Applying a well known result for attracting fixed points of biholomorphisms [4, 6], we observe that one immediately obtains the following result: if (Mn, g) is a complete non-compact gradient Kähler-Ricci soliton which is either steady with positive Ricci curvature so that the scalar curvature attains its maximum at some point, or expanding with non-negative Ricci curvature, then M is biholomor...

We show that a closed four-manifold with \(4\frac{1}{2}\)-positive curvature operator of the second kind is diffeomorphic to spherical space form. The assumption sharp as both \({\mathbb{CP}\mathbb{}}^2\) and \({\mathbb {S}}^3 \times {\mathbb {S}}^1\) have \(4\frac{1}{2}\)-nonnegative kind. In higher dimensions \(n\ge 5\), we Riemannian manifolds are homeomorphic forms. These results proved by ...

We study Ollivier-Ricci curvature, a discrete version of Ricci curvature, which has gained popularity over the past several years and has found applications in diverse fields. However, the Ollivier-Ricci curvature requires an optimal mass transport problem to be solved, which can be computationally expensive for large networks. In view of this, we propose two alternative measures of curvature t...

Let (Mn; g) be a compact Riemannian manifold with Ric (n 1). It is well known that the bottom of spectrum 0 of its unverversal covering satis…es 0 (n 1) =4. We prove that equality holds i¤ M is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy. 1. Introduction Complete Riemannian manifolds with nonnegative Ricci curvature have been intensively studied by many people and...

In this paper we classify pseudosymmetric and Ricci-pseudosymmetric (κ, μ)-contact metric manifolds in the sense of Deszcz. Next we characterize Weyl-pseudosymmetric (κ, μ)-contact metric manifolds.

Static space times with maximal symmetric transverse spaces are classified according to their Ricci collineations. These are investigated for non-degenerate Ricci tensor (det.(Rα) 6= 0). It turns out that the only collineations admitted by these spaces can be ten, seven, six or four. Some new metrics admitting proper Ricci collineations are also investigated. PACS numbers: 04.20.-q, 04.20.Jb

We characterize the conjugate linearized Ricci flow on closed three– manifolds of bounded geometry and discuss its properties. In particular, we express the evolution of the metric and of its Ricci tensor in terms of the backward heat kernel of the conjugate linearized Ricci flow. These results provide various conservation laws and monotonicity formulas for the linearized flow.

By extending Koiso’s examples to the non-compact case, we construct complete gradient Kähler-Ricci solitons of various types on certain holomorphic line bundles over compact Kähler-Einstein manifolds. Moreover, a uniformization result on steady gradient Kähler-Ricci solitons with nonnegative Ricci curvature is obtained under additional assumptions.

The aim of the present paper is to study a Bochner Ricci semi-symmetric quasi-Einstein Hermitian manifold (QEH)n, a Bochner Ricci semi-symmetric generalised quasi-Einstein Hermitian manifold G(QEH)n and a Bochner Ricci semisymmetric pseudo generalised quasi-Einstein Hermitian manifold P (GQEH)n.

We prove the existence and uniqueness of the weak Kähler-Ricci flow on projective varieties with log terminal singularities. It is also shown that the weak Kähler-Ricci flow can be uniquely continued through divisorial contractions and flips if they exist. We then propose an analytic version of the Minimal Model Program with Ricci flow.