Curvature

Abstract: Curvature is the most important concept of Riemannian geometry. It quantifies the local deviation of a Riemannian manifold from being Euclidean.  In the 20th century, various curvature concepts have been developed in metric geometry that no longer utilize infinitesimal calculations, but the geometry of triangles. These concepts require that any two points can be connected by a geodesic arc. With Parvaneh Joharinad, we have developed a more general curvature concept based upon intersection patterns of balls. This concept is useful in data analysis and machine learning.