Scalar curvature

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In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near 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 raise an index to obtain a (1,1)-valent tensor in order to take the trace. In terms of local coordinates one can write

S = gijRij

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

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

where \Gamma^a_{bc} are the Christoffel symbols of the metric.

Unlike the Riemann curvature tensor or the Ricci tensor, which both can be naturally be defined for any affine connection, the scalar curvature is entirely special to the realm of Riemannian geometry; its very definition involves the metric in an inextricable fashion.

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[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:

S = \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 in terms of the scalar curvature and metric area form. In any coordinate system, one thus has:

2R_{1212} \,= S \det (g_{ij}) = S[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:

  1. the Riemann curvature tensor: R_{ijk}^l or Rabcd
  2. the Ricci tensor: Rij
  3. the scalar curvature: R

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.

[edit] See also

v • d • e
Different notions of curvature defined in differential geometry
Differential geometry of curves
Differential geometry of surfaces
Riemannian geometry
Curvature of connections