The Ricci curvature tensor is a rank , symmetric tensor that arises naturally in pseudo-Riemannian geometry. Let be a smooth, -dimensional pseudo-Riemannian manifold, and let denote the corresponding Riemann curvature tensor. The Ricci tensor is commonly defined as the following contraction of the full curvature tensor:
The index symmetry of , 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 form. To that end for vector-fields we will write
Contracting the Ricci tensor, we obtain an important scalar invariant
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
denote the sectional curvature of along the plane spanned by vectors . Fix a point and a tangent vector , and let
In this way, the quantity
describes the average value of the sectional curvature for all planes in that contain . It is possible to show that
thereby giving us the desired geometric interpretation.
Decomposition of the curvature tensor.
For , the Ricci tensor can be characterized in terms of the decomposition of the full curvature tensor into three covariantly defined summands, namely
The 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 and correspond to the trace-free part of the Ricci curvature tensor, and to the Ricci scalar. Indeed, we can recover and from and as follows:
The Ricci tensor also plays an important role in the theory of general relativity. In this keystone application, is a 4-dimensional pseudo-Riemannian manifold with signature . 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 manifold is Ricci flat if and only if locally, the manifold, is conformally equivalent to flat space.
|Date of creation||2013-03-22 15:02:38|
|Last modified on||2013-03-22 15:02:38|
|Last modified by||rmilson (146)|
|Defines||Weyl curvature tensor|