Curvature Invariants

A direct, bundle-theoretic method for defining and extending local isometries out of curvature data is developed. As a by-product, conceptual direct proofs of a classical result of Singer and a recent result of the authors are derived. A classical result of I. M. Singer [11] states that a Riemannian manifold is locally homogeneous if and only if its Riemannian curvature tensor together with its covariant derivatives up to some index k+1 are independent of the point (the integer k is called the Singer invariant). More precisely Theorem 1 (Singer [11]). Let M be a Riemannian manifold. Then M is locally homogeneous if and only if for any p, q ∈ M there is a linear isometry F : TpM → TqM such that F ∇Rq = ∇ Rp ,