Index of Quasiconformally Symmetric Semi-Riemannian Manifolds

In 1923, Eisenhart 1 gave the condition for the existence of a second-order parallel symmetric tensor in a Riemannian manifold. In 1925, Levy 2 proved that a second-order parallel symmetric nonsingular tensor in a real-space form is always proportional to the Riemannian metric. As an improvement of the result of Levy, Sharma 3 proved that any second-order parallel tensor not necessarily symmetric in a real-space form of dimension greater than 2 is proportional to the Riemannian metric. In 1939, Thomas 4 defined and studied the index of a Riemannian manifold. A set of metric tensors a metric tensor on a differentiable manifold is a symmetric nondegenerate parallel 0, 2 tensor field on the differentiable manifold {H1, . . . ,H } is said to be linearly independent if