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Abstract
In a M. Prvanović’s paper [5], we can find a new curvature-like tensor in an almost Hermitian manifold.
In this paper, we define a new curvature-like tensor, named contact holomorphic Riemannian, briefly (CHR), curvature tensor in an almost contact
Riemannian manifold. Then, using this tensor, we mainly research (CHR)-
curvature tensor in a Kenmotsu and a Sasakian manifold. We introduce
the flatness of a (CHR)-curvature tensor and show that a Kenmotsu and
a Sasakian manifold with a flat (CHR)-curvature tensor is flat, see Theorems
3.1 and 4.1. Next, we introduce the notion of an (CHR)-n-Einstein in
an almost contact Riemannian manifold. In particular, in a Sasakian or a
Kenmotsu manifold, a (CHR)-n-Einstein manifold is n-Einstein, see Theorem
5.3. Finally, from this tensor, we introduce a notion of a (CHR)-space
form in an almost contact Riemannian manifold. In particular, if a Kenmotsu
and a Sasakian manifold are (CHR)-space form, then the (CHR)-curvature
tensor satisfies a special equation, see Theorems 6.2 and 7.1.
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References
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5. Mileva Prvanovic. Conformally invariant tensors of an almost Hermitian manifold as¬sociated with the holomorphic curvature tensor. Journal of Geometry, 103(89):89-101, 2012.
6. Kentaro Yano, Masahiro Kon. Structures on Manifolds, volume 3 of Series in pure mathematics. World Scientific, 1984.