Wrinkled Embeddings

TL;DR

This paper introduces the concept of wrinkled embeddings, a topological embedding with controlled singularities, and proves that any rotation of the tangent plane field of a smooth submanifold can be approximated by homotopies of these wrinkled embeddings.

Contribution

It establishes that any tangent plane rotation of a smooth submanifold can be approximated by homotopies of wrinkled embeddings, extending the flexibility of embeddings with controlled singularities.

Findings

Any rotation of the tangent plane field can be approximated by wrinkled embeddings.

Wrinkled embeddings have cuspidal corners on a set of (n-1)-dimensional spheres.

The method extends the classical embedding theory to include controlled singularities.

Abstract

A {\it wrinkled embedding} $f:V^n\to W^m$ is a topological embedding which is a smooth embedding everywhere on $V$ except a set of $(n-1)$-dimensional spheres, where $f$ has cuspidal corners. In this paper we prove that any rotation of the tangent plane field $TV\subset TW$ of a {\it smoothly embedded} submanifold $V\subset W$ can be approximated by a homotopy of {\it wrinkled embeddings} $V\to W$.