Parallelism in K-contact geometry

On a 2n+1-dimensional K-contact manifold, there are no nontrivial parallel forms except of degrees 0 and 2n+1.     

Ricci solitons and geometry of four dimensional Einstein-like neutral Lie groups

Periodica Mathematica Hungarica - We classify left-invariant Einstein-like metrics of neutral signature, over four-dimensional Lie groups. Several geometric properties such as being conformally...     

Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry

The aim of this paper is to study two problems in the framework of paracontact geometry of dimension \(3\) , namely, the class of parallel symmetric tensor fields of \((0, 2)\) -type and possible Lorentz Ricci solitons. We search for two types of Ricci solitons: the first when its potential vector field is exactly the characteristic vector field \(\xi \) of the paracontact structure and the second when the potential vector field is a paracontact-holomorphic one. For the former case we find all variants of Ricci solitons, expanding, steady and shrinking, and the fact that \(\xi \) is a conformal Killing vector field. A class of examples is completely discussed.     

Homogeneous Ricci almost solitons

It is shown that a locally homogeneous proper Ricci almost soliton is either of constant sectional curvature or locally isometric to a product R×N(c), where N(c) is a space of constant curvature.     

Ricci solitons and odd-dimensional spheres

As a real hypersurface in a complex space, we prove two criterion inequalities for an odd-dimensional sphere in terms of the shape operator, the Reeb vector field and its associated 1-form. Also, we determine a real hypersurface in a complex space which admits a Ricci soliton with the Reeb vector field the potential vector field.     

Liouville and geodesic Ricci solitons

On a tangent bundle endowed with a pseudo-Riemannian metric of complete lift type two classes of Ricci solitons are obtained: a 1-parameter family of shrinking Liouville Ricci solitons if the base manifold is Ricci flat and a steady geodesic Ricci soliton if the base manifold is flat. A nonexistence result of geodesic Ricci solitons for the tangent bundle of a non-flat space form is also provided. To cite this article: M. Crasmareanu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Maximum principles and gradient Ricci solitons

It is shown that the Omori-Yau maximum principle holds true on complete gradient shrinking Ricci solitons both for the Laplacian and the f-Laplacian. As an application, curvature estimates and rigidity results for shrinking Ricci solitons are obtained. Furthermore, applications of maximum principles are also given in the steady and expanding situations.     

Three-dimensional Lorentzian homogeneous Ricci solitons

We study three-dimensional Lorentzian homogeneous Ricci solitons, proving the existence of shrinking, expanding and steady Ricci solitons. For all the non-trivial examples, the Ricci operator is not diagonalizable and has three equal eigenvalues.     

Rigidity of shrinking Ricci solitons

We show that a compact Ricci soliton is rigid if and only if the Weyl conformal tensor is harmonic. In the complete noncompact case we prove the same result assuming that the curvature tensor has at most exponential growth and the Ricci tensor is bounded from below.     

Bach-flat gradient steady Ricci solitons

In this paper we prove that any n-dimensional (n ≥ 4) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton. In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in Cao and Chen (Trans Am Math Soc 364:2377–2391, 2012) and Catino and Mantegazza (Ann Inst Fourier 61(4):1407–1435, 2011).     

On the Ricci curvature of steady gradient Ricci solitons

Assume (Mn,g) is a complete steady gradient Ricci soliton with positive Ricci curvature. If the scalar curvature approaches 0 towards infinity, we prove that , where O is the point where R obtains its maximum and γ(s) is a minimal normal geodesic emanating from O. Some other results on the Ricci curvature are also obtained.     

A Myers-type theorem and compact Ricci solitons

Let the Ricci curvature of a compact Riemannian manifold be greater, at every point, than the Lie derivative of the metric with respect to some fixed smooth vector field. It is shown that the fundamental group then has only finitely many conjugacy classes. This applies, in particular, to all compact shrinking Ricci solitons.

     

Expanding Ricci solitons asymptotic to cones

We show that an expanding gradient Ricci soliton which is asymptotic to a cone at infinity in a certain sense must be rotationally symmetric.     

Ricci solitons on Lorentzian Walker three-manifolds

We investigate Ricci solitons on Lorentzian three-manifolds (M,g _f ) admitting a parallel degenerate line field. For several classes of these manifolds, described in terms of the defining function f, the existence of non-trivial Ricci solitons is proved.     

Remarks on non-compact gradient Ricci solitons

In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and L ^p -Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scalar curvature estimates for gradient Ricci solitons under L ^p conditions on the relevant quantities.