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

The object of the present paper is to obtain a necessary condition for a three dimensional invariant submanifold of a Kenmotsu manifold to be totally geodesic. Besides this we study an invariant submanifold of Kenmotsu manifolds satisfying Q(α, R) = 0 and Q(S, α) = 0, where S, R are the Ricci tensor and curvature tensor respectively and α is the second fundamental form. Finally, we construct an example to verify our results.

On invariant submanifolds of Kenmotsu manifolds

In the present paper, <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mtext>LP</mtext> </math> -Kenmotsu manifolds admitting <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>η</mi> </math> -Ricci solitons have been studied. Moreover, some results for <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>η</mi> </math> -Ricci solitons in LP-Kenmotsu manifolds in the spacetime of general relativity have also been proved. Through a nontrivial example, we have given a proof for the existence of η-Ricci solitons in a 5-dimensional LP-Kenmotsu manifold. 

LP-Kenmotsu Manifolds Admitting η-Ricci Solitons and Spacetime

In this paper, it is proved that a trans-Sasakian 3-manifold is locally
 symmetric if and only if it is locally isometric to the sphere space S3(c2),
 the hyperbolic space H3(-c2), the Euclidean space R3, the product space R x
 S2(c2) or R x H2(-c2), where c is a nonzero constant. Some examples are
 constructed to illustrate main results. We also give some new conditions for
 a compact trans-Sasakian 3-manifold to be proper.

/pdf/ricci-tensors-on-trans-sasakian-3-manifolds-1obkgwh4nf.pdf

Ricci tensors on trans-Sasakian 3-manifolds

The object of the present paper is to study 3-dimensional trans-Sasakian manifolds with conservative curvature tensor and also 3-dimensional conformally flat trans-Sasakian manifolds. Next we consider compact connected -Einstein 3-dimensional trans-Sasakian manifolds. Finally, an example of a 3-dimensional trans-Sasakian manifold is given, which verifies our results.

http://www.ndsl.kr/soc_img/society/kms/DBSHCJ/2012/v27n4/DBSHCJ_2012_v27n4_795.pdf

On a class of three-dimensional trans-sasakian manifolds

The quest of the offering article is to investigate \emph{almost Riemann soliton} and \emph{gradient almost Riemann soliton} in a non-cosymplectic normal almost contact metric manifold $M^3$. Before all else, it is proved that if the metric of $M^3$ is Riemann soliton with divergence-free potential vector field $Z$, then the manifold is quasi-Sasakian and is of constant sectional curvature -$\lambda$, provided $\alpha,\beta =$ constant. Other than this, it is shown that if the metric of $M^3$ is \emph{ARS} and $Z$ is pointwise collinear with $\xi $ and has constant divergence, then $Z$ is a constant multiple of $\xi $ and the \emph{ARS} reduces to a Riemann soliton, provided $\alpha,\;\beta =$constant. Additionally, it is established that if $M^3$ with $\alpha,\; \beta =$ constant admits a gradient \emph{ARS} $(\gamma,\xi,\lambda)$, then the manifold is either quasi-Sasakian or is of constant sectional curvature $-(\alpha^2-\beta^2)$. At long last, we develop an example of $M^3$ conceding a Riemann soliton.

A note on Almost Riemann Soliton and gradient almost Riemann soliton.

The purpose of the present paper is to investigate the almost quasi- Yamabe soliton and gradient almost quasi-Yamabe solitons under the framework of three-dimensional normal almost paracontact metr...

Almost quasi-Yamabe solitons and gradient almost quasi-yamabe solitons in paracontact geometry

: In this paper, we classify 3-dimensional Riemannian manifolds endowed with a special type of vector field if the Riemannian metrices are Ricci-Yamabe solitons and gradient Ricci-Yamabe solitons, respectively. Finally, we construct an example to illustrate our result.

Ricci-Yamabe solitons and 3-dimensional Riemannian manifolds

The purpose of the offering exposition is to characterize gradient Yamabe, gradient Einstein and gradient m-quasi Einstein solitons within the framework of 3-dimensional para-Kenmotsu manifolds. Fi...

Krishnendu De

Papers

On a class of three-dimensional trans-sasakian manifolds

A note on Almost Riemann Soliton and gradient almost Riemann soliton.

Almost quasi-Yamabe solitons and gradient almost quasi-yamabe solitons in paracontact geometry

Ricci-Yamabe solitons and 3-dimensional Riemannian manifolds

A note on gradient solitons on para-Kenmotsu manifolds