Geodesic normal coordinates and natural tensors for pseudo-Riemannian submanifolds

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.