Invariants over curvature tensor fields
The author presents a construction of higher degree polynomial \(O(n)\)-invariants over the space of curvature tensor fields. Let \({\mathcal C}\) be the space of the curvature tensors on the real \(n\)-dimensional space \(V\), endowed with a dot product \(g\) (the elements of \({\mathcal C}\) are \(4\)-linear forms on \(V\), having the usual properties of the Riemann Christoffel tensor of a Riemannian manifold). The space \({\mathcal C}^*\) of linear forms on \({\mathcal C}\) has a unique (defined up to a constant factor) element invariant under the usual action of \(O(n)\) on \({\mathcal C}\). This is the usual scalar curvature. The author presents an explicit approach to the \(O(n)\)-module structure of \({\mathcal C}^*\). He obtains some replicative properties of the irreducible submodules of \({\mathcal C}^*\). Using these properties, there are obtained some polynomials defined over \({\mathcal C}\), which are \(O(n)\)-invariant. In the cases where \(n=3\) and \(n=4\), there are obtained the generator sets of \(O(n)\)-invariants over \({\mathcal C}\).
classification of curvature tensor fields
scientific article; zbMATH DE number 1203898