On quasi-conformally flat weakly Ricci symmetric manifolds

The object of the present paper is to study quasi-conformally flat weakly Ricci symmetric manifolds.      

On Ricci flat warped products with a quarter-symmetric connection

In this paper, we have computed the warping functions for a Ricci flat Einstein multiply warped product spaces M with a quarter-symmetric connection for different dimensions of M [i.e; (1). dimM =  2, (2). dimM =  3, (3). \({dim M \geq 4}\)] and all the fibers are Ricci flat.     

On Riemannian 3-manifolds with distinct constant Ricci eigenvalues

The first author was partly supported by the grant GAR 201/93/0469     

On Ricci eigenvalues of locally homogeneous Riemannian 3-manifolds

We find the necessary and sufficient conditions for three constants ₁, ₂, ₃ ³ to be the principal Ricci curvatures of some 3-dimensional locally homogeneous Riemannian space.The first author was supported by the grant GAR 201/93/0469; the second author was supported by the grant SFS, Project #0401.     

On Gradient Ricci Solitons

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work in M. Fernandez-Lopez and E. Garcia-Rio, Rigidity of shrinking Ricci solitons in Math. Z. (2011) classify complete gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic functions on gradient shrinking Kähler and gradient steady Ricci solitons. Consequences to the structure of shrinking and steady solitons at infinity are also discussed.     

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.     

On the Ricci operator of locally homogeneous Lorentzian 3-manifolds

We determine the admissible forms for the Ricci operator of three-dimensional locally homogeneous Lorentzian manifolds.      

Ricci flat metrics with bidimensional null orbits and non-integrable orthogonal distribution

We study Ricci flat 4-metrics of any signature under the assumption that they allow a Lie algebra of Killing fields with 2-dimensional orbits along which the metric degenerates and whose orthogonal distribution is not integrable. It turns out that locally there is a unique (up to a sign) metric which satisfies the conditions. This metric is of signature (++−−) and, moreover, homogeneous possessing a 6-dimensional symmetry algebra.     

On the projective Ricci curvature

Science China Mathematics - The notion of the Ricci curvature is defined for sprays on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci...     

2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension

V.A. Rohlin asked in 1949 whether 2-fold mixing implies 3-fold mixing for a stationary process (ξ_i )_i2ℤ, and the question remains open today. In 1978, F. Ledrappier exhibited a counterexample to the 2-fold mixing implies 3-fold mixing problem, the socalled 3-dot system, but in the context of stationary random fields indexed by ℤ². In this work, we first present an attempt to adapt Ledrappier's construction to the onedimensional case, which finally leads to a stationary process which is 2-fold but not 3-fold mixing conditionally to the σ-algebra generated by some factor process. Then, using arguments coming from the theory of joinings, we will give some strong obstacles proving that Ledrappier's counterexample can not be fully adapted to one-dimensional stationary processes.     

Stable bundles on 3-fold hypersurfaces

Using monads, we construct a large class of stable bundles of rank 2 and 3 on 3-fold hypersurfaces, and study the set of all possible Chern classes of stable vector bundles.