Geodesic normal coordinates and natural tensors for pseudo-Riemannian submanifolds

2024/11/14 by C. Robin Graham, Graham, C Robin, Tzu-Mo Kuo +1
Mathematics · Medicine · Physics and Astronomy · #53A55 (Primary) 53A07 #53B25 #53B30 (Secondary) #Advanced Differential Geometry Research #Bone health and osteoporosis research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2411.09679

openalex publication_date 2024/11/14 · openalex created_date 2024/11/17 · openalex updated_date 2026/07/28

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.

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