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Ricci curvature

In differential geometry, Ricci curvature is 2-valent tensor, obtained as a trace of the full curvature tensor. It can be thought of as a Laplacian of the Riemannian metric tensor in the case of Riemannian manifolds.

In dimension 2 and 3, the curvature tensor is completely determined by the Ricci curvature. One can think of Ricci curvature on a Riemannian manifold, as being an operator on the tangent space. If this operator is just multiplication by a constant, then we have an Einstein manifold. The Ricci curvature is proportional to the metric tensor in this case.

Ricci curvature can be explained in terms of the sectional curvature in the following way: for a unit vector v, is sum of the sectional curvatures of all the planes spanned by the vector v and a vector from an orthonormal frame containing v (there are n-1 such planes). Here R(v) is Ricci curvature as a linear operator on the tangent plane, and <.,.> is metric scalar product. Ricci curvature contains the same information as all such sums over all unit vectors. In dimensions 2 and 3 this is the same as specifying all sectional curvatures or the curvature tensor, but in higher dimensions Ricci curvature contains less information. For instance, Einstein manifolds do not have to have constant curvature in dimensions 4 and up.

Applications of Ricci curvature tensor

The Ricci-curvature can be used to define Chern classes of a manifold, which are topological invariants independent of the metric.

Ricci curvature is also used in Ricci flow, where metric is deformed in the direction of the Ricci curvature. Hence, Einstein metrics are limits of this flow. On surfaces, the flow produces metric of constant curvature, and uniformization theorem for surfaces follows. In dimension 3, Ricci flow can be used to split a manifold into parts having constant curvature.

Ricci curvature plays an important role in general relativity, where it is the key term in Einstein equations.

Volume growth and global topology of positive Ricci curvature

If Ricci curvature is bounded from below on a complete Riemannian manifold by (n-1)k, then its diameter is bounded by , and manifold has to have a finite fundamental group (Mayers estimate). The volume of a ball is smaller or equal to the volume of a ball inside a spere of the constant curvature k, and as radius of a ball goes to zero, the ratio of the two volumes increasingly approaches one (relative volume comparison). If the diameter is equal to , then the manifold is isometric to a sphere of a constant curvature k (maximal diameter rigidity).