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:

$S = \mbox{tr}_g\,\operatorname{Ric}$

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

S = g i j R i j

where

$\operatorname{Ric} = R_{ij}\,dx^i\otimes dx^j$

Given a coordinate system and a metric tensor, scalar curvature can be expressed as follows

$R = g^{ab} (\Gamma^c_{ab,c} - \Gamma^c_{ac,b} + \Gamma^c_{ab}\Gamma^d_{cd} - \Gamma^d_{ac} \Gamma^c_{bd})$

1 Direct geometric interpretation
2 2 dimensions
3 Traditional notation
4 See also

[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 $(M, g)$ . 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

$\frac{\operatorname{Vol} (B_\varepsilon(p) \subset M)}{\operatorname{Vol} (B_\varepsilon(0)\subset {\mathbb R}^n)}= 1- \frac{S}{6(n+2)}\varepsilon^2 + O(\varepsilon^4)$

Thus, the second derivative of this ratio, evaluated at radius ε = 0, is exactly minus the scalar curvature divided by 3(n + 2).

Boundaries of these balls are (n-1) dimensional spheres with radii $ε$ ; their areas satisfy the following equation:

$\frac{\operatorname{Area} (\partial B_\varepsilon(p) \subset M)}{\operatorname{Area} (\partial B_\varepsilon(0)\subset {\mathbb R}^n)}= 1- \frac{S}{6n}\varepsilon^2 + O(\varepsilon^4)$

[edit] 2 dimensions

In 2 dimensions, scalar curvature is exactly twice the Gauss curvature:

$R = \frac{2}{\rho_1\rho_2}$

where $\rho_1,\,\rho_2$ are principal radii of the surface. For example, scalar curvature of a sphere with radius r is equal to $2/r^2\,$ . More generally, scalar curvature of an n-sphere with a radius r is $n(n-1)/r^2\,$ .

2-dimensional Riemann tensor has only one independent component and it can be easily expressed through scalar curvature:

$2R_{1212} \,= Rg = R(g_{11}g_{22}-(g_{12})^2)$

[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.