Riemannian 3-metrics with a generic Codazzi Ricci tensor

We determine a large family of explicit metrics, defined on open subsets of \mathbb R ³{\mathbb R ^3} , having a Codazzi Ricci tensor and three distinct Ricci eigenvalues.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

Estimation of Ricci tensor on certain Fano manifolds

We give time existence to some Monge-Ampère equations on certain Fano manifolds which cannot carry Einstein-K?hler metrics. These solutions allow us to obtain an estimation of Ricci tensors on our manifolds.

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Résumé. On met en évidence un intervalle de temps pour lequel des équations de Monge-Ampère sur certaines variétés de Fano, qui ne possèdent pas de métriques d'Einstein-K?hler, admettent des solutions. Ces solutions permettent d'obtenir des métriques dont on sait minorer le tenseur de Ricci.

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Received June 22, 1998; in final form January 25, 1999     

The Ricci tensor of almost parahermitian manifolds

We study the pseudoriemannian geometry of almost parahermitian manifolds, obtaining a formula for the Ricci tensor of the Levi–Civita connection. The formula uses the intrinsic torsion of an underlying \(\mathrm {SL}(n,\mathbb {R})\)-structure; we express it in terms of exterior derivatives of some appropriately defined differential forms. As an application, we construct Einstein and Ricci-flat examples on Lie groups. We disprove the parakähler version of the Goldberg conjecture and obtain the first compact examples of a non-flat, Ricci-flat nearly parakähler structure. We study the paracomplex analogue of the first Chern class in complex geometry, which obstructs the existence of Ricci-flat parakähler metrics.     

Eigenvalue estimate on complete noncompact Riemannian manifolds and applications

We obtain some sharp estimates on the first eigenvalues of complete noncompact Riemannian manifolds under assumptions of volume growth. Using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures.

     

Smoothing metrics on closed Riemannian manifolds through the Ricci flow

Under the assumption of the uniform local Sobolev inequality, it is proved that Riemannian metrics with an absolute Ricci curvature bound and a small Riemannian curvature integral bound can be smoothed to having a sectional curvature bound. This partly extends previous a priori estimates of Li (J Geom Anal 17:495–511, 2007; Adv Math 223:1924–1957, 2010).     

On the fundamental group of Riemannian manifolds with nonnegative Ricci curvature

In 1968 Milnor conjectured that the fundamental group of any complete Riemannian manifold with nonnegative Ricci curvature is finitely generated. In this paper we obtain two results concerning Milnor’s conjecture. We first prove that the generators of fundamental group can be chosen so that it has at most logarithmic growth. Secondly we prove that the conjecture is true if additional the volume growth satisfies certain condition.