Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

Integral formulas in Riemannian geometry

In this paper, we study para-Sasakian manifold (M,g) whose metric g is an η-Ricci soliton (g,V ) and almost η-Ricci soliton. We prove that, if g is an η-Ricci soliton, then either M is Einstein and...

η-Ricci solitons and almost η-Ricci solitons on para-Sasakian manifolds

Abstract Let (M, g) be a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold of dimension 2n + 1. In this paper, we prove that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation. Moreover, two concrete examples of (κ, μ)′-almost Kenmotsu 3-manifolds admitting a Killing vector field and strict infinitesimal contact transformation are given.

https://www.degruyter.com/document/doi/10.1515/math-2019-0056/pdf

*-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds

Abstract We prove that if an η \eta -Einstein para-Kenmotsu manifold admits a conformal η \eta -Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a conformal η \eta -Ricci soliton is Einstein if its potential vector field V V is infinitesimal paracontact transformation or collinear with the Reeb vector field. Furthermore, we prove that if a para-Kenmotsu manifold admits a gradient conformal η \eta -Ricci almost soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits conformal η \eta -Ricci soliton and satisfy our results. We also have studied conformal η \eta -Ricci soliton in three-dimensional para-cosymplectic manifolds.

/pdf/geometry-of-conformal-e-ricci-solitons-and-conformal-e-ricci-2pjjz7h3.pdf

Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry

We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $$h:=\frac{1}{2}\pounds _\xi \varphi $$
 and $$\ell := R(\cdot ,\xi )\xi $$
 , emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $$\xi $$
 -sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost CR structure $$(\mathcal {H}(M), J, \theta )$$
 corresponding to almost contact pseudo-metric manifold M to be CR manifold. Finally, we prove that a contact pseudo-metric manifold $$(M, \varphi ,\xi ,\eta ,g)$$
 is Sasakian pseudo-metric if and only if the corresponding nondegenerate almost CR structure $$(\mathcal {H}(M), J)$$
 is integrable and J is parallel along $$\xi $$
 with respect to the Bott partial connection.

Certain results on almost contact pseudo-metric manifolds

We analyze the -manifolds endowed with some symmetric properties, focusing on Ricci tensor and the 1-form . We study some properties of special Weakly Ricci-Symmetric -manifolds and also shown that Weakly -Ricci Symmetric -manifold is an -Einstein manifold.

Some Symmetric Properties on (LCS) n -manifolds

Consider a gradient Einstein-type metric in the setting of Kcontact manifolds and (κ, μ)-contact manifolds. First, it is proved that, if a complete K-contact manifold admits a gradient Einstein-type metric, then M is compact, Einstein, Sasakian and isometric to the unit sphere S2n+1. Next, it is proved that, if a non-Sasakian (κ, μ)-contact manifolds admits a gradient Einstein-type metric, then it is flat in dimension 3, and for higher dimension, M is locally isometric to the product of a Euclidean space En+1 and a sphere Sn(4) of constant curvature +4.

https://www.koreascience.or.kr:443/article/JAKO202013461499310.pdf

Gradient Einstein-type contact metric manifolds

First we consider almost Kenmotsu manifolds which satisfy Codazzi condition for and , and we prove that in such cases the tensor vanishes. Next, we prove that an almost Kenmotsu manifold having constant -sectional curvature which is locally symmetric is a Kenmotsu manifold of constant curvature . We also prove that, for a -almost Kenmotsu manifold of with , every conformal vector field is Killing. Finally, we prove that if is a -almost Kenmotsu manifold with and , then the vector field which leaves the curvature tensor invariant is Killing.

Some results on almost Kenmotsu manifolds

The aim of this article is to study the Riemann soliton and gradient almost Riemann soliton on certain class of almost Kenmotsu manifolds. Also some suitable examples of Kenmotsu and $(\kappa,\mu)'$-almost Kenmotsu manifolds are constructed to justify our results.

V. Venkatesha

Papers

Certain results on almost contact pseudo-metric manifolds

Some Symmetric Properties on (LCS) n -manifolds

Gradient Einstein-type contact metric manifolds

Some results on almost Kenmotsu manifolds

Riemann solitons and almost Riemann solitons on almost Kenmotsu manifolds