On the mean curvature of Nash isometric embeddings

TL;DR

This paper investigates how the geometry of a Riemannian manifold influences the mean curvature vector of its Nash isometric embeddings into Euclidean space, revealing geometric restrictions on the embedding's curvature.

Contribution

It provides new insights into the relationship between the intrinsic geometry of manifolds and the extrinsic mean curvature constraints of their isometric embeddings.

Findings

Geometry imposes restrictions on the mean curvature vector

Nash embeddings can be characterized by curvature properties

New bounds or conditions on mean curvature for embeddings

Abstract

J. Nash proved that the geometry of any Riemannian manifold M imposes no restrictions to be embedded isometrically into a (fixed) ball B_{\mathbb{R}^{N}}(1) of the Euclidean space R^N. However, the geometry of M appears, to some extent, imposing restrictions on the mean curvature vector of the embedding.