Ricci curvature

Ricci curvature is a tensor obtained by tracing the Riemann curvature tensor. It captures how geodesic families and small-volume growth differ from their Euclidean behavior in each direction.

Riemann curvature tensor: A tensor describing the curvature of a Riemannian manifold through the change of tangent vectors under parallel transport. Ricci curvature is its trace over two tensor indices.
Riemannian manifold: A smooth manifold equipped with a smoothly varying positive-definite inner product on each tangent space. Ricci curvature measures intrinsic geometry on this kind of manifold.
Ricci flow: A geometric evolution equation that changes a Riemannian metric in proportion to minus twice its Ricci tensor. The flow uses Ricci curvature to smooth and reshape geometric irregularities.
Scalar curvature: The scalar obtained by tracing the Ricci tensor, measuring total curvature at a point. It compresses Ricci curvature's directional information into a single number.
Sectional curvature: The Gaussian curvature of a two-dimensional tangent plane within a Riemannian manifold. Ricci curvature sums sectional curvatures over planes containing a chosen direction.
Riemannian metric: A smoothly varying inner product on tangent spaces that defines lengths, angles, and distances. The metric determines the connection and curvature from which Ricci curvature is formed.
Hamilton's cigar soliton: A steady two-dimensional Ricci soliton with a complete metric shaped like a cigar. It is an explicit model showing how Ricci flow can preserve a translating shape.
Weyl curvature tensor: The trace-free part of the Riemann curvature tensor in dimensions at least four. It captures conformal curvature that Ricci curvature's traces do not detect.
Jacobi field: A vector field describing the infinitesimal separation of neighboring geodesics in a variation. Its evolution along geodesics reveals how Ricci curvature focuses geodesic families.
Levi-Civita connection: The unique torsion-free connection compatible with a Riemannian metric. Its curvature tensor supplies the quantity that is traced to obtain Ricci curvature.

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