All order covariant tubular expansion

TL;DR

This paper derives a comprehensive all-order expansion of Fermi normal coordinates around submanifolds in (pseudo-)Riemannian manifolds, providing explicit formulas for curvature contributions and confirming consistency with existing integral theorems.

Contribution

It generalizes previous Riemann normal coordinate expansions to all orders for Fermi normal coordinates around submanifolds, with explicit closed-form curvature expansion coefficients.

Findings

Derived all-order FNC expansion of vielbein with closed-form coefficients

Confirmed consistency with existing integral theorems for the metric

Extended previous work on Riemann normal coordinate expansion

Abstract

We consider tubular neighborhood of an arbitrary submanifold embedded in a (pseudo-)Riemannian manifold. This can be described by Fermi normal coordinates (FNC) satisfying certain conditions as described by Florides and Synge in \cite{FS}. By generalizing the work of Muller {\it et al} in \cite{muller} on Riemann normal coordinate expansion, we derive all order FNC expansion of vielbein in this neighborhood with closed form expressions for the curvature expansion coefficients. Our result is shown to be consistent with certain integral theorem for the metric proved in \cite{FS}.