Searching for the dynamical foundations of Havrda-Charvát/Daróczy/ Cressie-Read/Tsallis non-additive entropy, we come across a covariant quantity called, alternatively, a generalized Ricci curvature, an N -Ricci curvature or a Bakry-Émery-Ricci curvature in the configuration/phase space of a system. We explore some of the implications of this tensor and its associated curvature and present a co...

We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.

We give new estimates on the lower bounds for the first closed and Neumann eigenvalues for the compact manifolds with positive Ricci curvature in terms of the diameter and the lower bound of Ricci curvature.

We give an estimate on the lower bound of the first non-zero eigenvalue of the Laplacian for a closed Riemannian manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature.

It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems canno...

Following work of Ecker (Comm Anal Geom 15:1025–1061, 2007), we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-withboundary. We compute its variational properties and its time derivative under Perelman’s modified Ricci flow. The answer has a boundary term which involves an extension of Hamilton’s differential Harnack expression for the mean curvature flow in Euclid...

The famous Frankel conjecture asserts that any compact Kähler manifold with positive bisectional curvature must be biholomorphic to CP n. This conjecture was settled affirmatively in early 1980s by two groups of mathematicians independently: Siu-Yau[16] via differential geometry method and Morri [15] by algebraic method. There are many interesting papers following this celebrated work; in parti...

One of the most interesting questions in Riemannian geometry is that of deciding whether a manifold admits curvatures of certain kinds. More specifically, one might want to know whether some given manifold admits a canonical metric, i.e. one with constant curvature of some form (sectional curvature, scalar curvature, etc.). (This will in fact have many topological implications.). One such probl...

On a compact Riemannian manifold with boundary, we study how Ricci curvature of the interior affects the geometry of the boundary. First we establish integral inequalities for functions defined solely on the boundary and apply them to obtain geometric inequalities involving the total mean curvature. Then we discuss related rigidity questions and prove Ricci curvature rigidity results for manifo...

We prove that the connected sums CP2#CP2 and CP2#CP2#CP2 admit self-dual metrics with positive Ricci curvature. Moreover, every self-dual metric of positive scalar curvature on CP2#CP2 is conformal to a metric with positive Ricci curvature. ∗Supported in part by NSF grant DMS-9204093