Principal curvatures and parallel surfaces of wave fronts

TL;DR

This paper establishes criteria for principal curvatures to be smooth at singular points of wave fronts, and explores their implications for parallel surfaces and related geometric invariants.

Contribution

It introduces new criteria for smoothness of principal curvatures at singularities and links these to the behavior of parallel surfaces and geometric invariants.

Findings

Criteria for bounded principal curvatures at singular points

Analysis of singularities of parallel surfaces

Relations between singularities and geometric invariants

Abstract

We give criteria for which a principal curvature becomes a bounded $C^\infty$-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularities to some geometric invariants of fronts.