We obtain the evolution equations for the Riemann tensor, the Ricci tensor and the scalar curvature induced by the mean curvature flow. The evolution for the scalar curvature is similar to the Ricci flow, however, negative, rather than positive, curvature is preserved. Our results are valid in any dimension.

Introduction. In [5], J. Milnor cited "understanding the Ricci tensor Rik = J^ Rt'kl 9J as a fundamental problem of present-day mathematics. A basic issue, then, is to determine which symmetric covariant tensors of rank two can be Ricci tensors of Riemannian metrics. The definition of Ricci curvature casts the problem of finding a metric g which realizes a given Ricci curvature R as one of solv...

where Rαβ(x, t) denotes the Ricci curvature tensor of the metric gαβ(x, t). One of the main problems in differential geometry is to find canonical structure on manifolds. The Ricci flow introduced by Hamilton [8] is an useful tool to approach such problems. For examples, Hamilton [10] and Chow [7] used the convergence of the Ricci flow to characterize the complex structures on compact Riemann s...

A Ricci surface is a Riemannian 2-manifold (M, g) whose Gaussian curvature K satisfies K∆K+g(dK, dK)+4K = 0. Every minimal surface isometrically embedded in R is a Ricci surface of non-positive curvature. At the end of the 19 century Ricci-Curbastro has proved that conversely, every point x of a Ricci surface has a neighborhood which embeds isometrically in R as a minimal surface, provided K(x)...

In this paper, we prove that the Lp essential spectra of the Laplacian on functions are [0,+∞) on a noncompact complete Riemannian manifold with non-negative Ricci curvature at infinity. The similar method applies to gradient shrinking Ricci soliton, which is similar to non-compact manifold with non-negative Ricci curvature in many ways. © 2010 Elsevier Inc. All rights reserved.

We study complete noncompact long time solutions (M, g(t)) to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. Ri̄ ≥ cRgi̄ at (p, t) for all t for some c > 0, then there always exists a local gradient Kähler-Ricci soliton limit around p after possibly rescaling g(t) alon...

This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn’t imply flatness. We show that, in dimensions greater than 4, if a Ricci flow invariant condition is weaker than “Einstein w...

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.

‎We study curvature properties of four-dimensional Lorentzian manifolds with two-symmetry property‎. ‎We then consider Einstein-like metrics‎, ‎Ricci solitons and homogeneity over these spaces‎‎.

These notes are the planned contents of my lectures. Some parts could be only briefly explained or skipped due to the lack of time or possible overlap with other lectures. The aim of these lectures is to review the recent development on the relation between optimal transport theory and Riemannian geometry. Ricci curvature is the key ingredient. Optimal transport theory provides a good character...