pointed out that a key technical element in applying Nash's Theorem effectively is finding useful relationships between intrinsic and extrinsic quantities characterising immersions. And such relations seem to be rather few, at least few enough to present us with a technical challenge in applying Nash's Theorem. One technical difficulty is presented by the passing through the narrow gateway provided by Gauss' equation, and that's why in order to obtain some new results it might be useful to include additional natural conditions. A turning point in the history of the question we pursue was an enlightening paper written by B.-Y. Chen in 1993, which paved the way for a deeper understanding of the meaning of the Riemannian inequalities between intrinsic and extrinsic quantities. Our present discussion invites a reflection on whether we could hope to characterise submanifolds by using mainly extrinsic quantities.