 Ricci curvature

Ricci curvature is the trace of the Riemann curvature tensor in its first and third arguments. For a unit tangent vector $X$ and an orthonormal basis $X,e_2,\dots ,e_n$,

Ricci curvature is a geometric concept that arises in the study of Riemannian and pseudo-Riemannian manifolds within the field of differential geometry. It measures how much the shape of a manifold deviates from being flat in a particular way, focusing on how volumes are distorted by the curvature of the space. To define Ricci curvature, we start with the Riemann curvature tensor, which encapsulates all the geometrical information about the curvature of a manifold.