Gauss-Bonnet for Form Curvatures

TL;DR

This paper explores curvatures on simplicial complexes that satisfy Gauss-Bonnet, showing they can be expressed as expectations of indices and deformed via wave dynamics while preserving topological invariants.

Contribution

It introduces a class of curvatures supported on k-dimensional parts of complexes that satisfy Gauss-Bonnet and can be deformed through wave dynamics without losing this property.

Findings

Curvatures can be expressed as expectations of Poincare-Hopf indices.

Wave dynamics can deform these curvatures while preserving Gauss-Bonnet.

The framework applies to both discrete and continuous time systems.

Abstract

We look at curvatures that are supported on k-dimensional parts of a simplicial complex G. These curvature all satisfy the Gauss-Bonnet theorem, provided that the k-dimensional simplices cover $G$. Each of these curvatures can be written as an expectation of Poincare-Hopf indices. Linear or non-linear wave dynamics with discrete or continuous time allow to deform these curvatures while keeping the Gauss-Bonnet property.