Intrinsic properties of surfaces with singularities

TL;DR

This paper introduces two classes of positive semi-definite metrics on 2-manifolds, characterizes their intrinsic invariants related to singularities, and establishes Gauss-Bonnet type formulas for these metrics.

Contribution

It defines and distinguishes Kossowski and Whitney metrics, characterizes their intrinsic invariants, and proves Gauss-Bonnet formulas for each class on compact 2-manifolds.

Findings

Kossowski metrics correspond to wave fronts with cuspidal edges and swallowtails.

Whitney metrics relate to surfaces with cross cap singularities.

Gauss-Bonnet formulas are established for both metric classes.

Abstract

In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in $\boldsymbol{R}^3$ are Kossowski metrics, and the pull-back metrics of surfaces consisting only of cross cap singularities are Whitney metrics. Since the singular sets of Kossowski metrics are the union of regular curves on the domains of definitions, and Whitney metrics admit only isolated singularities, these two classes of metrics are disjoint. In this paper, we give several characterizations of intrinsic invariants of cuspidal edges and cross caps in these classes of metrics. Moreover, we prove Gauss-Bonnet type formulas for Kossowski metrics and for Whitney metrics on compact 2-manifolds.