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

/pdf/constraining-anisotropic-cosmological-model-in-f-mathcal-r-bko9vbfa.pdf

Constraining Anisotropic Cosmological Model in $f(\mathcal{R},\mathscr{L}_m)$ Gravity

In this paper, we study an almost coKahler manifold admitting certain metrics such as $$*$$
 -Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKahler 3-manifold (M, g) admitting a $$*$$
 -Ricci soliton (g, X) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKahler $$(\kappa ,\mu )$$
 -almost coKahler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f. Finally, we prove that a $$(\kappa , \mu )$$
 -almost coKahler manifold (M, g) is coKahler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKahler manifolds which are non-coKahler.

Certain types of metrics on almost coKähler manifolds

Let M be a real hypersurface of a complex space form of constant curvature c. In this paper, we study the hypersurface M which admits a nontrivial solution to Fischer–Marsden equation, that is, the induce metric g of M satisfies $$ Hess _g(\nu )=(\varDelta _g \nu )g+\nu S_g$$
 , where $$\nu $$
 is a nontrivial function. We prove that there does not exist a complete Hopf real hypersurface in a non-flat complex space form satisfying Fischer–Marsden equation. Finally, we show that a complete real hypersurface with $$A\xi =\alpha \xi $$
 , $$\alpha \ne 0$$
 , of a complex Euclidean space $${\mathbb {C}}^n$$
 satisfying Fischer–Marsden equation is locally congruent to a sphere or $${\mathbb {S}}^1 \times {\mathbb {R}}^{2n-2}$$
 .

Real hypersurfaces of complex space forms satisfying Fischer–Marsden equation

The object of the present paper is to study some properties of W2curvature tensor in an Lorentzian para-Sasakian manifolds.

/pdf/on-lorentzian-para-sasakian-manifolds-satisfying-w2-ooq49w66uv.pdf

On Lorentzian Para-Sasakian Manifolds Satisfying W2- Curvature Tensor

https://link.springer.com/content/pdf/10.1186/s42787-019-0022-0.pdf

V. Venkatesha

Papers

Constraining Anisotropic Cosmological Model in $f(\mathcal{R},\mathscr{L}_m)$ Gravity

Certain types of metrics on almost coKähler manifolds

Real hypersurfaces of complex space forms satisfying Fischer–Marsden equation

On Lorentzian Para-Sasakian Manifolds Satisfying W2- Curvature Tensor

On a class of generalized φ-recurrent Sasakian manifold