Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts

TL;DR

This paper introduces coherent tangent bundles as an intrinsic way to study wave fronts, deriving new Gauss-Bonnet formulas that reveal novel topological and geometric insights.

Contribution

It defines coherent tangent bundles for wave fronts and establishes four Gauss-Bonnet formulas by interchanging fundamental forms, providing new tools for geometric analysis.

Findings

Derived four Gauss-Bonnet formulas for wave fronts.

Discovered new topological and geometric properties of wave fronts.

Revealed the dual role of fundamental forms in wave front geometry.

Abstract

We give a definition of `coherent tangent bundles', which is an intrinsic formulation of wave fronts. In our application of coherent tangent bundles for wave fronts, the first fundamental forms and the third fundamental forms are considered as induced metrics of certain homomorphisms between vector bundles. They satisfy the completely same conditions, and so can reverse roles with each other. For a given wave front of a 2-manifold, there are two Gauss-Bonnet formulas. By exchanging the roles of the fundamental forms, we get two new additional Gauss-Bonnet formulas for the third fundamental form. Surprisingly, these are different from those for the first fundamental form, and using these four formulas, we get several new results on the topology and geometry of wave fronts.