Sergio Console and Carlos Olmos

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 [7] 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 [7]). 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 , for any s ≤ k + 1. An alternate proof with a more direct approach was given in [5]. Out of the curvature tensor and its covariant derivatives one can construct scalar invariants, like for instance the scalar curvature. In general, any polynomial function in the components of the curvature tensor and its covariant derivatives which does not depend on the choice of the orthonormal basis at the tangent space of each point is a a scalar Weyl invariant or a scalar curvature invariant. By Weyl theory of invariants, a scalar Weyl invariant is a linear combination of complete traces of tensors 〈∇1R, 〉 . . . 〈∇lR, 〉, (m1, . . .ms ≥ 0, ∇ R = R). Prüfer, Tricerri and Vanhecke studied the interplay among local homogeneity and these curvature invariants. Using Singer’s Theorem, they got the following Theorem 2 (Prüfer, Tricerri and Vanhecke [6]). Let M be an n-dimensional Riemannian manifold. Then M is locally homogeneous if and only if all scalar Weyl invariants of order s with s ≤ n(n−1) 2 are constant. More in general for a non-homogeneous Riemannian manifold one can look at the regular level sets of scalar Weyl invariants. In a recent paper [2], we proved the following theorem, which generalizes the above results 1991 Mathematics Subject Classification. Primary 53C30; Secondary 53C21.