A formal treatment of Killing 1-form and 2-Killing 1-form on Riemannian Poisson manifold, Riemannian Poisson warped product space are presented. In this way, we obtain Bochner type result on compact Riemannian Poisson manifold, compact Riemannian Poisson warped product space for Killing 1-form and 2-Killing 1-form. Finally, we give the characterization of a 2-Killing 1-form on (R 2 , g, Π).
We study Einstein warped space with a quarter symmetric connection. As a result, first, we find basic results on curvature, Ricci and scalar tensors with respect to the quarter symmetric connection. Moreover, we prove some results corresponding to second order quarter symmetric connection. Finally, we prove that if M is an Einstein warped space with nonpositive scalar curvature and compact base with respect to quarter symmetric connection and the warping function satisfy some condition then M is simply a Riemannian product space.
In this article, we introduce the sectional curvature in contravariant warped product spaceAfter that we find the sectional curvature of M for which M 1 and M 2 are Poisson manifolds of positive sectional curvatures. In dual space of M , we introduce the notion of null, spacelike, timelike 1− forms and then by using these forms, qualar curvature is defined. Finally, as an examples we obtain the sectional curvatures for M 1 = H 2 1 , M 2 = S 2 0 , E 2 2 and qualar curvature for M .
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