Scalar curvature
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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>
[edit] Direct geometric interpretation
When the scalar curvature is positive at a point, the volume of a small ball about the point has smaller volume than a ball of the same radius in Euclidean space. On the other hand, when the scalar curvature is negative at a point, the volume of a small ball is instead larger than it would be in Euclidean space.
This can be made more quantitative, in order to characterize the precise value of the scalar curvature S at a point p of a Riemannian n-manifold <math>(M,g)</math>. Namely, the ratio of the n-dimensional volume of a ball of radius ε in the manifold to that of a corresponding ball in Euclidean space is given, for small ε, by
<math> \frac{Vol (B_\varepsilon(p) \subset M)}{Vol (B_\varepsilon(0)\subset {\mathbb R}^n)}= 1- \frac{S}{6(n+2)}\varepsilon^2 + O(\varepsilon^4)</math>
Thus, the second derivative of this ratio, evaluated at radius ε=0, is exactly minus the scalar curvature divided by 3(n+2).
[edit] Traditional notation
Among those who use index notation for tensors, it is common to use the letter R to represent three different things:
- the Riemann curvature tensor
- the Ricci tensor
- the scalar curvature
These three are then 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.
