Linear Algebra of Curvature Tensors and Their Covariant Derivatives

Abstract

Fix a point on a Riemannian manifold, and consider the tangent space V at the point equipped with inner product g. The Riemann curvature tensor R and its first covariant derivative ∇R at the point are tensors in and . If we take all the known symmetries of these tensors we can define subspaces and such that R ∊ Curv and ∇R ∊ ∇Curv. Also, the orthogonal group O(g) acts naturally on all these spaces. The two fundamental problems of the linear algebra of the spaces Curv and ∇Curv are: (1) find the decomposition into irreducible representations of O(g), with corresponding projection operators, (2) give a description of the structure of the O(g) orbits, by means of orbit invariant functions and a canonical form for elements of each orbit.