Scalar curvature

From Free net encyclopedia

In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. It assigns to each point on a Riemannian manifold a single real number characterizing the intrinsic curvature of the manifold at that point.

In two dimensions the scalar curvature completely characterizes the curvature of a Riemannian manifold. In dimensions ≥ 3, however, more information is needed. See curvature of Riemannian manifolds for a complete discussion.

The scalar curvature usually denoted by S (other notation are Sc, R). It is defined as the trace of the Ricci curvature tensor with respect to the metric:

<math>S = \mbox{tr}_g\,Ric</math>

The trace depends on the metric since the Ricci tensor is a (0,2)-valent tensor; one must first contract with the metric to obtain a (1,1)-valent tensor in order take the trace (see musical isomorphisms). In terms of local coordinates one can write

<math>S = g^{ij}R_{ij}</math>

where

<math>Ric = R_{ij}\,dx^i\otimes dx^j</math>

On R-notation

Among those who use index notation for tensors, it is common to use the letter R to represent three different things:

1. the Riemann curvature tensor
2. the Ricci tensor
3. the Ricci scalar

The three are distinguished from each other by their number of indices: the Riemann tensor has four indices, the Ricci tensor has two indices, and the Ricci scalar has zero indices. Those not using an index notation usually reserve R for the full Riemann curvature tensor.es:Escalar de curvatura de Ricci fr:Scalaire de Ricci