Curvature tensions in sasakian, p-sasakian, lp-sasakian and b-einstein sasakian manifolds

Abstract

In chapter 1, the preliminaries and definitions are introduced. The notion on manifolds, differentiable manifolds, tensor and vector fields, connections and complex manifolds and curvature tensors are introduced. The spaces to be studied namely, Sasaki an, Para- Sasakian, LP-Sasakian and 77-Einstein Sasakian are defined. The literature review is also included in this chapter. In Chapter 2, properties and representations of the W3 and W5 curvature tensor are studied in a Sasakian Manifold. The results obtained include the representation of the W5 curvature tensor in a W5-symmetric Sasakian manifold. It is also proved that W3-fiat Sasakian manifold is and TJ- Einstein Manifold and the representation of the Riemann curvature tensor in such a manifold is obtained. Expressions of the Ricci tensor and the Riemann curvature tensor in a W5 fiat sasakian manifold are derived. It is further shown that a W5-fiat Einstein Sasakian manifold is a fiat manifold. In Chapter 3, properties and representations of the W3 and W5 curvature tensor are studied in a Para-Sasakian Manifold. Characterisations of the W5 and the W3 curvature tensors under various conditions are derived. It is shown that W3-fiat P-Sasakian manifold is 77-Einstein and an expression of the Riemann curvature tensor is obtained. It is shown that W5-fiat P-Sasakian manifold is an Einstein manifold and a corresponding expression of the Riemann curvature tensor is obtained. The Ricci tensor is also considered in a PSasakian manifold satisfying. W5 . S = O. An expression for the W5 curvature tensor in a P-Sasakian manifold satisfying W5 (C X) . W5 = 0 is derived and it is further proved that such a manifold is Ricci fiat. It is also shown that a P- Sasakian manifold satisfying W5 (~, X) . S = 0 or W5 (~, X) . R = 0 is an TJ-Einstein Manifold. In chapter 4, we study Lorentzian Para Sasakian manifolds that are ¢- W3 fiat, ¢- W5 fiat, W3-fiat and W5-fiat. It is shown that LP-Sasakian manifolds that are ¢ - W3 fiat, ¢ - W5 fiat or W3-fiat are TJ-Einstein. An expression for the Riemann curvature tensor and the Ricci tensor in a W5-fiat LP-Sasakian manifold is derived and further more, it is proved that such a manifold is a manifold of negative constant scalar curvature In chapter 5, properties of the W3 curvature tensors along with its symmetric and skew symmetric parts are studied in an TJ- Einstein Sasakian manifold. An expression for the W3 curvature tensor and its symmetric and skew symmetric part in a W3-symmetric TJ-Einstein Sasakian manifold is derived. It is shown that a W3-fiat 77-Einstein Sasakian manifold is an Einstein Manifold . It is further shown that such a manifold is isometric to the unit sphere and is a manifold of negative contant scalar curvature