Let (M, g) be a compact n-dimensional (n 2) manifold with nonnegative Ricci curvature, and if n 3, then we assume that (M, g) × R has nonnegative isotropic curvature. The lower bound of the Ricci flow’s existence time on (M, g) is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows’ maximal existence times, which was first proved by E. Cabeza...

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvatu...

Copyright q 2010 Zisheng Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to radially symmetric Ricci curvature lower bound...

On closed Riemannian manifolds with Bakry-Émery Ricci curvature bounded from below and gradient of the potential function, we obtain lower bounds for all positive eigenvalues Beltrami-Laplacian instead weighted Laplacian. The bound k th eigenvalue depends on k, curvature, dimension diameter upper manifold, but volume manifold is not involved.

The aim of the present paper is to bridge the gap between the Bakry-Émery and the Lott-Sturm-Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds. We start from a strongly local Dirichlet form E admitting a Carré du champ Γ in a Polish measure space (X,m) and a canonical distance dE that induces the original topology of X. We first characterize the distin...

We survey work of Lott-Villani and Sturm on lower Ricci curvature bounds for metric-measure spaces. An intriguing question is whether one can extend notions of smooth Riemannian geometry to general metric spaces. Besides the inherent interest, such extensions sometimes allow one to prove results about smooth Riemannian manifolds, using compactness theorems. There is a good notion of a metric sp...

In this note we discuss the fundamental groups and diameters of positively Ricci curved n-manifolds. We use a method combining the results about equivarient Hausdorff convergence developed by Fukaya and Yamaguchi with the Ricci version of splitting theorem by Cheeger and Colding to give new information on the topology of compact manifolds with positive Ricci curvature. Moreover, we also obtain ...

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.

In this article, we derive Chen’s inequalities involving ?-invariant ?M, Riemannian invariant ?(m1,?,mk), Ricci curvature, ?k(2?k?m), the scalar curvature and squared of mean for submanifolds generalized Sasakian-space-forms endowed with a quarter-symmetric connection. As an application obtain inequality, first derived Chen inequality bi-slant submanifold Sasakian-space-forms.