We use a local argument to prove if an $r$-dimensional torus acts isometrically and effectively on connected $n$-dimensional manifold which has positive $k^\mathrm{th}$-intermediate Ricci curvature at some point, then $r \leq \lfloor \frac{n+k}{2} \rfloor$. This symmetry rank bound generalizes those established by Grove Searle for sectional Wilking quasipositive curvature. As consequence, we sh...

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.

In this work, we obtain a existence criteria for the longtime K\ahler Ricci flow solution. Using result, generalize result by Wu-Yau on of Einstein metric to case with possibly unbounded curvature. Moreover, negative scalar curvture must be unique up scaling.

We derive matter collineations for some static spherically symmetric spacetimes and compare the results with Killing, Ricci and Curvature symmetries. We conclude that matter and Ricci collineations are not, in general, the same.

Suppose M is a compact n-dimensional manifold, n ≥ 2, with a metric gij(x, t) that evolves by the Ricci flow ∂tgij = −2Rij in M × (0, T ). We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality. It is known that Ricci flow is a very powerful tool in understanding the geometry and structure of manifol...

The Ricci curvature criterion is used for the investigation of the relative instability of several configurations of N-body gravitating systems. It is shown, that the systems with double massive centers are more unstable than the homogeneous systems and those with one massive center. In general this shows the efficiency of the Ricci curvature method introduced by Gurzadyan and Kocharyan (1987) ...

Abstract The Ricci curvature criterion is used for the investigation of the relative instability of several configurations of N-body gravitating systems. It is shown, that the systems with double massive centers are more unstable than the homogeneous systems and those with one massive center. In general this shows the efficiency of the Ricci curvature method introduced by Gurzadyan and Kocharya...

We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path-connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the threeball) is contractible. As an application, using results of Maximo, Nunes, and Smith [MNS], we show the existence of properly embedd...

In this paper, we study the dilation limit of solutions to the Ricci flow on manifolds with nonnegative curvature operator. We first show that such a dilation limit must be a product of a compact ancient Type I solution of the Ricci flow with flat factors. Then we show under the Type I normalized Ricci flow, the compact factor has a subsequence converge to a Ricci soliton.

In this short paper we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result Böhm and Wilking have for dimensions twelve and above. Moreover, the manifolds constructed here are...