We classify the paracontact Riemannian manifolds that their Riemannian curvature satisfies in the certain condition and we show that this classification is hold for the special cases semi-symmetric and locally symmetric spaces. Finally we study paracontact Riemannian manifolds satisfying R(X, ξ).S = 0, where S is the Ricci tensor.

In the present paper, we investigate the necessary and sufficient condition of a given Finsler metric to be Einstein. The considered Einstein Finsler metric in the study describes all different kinds of Einstein metrics which are pointwise projective to the given one.

From the ‡Department of Physiology and Biophysics, Albert Einstein College of Medicine, Bronx, New York 10461, the **Department of Medicine, Division of Hematology, Albert Einstein College of Medicine, Bronx, New York 10461, the ‡‡Department of Anatomy and Structural Biology, Albert Einstein College of Medicine, Bronx, New York 10461, and ¶INSERM, Unité 473, 84 rue du Général Leclerc, 94276 Le ...

In this paper we examine homogeneous Einstein–Weyl structures and classify them on compact irreducible symmetric spaces. We find that the invariant Einstein–Weyl equation is very restrictive: Einstein–Weyl structures occur only on those spaces for which the isotropy representation has a trivial component, for example, the total space of a circle bundle.

Einstein’s research manuscripts provide important insights into his exceptional creativity. At the same time, they can present difficulties for a publication in the documentary edition of the Collected Papers of Albert Einstein (CPAE). The problems are illustrated by discussing how some important examples of Einstein’s research manuscripts have been included in previous volumes of the CPAE seri...

This review article has grown out of notes for the three lectures the second author presented during the XXIV-th Winter School of Geometry and Physics in Srni, Czech Republic, in January of 2004. Our purpose is twofold. We want give a brief introduction to some of the techniques we have developed over the last 5 years while, at the same time, we summarize all the known results. We do not give a...

We show that every K-contact Einstein manifold is Sasakian-Einstein and discuss several corollaries of this result.

We numerically calculate Perelman’s entropy for a variety of canonical metrics on CP-bundles over products of Fano Kähler-Einstein manifolds. The metrics investigated are Einstein metrics, Kähler-Ricci solitons and quasi-Einstein metrics. The calculation of the entropy allows a rough picture of how the Ricci flow behaves on each of the manifolds in question.

We investigate Chern number inequalities on Kähler-Einstein manifolds and their relation to uniformization. For Kähler-Einstein manifolds with c1 > 0, we prove certain Chern number inequalities in the toric case. For Kähler-Einstein manifolds with c1 < 0, we propose a series of Chern number inequalities, interpolating Yau’s and Miyaoka’s inequalities.