Related papers: On the Completeness of Gradient Ricci Solitons

Gradient almost para-Ricci-like solitons on para-Sasaki-like Riemannian $\Pi$-manifolds are studied. It is proved that these objects have constant soliton coefficients. For the soliton under study is shown that the corresponding scalar…

Ricci solitons are natural generalizations of Einstein metrics. They are also special solutions to Hamilton's Ricci flow and play important roles in the singularity study of the Ricci flow. In this paper, we survey some of the recent…

In this paper, we continue to study the generalized Ricci flow. We give a criterion on steady gradient Ricci soliton on complete and noncompact Riemannian manifolds that is Ricci-flat, and then introduce a natural flow whose stable points…

Curvature properties of a metric connection with totally skew-symmetric torsion are investigated. It is shown that if either the 3-form $T$ is harmonic, $dT=\delta T=0$ or the curvature of the torsion connection $R\in S^2\Lambda^2$ then the…

In this paper we consider $M = B\times_{f}F$ warped product gradient Ricci solitons. We proved that the potential function depends only on the base and the fiber $F$ is necessarily Einstein manifold. We provide all such solutions in the…

We prove that a shrinking gradient Ricci soliton which agrees to infinite order at spatial infinity with one of the standard cylindrical metrics on $S^k\times \RR^{n-k}$ for $k\geq 2$ along some end must be isometric to the cylinder on that…

We establish integral curvature estimates for complete gradient shrinking Sasaki-Ricci solitons. As an application, we show that any such soliton with harmonic Weyl tensor must be a finite quotient of a sphere. This result can be regarded…

In this article, we investigate the geometry of $4$-dimensional compact gradient Ricci solitons. We prove that, under an upper bound condition on the range of the potential function, a $4$-dimensional compact gradient Ricci soliton must…

In this paper we have investigated some aspects of gradient $\rho$-Einstein Ricci soliton in a complete Riemannian manifold. First, we have proved that the compact gradient $\rho$-Einstein soliton is isometric to the Euclidean sphere by…

In this article, we investigate global norm of potential vector field in Ricci soliton. In particular, we have deduced certain conditions so that the potential vector field has finite global norm in expanding Ricci soliton. We have also…

We show that Lorentzian manifolds whose isometry group is of dimension at least $\frac{1}{2}n(n-1)+1$ are expanding, steady and shrinking Ricci solitons and steady gradient Ricci solitons. This provides examples of complete locally…

We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. By using that this ODE is the…

Let $F$ be a left invariant Randers metric on a simply connected nilpotent Lie group $N$, induced by a left invariant Riemannian metric ${\hat{\textbf{\textit{a}}}}$ and a vector field $X$ which is…

A lower-bound estimate of injectivity radius for complete Riemannian manifolds is discussed in a pure geometric viewpoint and is applied to study tangent cones at infinity of certain gradient Ricci solitons. We also study the asymptotic…

A Ricci soliton $(M,g,v,\lambda)$ on a Riemannian manifold $(M,g)$ is said to have concurrent potential field if its potential field $v$ is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian…

We show that a shrinking Ricci soliton with positive sectional curvature must be compact. This extends a result of Perelman in dimension three and improves a result of Naber in dimension four, respectively.

I In this paper, first we study a complete smooth metric measure space $(M^n,g, e^{-f}dv)$ with the ($\infty$)-Bakry-\'Emery Ricci curvature $\textrm{Ric}_f\ge \frac a2g$ for some positive constant $a$. It is known that the spectrum of the…

Ricci-like solitons on Sasaki-like almost contact B-metric manifolds are the object of study. Cases, where the potential of the Ricci-like soliton is the Reeb vector field or pointwise collinear to it, are considered. In the former case,…

A contact 3-manifold $M$ admitting a transversal Ricci soliton $(g,v,\lambda)$ is either Sasakian or locally isometric to one of the Lie groups SU(2), $SL(2,R)$, E(2), E(1,1) with a left invariant metric.

We descrive examples of metrics in the conformal class $[g]$ on complete conformally flat Riemannian manifolds $(M,g].$ These metrics have a constant scalar curvature and an harmonic curvature with non parallel Ricci tensor.