Numerical calculation method for function integration on submanifolds of $\mathbb{R}^n$ or compact Riemannian manifolds

TL;DR

This paper introduces a numerical method for calculating integrals of functions over submanifolds and hypersurfaces in Euclidean and Riemannian manifolds, enabling digital representation of volume elements.

Contribution

The paper develops a novel numerical approach for function integration on submanifolds in Euclidean and Riemannian settings, extending existing methods.

Findings

Effective digital representation of volume elements

Accurate numerical integration on submanifolds

Extension to Riemannian manifolds

Abstract

In this paper, we present a method for digitally representing the "volume element" and calculating the integral of a function on compact hypersurfaces with or without boundary, and low-dimensional submanifolds in $\mathbb{R}^n$. We also extend such calculation to hypersurfaces in compact Riemannnian manifolds.