Geodesic normal coordinates and natural tensors for pseudo-Riemannian submanifolds

TL;DR

This paper develops a specialized coordinate system for pseudo-Riemannian submanifolds, expressing metric Taylor coefficients as universal polynomials of curvature and second fundamental form derivatives, and characterizes natural submanifold tensors and operators.

Contribution

It introduces geodesic normal coordinates adapted to submanifolds and provides a universal polynomial expression for metric coefficients, advancing the understanding of natural tensors and operators in pseudo-Riemannian geometry.

Findings

Taylor coefficients of the metric are universal polynomials

Natural submanifold tensors are linear combinations of curvature derivatives

Characterization of natural submanifold differential operators

Abstract

We construct a version of geodesic normal coordinates adapted to a submanifold of a pseudo-Riemannian manifold and show that the Taylor coefficients of the metric in these coordinates can be expressed as universal polynomials in the components of the covariant derivatives of the background curvature tensor and the covariant derivatives of the second fundamental form. We formulate a definition of natural submanifold tensors and show that these are linear combinations of contractions of covariant derivatives of the background curvature tensor and covariant derivatives of the second fundamental form. We also describe how this result gives a similar characterization of natural submanifold differential operators.