On the complete decomposition of curvature tensors of Riemannian manifolds with symmetric connection
Let V be a real vector space of dimension n with inner product g and let \({\mathcal R}(V)\) denote the vector space formed by all the algebraic curvature tensors having the same symmetries as the curvature tensor associated to a symmetric connnection on a Riemannian manifold. The main part of the paper consists in deriving complete decompositions of \({\mathcal R}(V)\) under the action of SO(n). The dimensions of the factors are determined and the projections of an element of \({\mathcal R}(V)\) on these factors are derived. The consideration of the norms of these projections leads to (in)equalities for quadratic invariants which may be used in Riemannian geometry to characterize special spaces. Several examples of applications are given with focus on projective curvature tensors and projective transformations and on locally affine symmetric manifolds.
scientific article; zbMATH DE number 4203187
scientific article