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

Structures on Manifolds

S W Hawking and G F R Ellis London: Cambridge University Press 1973 pp xi + 391 price £10 Much progress has been made in recent years in understanding the global consequences of the general theory of relativity. Drs Hawking and Ellis have given us a coherent account of this research and of two remarkable predictions which have emerged from it.

http://iopscience.iop.org/article/10.1088/0031-9112/25/6/029/pdf

The Large Scale Structure of Space–Time

Abstract In this paper, we consider *-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if (M, g) is a Kenmotsu manifold and g is a *-Ricci soliton, then soliton constant λ is zero. For 3-dimensional case, if M admits a *-Ricci soliton, then we show that M is of constant sectional curvature –1. Next, we show that if M admits a *-Ricci soliton whose potential vector field is collinear with the characteristic vector field ξ, then M is Einstein and soliton vector field is equal to ξ. Finally, we prove that if g is a gradient almost *-Ricci soliton, then either M is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of M. We verify our result by constructing examples for both *-Ricci soliton and gradient almost *-Ricci soliton.

∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds

We prove that if a Sasakian metric is a ∗-Ricci Soliton, then it is either positive Sasakian, or null-Sasakian. Next, we prove that if a complete Sasakian metric is an almost gradient ∗-Ricci Solit...

∗-Ricci Soliton within the frame-work of Sasakian and (κ,μ)-contact manifold

In this paper, we consider the Fischer–Marsden conjecture within the frame-work of K-contact manifolds and $$(\kappa ,\mu )$$
 -contact manifolds. First, we prove that a complete K-contact metric satisfying $$\mathcal {L}^{*}_g(\lambda )=0$$
 is Einstein and is isometric to a unit sphere $$S^{2n+1}$$
 . Next, we prove that if a non-Sasakian $$(\kappa ,\mu )$$
 -contact metric satisfies $$\mathcal {L}^{*}_g(\lambda )=0$$
 , then $$ M^{3} $$
 is flat, and for $$n > 1$$
 , $$M^{2n+1}$$
 is locally isometric to the product of a Euclidean space $$E^{n+1}$$
 and a sphere $$S^n(4)$$
 of constant curvature $$+\,4$$
 .

The Fischer–Marsden conjecture and contact geometry

Certain contact metrics satisfying the Miao–Tam critical condition

In this paper, we consider the CPE conjecture in the frame-work of ${K}$-contact manifold and ${(\kappa, \mu)}$-contact manifold. First, we prove that a complete ${K}$-contact metric satisfying the CPE is Einstein and is isometric to a unit sphere ${S^{2n+1}}$. Next, we prove that if a non-Sasakian ${ (\kappa, \mu) }$-contact metric satisfies the CPE, then ${ M^{3} }$ is flat and for ${ n > 1 }$, ${ M^{2n+1} }$ is locally isometric to ${ E^{n+1} \times S^{n}(4)}$.

The Critical Point Equation And Contact Geometry

The aim of this article is to study the k-almost Ricci soliton and k-almost gradient Ricci soliton on contact metric manifold. First, we prove that if a compact K-contact metric is a k-almost gradient Ricci soliton then it is isometric to a unit sphere S2n+1. Next, we extend this result on a compact k-almost Ricci soliton when the flow vector field X is contact. Finally, we study some special types of k-almost Ricci soliton where the potential vector field X is point wise collinear with the Reeb vector field {\xi} of the contact metric structure.

Dhriti Sundar Patra

Papers

∗-Ricci Soliton within the frame-work of Sasakian and (κ,μ)-contact manifold

The Fischer–Marsden conjecture and contact geometry

Certain contact metrics satisfying the Miao–Tam critical condition

The Critical Point Equation And Contact Geometry

The $k$-almost Ricci solitons and contact geometry