Projective Curvature Tensor of C9-Manifolds

Abstract

This article determines on C₉− manifold only fifteen non-zero components of the projective curvature tensor. It proves a C₉− manifold of dimension > 3 with flat projective curvature tensor is just a cosymplectic manifold with flat Ricci tensor. Whereas, the C₉− manifold of dimension 3 with flat projective curvature tensor is Einstein manifold. On the other hand, it discovers fifteen projective invariant classes that denoted by P₁,P₂,...,P₁₅. It establishes some of these classes are equivalent on C₉− manifold and the C₉− manifold is Einstein manifold if it belongs to the class P1∩P11.

Keywords: C₉−manifold; projective curvature tensor; Einstein manifold.

2010 Mathematics Subject Classification. 53D15; 53C25