Behavior of corank one singular points on wave fronts

TL;DR

This paper studies the behavior of cuspidal edges near corank one singular points on wave fronts in three-dimensional space, providing intrinsic formulations and establishing Gauss-Bonnet-type formulas.

Contribution

It introduces an intrinsic approach to analyze corank one singular points on wave fronts and derives Gauss-Bonnet-type formulas for these singularities.

Findings

Characterization of cuspidal edges near corank one singular points

Intrinsic formulation of wave fronts in R^3

Gauss-Bonnet-type formulas for singular points

Abstract

Let $M^2$ be an oriented 2-manifold and $f:M^2\to R^3$ a $C^\infty$-map. A point $p\in M^2$ is called a singular point if $f$ is not an immersion at $p$. The map $f$ is called a front (or wave front), if there exists a unit $C^\infty$-vector field $\nu$ such that the image of each tangent vector $df(X)$ $(X\in TM^2)$ is perpendicular to $\nu$, and the pair $(f,\nu)$ gives an immersion into $R^3\times S^2$. In our previous paper, we gave an intrinsic formulation of wave fronts in $R^3$. In this paper, we shall investigate the behavior of cuspidal edges near corank one singular points and establish Gauss-Bonnet-type formulas under the intrinsic formulation.