Multi-step Fermi normal coordinates

TL;DR

This paper extends Fermi normal coordinates to multiple subspaces, allowing for multi-geodesic coordinate systems, and derives the connection and metric in terms of the Riemann tensor, generalizing previous results.

Contribution

It introduces a generalized multi-step Fermi normal coordinate system for manifolds with decomposed tangent spaces, expanding the applicability of Fermi coordinates.

Findings

Derived expressions for connection and metric as Riemann tensor integrals.

Recovered known results for Riemann normal coordinates and two-subspace cases.

Provided a new framework for multi-geodesic coordinate systems.

Abstract

We generalize the concept of Fermi normal coordinates adapted to a geodesic to the case where the tangent space to the manifold at the base point is decomposed into a direct product of an arbitrary number of subspaces, so that we follow several geodesics in turn to find the point with given coordinates. We compute the connection and the metric as integrals of the Riemann tensor. In the case of one subspace (Riemann normal coordinates) or two subspaces, we recover some results previously found by Nesterov, using somewhat different techniques.