Nonuniform Thickness and Weighted Distance

TL;DR

This paper explores nonuniform tubular neighborhoods of curves in Euclidean space using weighted distances, generalizing the normal exponential map, and classifies singularities within almost injectivity radius.

Contribution

It introduces new notions of injectivity radii, generalizes the thickness formula for nonuniform thickness, and classifies singularities using the Horizontal Collapsing Property.

Findings

Different notions of injectivity radii are introduced.

A generalized thickness formula for nonuniform thickness is derived.

Examples demonstrate the failure of standard injectivity radius to be upper semicontinuous.

Abstract

Nonuniform tubular neighborhoods of curves in Euclidean n-space are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtained for nonuniform thickness. All singularities within almost injectivity radius are classified by the Horizontal Collapsing Property. Examples are provided to show the distinction between the different types of injectivity radii, as well as showing that the standard differentiable injectivity radius fails to be upper semicontinuous on a singular set of weight functions.