Approximating Coarse Ricci Curvature on Metric Measure Spaces with Applications to Submanifolds of Euclidean Space

Coarse Ricci Curvature with Applications to Manifold Learning

Consider a sample of n points taken i.i.d from a submanifold of Euclidean space. This defines a metric measure space. We show that there is an explicit set of scales tn → 0 such that a coarse Ricci curvature at scale tn on this metric measure space converges almost surely to the coarse Ricci curvature of the underlying manifold.

RICCI CURVATURE OF SUBMANIFOLDS OF A SASAKIAN SPACE FORM

Involving the Ricci curvature and the squared mean curvature, we obtain basic inequalities for different kind of submaniforlds of a Sasakian space form tangent to the structure vector field of the ambient manifold. Contrary to already known results, we find a different necessary and sufficient condition for the equality for Ricci curvature of C-totally real submanifolds of a Sasakian space form...

Coarse Ricci curvature and the manifold learning problem

Abstract. We consider the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. We discuss applications of our construction to the manifold learning problem, specifically to the statistical problem of estimating the Ricci curvature of a su...

Shape Operator of Slant Submanifolds in Kenmotsu Space Forms

Ricci curvature of Markov chains on metric spaces

We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scaling, the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definit...