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.

Global topology and geometry of positive Ricci curvature

Mayers theorem states that if Ricci curvature is bounded from below on a complete Riemannian manifold by (n-1)k > 0, then its diameter , and manifold has to have a finite fundamental group. If the diameter is equal to , then the manifold is isometric to a sphere of a constant curvature k.

Bishop-Gromov inequality states that if Ricci curvature of a complete m-dimensional Riemannian manifold is ≥0 then the volume of a ball is smaller or equal to the volume of a ball of the same radius in Eulidean m-space. More over if denotes the volume of the ball with center p and radius in the manifold and denotes the volume of the ball of radius R in Euclidean m-space then function is nonincreasing. (The last inequality can be generalized to arbitary curvature bound and is the key point in the proof of Gromov's compactness theorem.)

Cheeger-Gromoll splitting theorem states that if a complete Riemannian manifold with Ricc ≥ 0 has a straight line (i.e. a both sides infinite minimizing geodesic) then it is isometric to a product space RL, where L is a Riemannian manifold.