Invariants over Curvature Tensor Fields

In this paper, we present constructions of higher-order polynomialO(n)-invariants over curvature tensor fields. These invariants are higher-order analogues of the scalar curvature. Our methods are based on certain replicative properties of theO(n)-module structure on a “curvature space” and a realization of a “curvature space” by the action of the symmetric tensor of the Lie algebraso(n) on symmetric matrices. By our methods, we are able to find a complete generator set of functional invariants over the curvature tensor fields of a manifold with lower dimensions.