M-projectively flat Sasakian manifolds. (English) Zbl 0631.53038

Ojha, R. H.

Indian J. Pure Appl. Math. 17, 481-484 (1986).

Let \(V_ n\) be a Sasakian manifold and M its M-projective curvature tensor. The author proves that \(V_ n\) is M-projectively flat if and only if it is an Einstein manifold (under the assumption that M is a recurrent tensor). The second important result of the paper states that the following three conditions are equivalent on a Sasakian manifold: (i) \(V_ n\) is M-projectively symmetric, (ii) \(V_ n\) is of constant curvature 1, (iii) \(R\circ R=0\) where the curvature transformation R acts on the tensor algebra as a derivation. We remark that a manifold satisfying the third condition is usually called a semi-symmetric space. These manifolds have been widely studied by I. Szabo [J. Differ. Geom. 17, 531-582 (1982; Zbl 0508.53025)].

Reviewer: L.Ornea

MSC:

53C25

Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords:

M-projective curvature tensor; Sasakian manifold; constant curvature; semi-symmetric space