A New Approach to Euler Calculus for Continuous Integrands

TL;DR

This paper proposes extending Euler calculus to continuous functions by integrating with respect to Gaussian curvature, enabling new applications of differential geometry in data analysis.

Contribution

It introduces a curvature-based extension of Euler calculus that satisfies a Fubini theorem and extends to a functor, broadening its applicability.

Findings

Defines a curvature calculus extending Euler calculus to continuous integrands.

Establishes a Fubini theorem for the new calculus.

Suggests potential applications in data analysis using differential geometry.

Abstract

Euler calculus is based on integrating simple functions with respect to the Euler characteristic. This paper makes the case for extending Euler calculus to continuous integrands by integrating with respect to (Gaussian) curvature. This requires a metric but is nevertheless defined within any O-minimal theory. It satisfies a Fubini theorem and extends to a functor. Euler calculus is the "adiabatic limit" of this "curvature calculus". All this suggests new applications of differential geometry to data analysis.