Curvature tensions in sasakian, psasakian, lpsasakian and beinstein 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, LPSasakian and 77Einstein 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 W5symmetric Sasakian manifold. It is also proved that W3fiat
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 W5fiat Einstein Sasakian manifold is a fiat manifold.
In Chapter 3, properties and representations of the W3 and W5 curvature tensor are
studied in a ParaSasakian Manifold. Characterisations of the W5 and the W3 curvature
tensors under various conditions are derived. It is shown that W3fiat PSasakian manifold
is 77Einstein and an expression of the Riemann curvature tensor is obtained. It is shown
that W5fiat PSasakian 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 PSasakian 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 TJEinstein Manifold.
In chapter 4, we study Lorentzian Para Sasakian manifolds that are ¢ W3 fiat, ¢ W5
fiat, W3fiat and W5fiat. It is shown that LPSasakian manifolds that are ¢  W3 fiat,
¢  W5 fiat or W3fiat are TJEinstein. An expression for the Riemann curvature tensor
and the Ricci tensor in a W5fiat LPSasakian 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 W3symmetric
TJEinstein Sasakian manifold is derived. It is shown that a W3fiat 77Einstein 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
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