Square-densities, and volume forms

TL;DR

This paper explores the relationship between volume forms and infinitesimal square-volumes of simplices in Riemannian manifolds, extending classical Heron's formula to a differential geometric context.

Contribution

It introduces a novel connection between Heron's formula and volume forms in Riemannian geometry, providing new insights into the structure of infinitesimal simplices.

Findings

Heron's formula can be generalized to relate volume forms and infinitesimal simplices.

A new geometric interpretation of volume forms in Riemannian manifolds.

Potential applications to differential geometry and geometric analysis.

Abstract

Heron's formula from antiquity, for the area of a triangle, is used to relate volume form and infinitesimal square-volume of certain infinitesimal simplices in a Riemannian manifold