Abstract
Let M be a trans-paraSasakian 3-manifold. In this paper, the necessary and sufficient condition for the Reeb vector field of a trans-paraSasakian 3-manifold to be harmonic is obtained. Also, it is proved that the Ricci operator of M is invariant along the Reeb flow if and only if M is a paracosymplectic manifold, an α-paraSasakian manifold or a space of negative constant sectional curvature.
It is known that there does not exist any Hopf hypersurface in complex Grassmannians of rank two of complex dimension 2m with constant sectional curvature for m≥3. The purpose of this article is to extend the above result, and without the Hopf condition, we prove that there does not exist any locally conformally flat real hypersurface for m≥3.
(m,?)-quasi-Einstein N(k)-contact metric manifolds have been studied and it
is established that if such a manifold is a (m,?)-quasi-Einstein manifold,
then the manifold is a manifold of constant sectional curvature k. Further
analysis has been done for gradient Einstein soliton, in particular.
Obtained results are supported by an illustrative example.
Kenmotsu geometry is a valuable part of contact geometry with nice
applications in other fields such as theoretical physics. In this article,
we study the statistical counterpart of a Kenmotsu manifold, that is,
Kenmotsu statistical manifold with some related examples. We investigate
some statistical curvature properties of Kenmotsu statistical manifolds. It
has been shown that a Kenmotsu statistical manifold is not a Ricci-flat
statistical manifold by constructing a counter-example. Finally, we prove a
very well-known Chen-Ricci inequality for statistical submanifolds in
Kenmotsu statistical manifolds of constant ?-sectional curvature by
adopting optimization techniques on submanifolds. This article ends with
some concluding remarks.
Real Hypersurfaces in Complex Grassmannians of Rank Two