A Closed Formula for the Riemann Normal Coordinate Expansion

TL;DR

This paper presents a new integral representation and a closed-form formula for the coefficients in the Riemann normal coordinate expansion, simplifying calculations in differential geometry.

Contribution

It introduces a novel integral representation and explicit formula for Riemann normal coordinate expansion coefficients, advancing mathematical tools in geometric analysis.

Findings

Derived an integral representation for expansion coefficients

Established a closed-form formula for all coefficients

Simplified computations in Riemannian geometry

Abstract

We derive an integral representation which encodes all coefficients of the Riemann normal coordinate expansion, and also a closed formula for those coefficients.