General Construction of Tubular Geometry

TL;DR

This paper develops a general method to describe tubular geometry around submanifolds in (pseudo)Riemannian spaces, providing explicit expansion coefficients and verifying higher-order terms, including spin connection contributions.

Contribution

It introduces a comprehensive approach to compute tubular expansion coefficients in arbitrary coordinates, extending previous methods and verifying higher-order geometric terms.

Findings

Explicit formulas for tubular expansion coefficients in general settings

Verification of the first non-trivial spin connection term in the expansion

Application of the method to loop space tubular geometry

Abstract

We consider the problem of locally describing tubular geometry around a submanifold embedded in a (pseudo)Riemannian manifold in its general form. Given the geometry of ambient space in an arbitrary coordinate system and equations determining the submanifold in the same system, we compute the tubular expansion coefficients in terms of this {\it a priori data}. This is done by using an indirect method that crucially applies the tubular expansion theorem for vielbein previously derived. With an explicit construction involving the relevant coordinate and non-coordinate frames we verify consistency of the whole method up to quadratic order in vielbein expansion. Furthermore, we perform certain (long and tedious) higher order computation which verifies the first non-trivial spin connection term in the expansion for the first time. Earlier a similar method was used to compute tubular geometry in loop space. We explain this work in the light of our general construction.