Ricci curvature is the trace of the Riemann curvature tensor in its first and third arguments. For a unit tangent vector and an orthonormal basis ,
Scalar curvature is the complete trace of the Riemann curvature tensor:An -manifold of constant sectional curvature has .
A Ricci-flat Riemannian manifold has identically zero Ricci tensor. Every flat Riemannian manifold is Ricci-flat, while the converse can fail in dimension at least four.
If a complete connected -dimensional Riemannian manifold satisfies for some , then its diameter is at most . It is therefore compact and has finite fundamental group.
A complete connected Riemannian manifold with nonnegative Ricci curvature that contains a line splits isometrically as .
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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.