Ricci tensor


The Ricci curvature tensor is a rank 2, symmetric tensorMathworldPlanetmath that arises naturally in pseudo-Riemannian geometry. Let (M,gij) be a smooth, n-dimensional pseudo-Riemannian manifold, and let Rijkl denote the corresponding Riemann curvature tensorMathworldPlanetmath. The Ricci tensor Rij is commonly defined as the following contractionPlanetmathPlanetmath of the full curvature tensor:


The index symmetryPlanetmathPlanetmath of Rij, so defined, follows from the symmetry properties of the Riemann curvature. To wit,


It is also convenient to regard the Ricci tensor as a symmetric bilinear formMathworldPlanetmath. To that end for vector-fields X,Y we will write


Related objects.

Contracting the Ricci tensor, we obtain an important scalar invariantMathworldPlanetmath


called the scalar curvature, and sometimes also called the Ricci scalar. Closely related to the Ricci tensor is the tensor


called the Einstein tensor. The Einstein tensor is also known as the trace-reversed Ricci tensor owing to the fact that


Another related tensor is


This is called the trace-free Ricci tensor, owing to the fact that the above definition implies that


Geometric interpretation.

In Riemannian geometry, the Ricci tensor represents the average value of the sectional curvatureMathworldPlanetmath along a particular direction. Let


denote the sectional curvature of M along the plane spanned by vectors u,vTxM. Fix a point xM and a tangent vectorMathworldPlanetmath vTxM, and let


denote the n-2 dimensional sphere of those unit vectorsMathworldPlanetmath at x that are perpendicularMathworldPlanetmathPlanetmath to v. Let μx denote the natural (n-2)-dimensional volume measure on TxM, normalized so that


In this way, the quantity


describes the average value of the sectional curvature for all planes in TxM that contain v. It is possible to show that


thereby giving us the desired geometric interpretationMathworldPlanetmathPlanetmath.

Decomposition of the curvature tensor.

For n3, the Ricci tensor can be characterized in terms of the decomposition of the full curvature tensor into three covariantly defined summands, namely

Fijkl =1n-2(Sjlgik+Sikgjl-Silgjk-Sjkgil),
Eijkl =1n(n-1)R(gjlgik-gilgjk),
Wijkl =Rijkl-Fijkl-Eijkl.

The Wijkl is called the Weyl curvature tensor. It is the conformally invariant, trace-free part of the curvature tensor. Indeed, with the above definitions, we have


The Eijkl and Fijkl correspond to the trace-free part of the Ricci curvature tensor, and to the Ricci scalar. Indeed, we can recover Sij and R from Eijkl and Fijkl as follows:

Sij =Fk,ikj
Eijij =R.


The Ricci tensor also plays an important role in the theory of general relativity. In this keystone application, M is a 4-dimensional pseudo-Riemannian manifold with signaturePlanetmathPlanetmathPlanetmathPlanetmath (3,1). The Einstein field equations assert that the energy-momentum tensor is proportional to the Einstein tensor. In particular, the equation


is the field equation for a vacuum space-time. In geometry, a pseudo-Riemannian manifold that satisfies this equation is called Ricci-flat. It is possible to prove that a manifoldMathworldPlanetmath is Ricci flat if and only if locally, the manifold, is conformally equivalent to flat space.

Title Ricci tensor
Canonical name RicciTensor
Date of creation 2013-03-22 15:02:38
Last modified on 2013-03-22 15:02:38
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 9
Author rmilson (146)
Entry type Definition
Classification msc 83C05
Defines scalar curvature
Defines Einstein tensor
Defines ricci scalar
Defines Weyl tensor
Defines Weyl curvature tensor