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Ricci curvature

In [differential geometry] , the ** Ricci curvature tensor ** , named after [Gregorio Ricci-Curbastro] , represents the amount by which the [volume element] of a [geodesic] [ball] in a curved [Riemannian manifold] deviates from that of the standard ball in [Euclidean space] . As such, it provides one way of measuring the degree to which the geometry determined by a given [Riemannian metric] might differ from that of ordinary Euclidean * n- * space. More generally, the Ricci tensor is defined on any [pseudo-Riemannian manifold] . Like the metric itself, the Ricci tensor is a [symmetric bilinear form] on the [tangent space] of the manifold.

The Ricci curvature is broadly applicable to modern [Riemannian geometry] and [general relativity] theory. In connection with the latter, it is up to an overall [trace] term, the portion of the [Einstein field equation] representing the geometry of [spacetime] , the other significant portion of which comes from the presence of matter and energy. In connection with the former, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. [comparison theorem] ) with the geometry of a constant curvature [space form] . If the Ricci tensor satisfies the vacuum Einstein equation, then the manifold is an [Einstein manifold] , which have been extensively studied (cf. ). In this connection, the [Ricci flow] equation governs the evolution of a given metric to an Einstein metric, the precise manner in which this occurs ultimately leads to the [solution of the PoincarĂ© conjecture] .

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