Integral formulas in Riemannian geometry

Structures on Manifolds

If M is a 3-dimensional contact metric manifold such that Qφ = φQ which admits a Yamabe soliton (g,V ) with the flow vector field V pointwise collinear with the Reeb vector field ξ, then we show th...

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

In this paper, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation admits a gradient $$\rho $$
 -Einstein soliton, then either $$M^{2n+1}$$
 is Einstein or the potential function is pointwise collinear with the Reeb vector field $$\xi $$
 on an open set $${\mathcal {O}}$$
 of $$M^{2n+1}$$
 . Moreover, we prove that if the metric of a $$(\kappa ,-2)'$$
 -almost Kenmotsu manifold with $$h'\ne 0$$
 admits a gradient $$\rho $$
 -Einstein soliton, then the manifold is locally isometric to the Riemannian product $${\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n$$
 and potential vector field is tangential to the Euclidean factor $${\mathbb {R}}^n$$
 . We show that there does not exist gradient $$\rho $$
 -Einstein soliton on generalized $$(\kappa ,\mu )$$
 -almost Kenmotsu manifold of constant scalar curvature. Finally, we construct an example for gradient $$\rho $$
 -Einstein soliton.

Gradient $$\rho $$ ρ -Einstein soliton on almost Kenmotsu manifolds

I. Symplectic Manifolds

On a type of almost Kenmotsu manifolds with nullity distributions

We prove that if a three-dimensional N(k)-paracontact metric manifold admits a Yamabe soliton $$(g,\xi )$$
 , then the scalar curvature is constant and the manifold is a paraSasakian manifold. Moreover, we show that if a three-dimensional N(k)-paracontact metric manifold admits a Yamabe soliton (g, V), then either the manifold is a space of constant curvature, or the flow vector field V is Killing.

Yamabe Solitons on Three-Dimensional N(k)-Paracontact Metric Manifolds

The aim of this paper is to characterize $3$-dimensional $N(k)$-paracontact metric manifolds satisfying certain curvature conditions. We prove that a $3$-dimensional $N(k)$-paracontact metric manifold $M$ admits a Ricci soliton whose potential vector field is the Reeb vector field $xi$ if and only if the manifold is a paraSasaki-Einstein manifold. Several consequences of this result are discussed. Finally, an illustrative example is constructed.

http://bims.iranjournals.ir/article_1040_a446023e2cc7956949c8c04a16793c63.pdf

On three-dimensional $N(k)$-paracontact metric manifolds and Ricci solitons

The purpose of this paper is to investigate the Fischer-Marsden conjecture on almost Kenmotsu manifolds. First, we prove that if a three-dimensional non-Kenmotsu (k, µ)′-almost Kenmotsu manifold sa...

The Fischer-Marsden conjecture on almost Kenmotsu manifolds

Abstract In this paper, we consider a quarter-symmetric metric connection in a P-Sasakian manifold. We investigate the curvature tensor and the Ricci tensor of a P-Sasakian manifold with respect to the quarter-symmetric metric connection. We consider semisymmetric P-Sasakian manifold with respect to the quarter- symmetric metric connection. Furthermore, we consider generalized recurrent P-Sasakian manifolds and prove the non-existence of recurrent and pseudosymmetric P-Sasakian manifolds with respect to the quarter-symmetric metric connection. Finally, we construct an example of a 5-dimensional P-Sasakian manifold admitting quarter-symmetric metric connection which verifies Theorem 4.1.

/pdf/quarter-symmetric-metric-connection-in-a-p-sasakian-manifold-3orattipa7.pdf

Krishanu Mandal

Papers

On a type of almost Kenmotsu manifolds with nullity distributions

Yamabe Solitons on Three-Dimensional N(k)-Paracontact Metric Manifolds

On three-dimensional $N(k)$-paracontact metric manifolds and Ricci solitons

The Fischer-Marsden conjecture on almost Kenmotsu manifolds

Quarter-symmetric metric connection in a P-Sasakian manifold