A visual introduction to Riemannian curvatures and some discrete generalizations

Yann Ollivier (1, 2)

1 LRI - Laboratoire de Recherche en Informatique
2 TAO - Machine Learning and Optimisation

We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the strategies used to define ana- logues of curvature in non-smooth or discrete spaces, beginning with Alexan- drov curvature and δ-hyperbolic spaces, and insisting on various notions of generalized Ricci curvature, which we briefly compare.

Domaines: Probabilités [math.PR]

Yann Ollivier. A visual introduction to Riemannian curvatures and some discrete generalizations. Galia Dafni, Robert McCann, Alina Stancu. Analysis and Geometry of Metric Measure Spaces: Lecture Notes of the 50th Séminaire de Mathématiques Supérieures (SMS), Montréal, 2011, AMS, pp. 197-219, 2013. ⟨hal-00858008⟩