Scalar curvature

In Riemannian geometry, the scalar curvature (or the 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. Specifically, the scalar curvature represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

In general relativity, the scalar curvature is the Lagrangian density for the Einstein–Hilbert action. The Euler–Lagrange equations for this Lagrangian under variations in the metric constitute the vacuum Einstein field equations, and the stationary metrics are known as Einstein metrics. The scalar curvature is defined as the trace of the Ricci tensor, and it can be characterized as a multiple of the average of the sectional curvatures at a point. Unlike the Ricci tensor and sectional curvature, however, global results involving only the scalar curvature are extremely subtle and difficult. One of the few is the positive mass theorem of Richard Schoen, Shing-Tung Yau and Edward Witten. Another is the Yamabe problem, which seeks extremal metrics in a given conformal class for which the scalar curvature is constant.

Definition

The scalar curvature is usually denoted by S (other notations 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 = g^{ij}R_{ij} = R^j_j

where R_ij are the components of the Ricci tensor in the coordinate basis:

\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^d_{ab}\Gamma^c_{cd} - \Gamma^d_{ac} \Gamma^c_{bd}) = 2g^{ab} (\Gamma^c_{a[b,c]} + \Gamma^d_{a[b}\Gamma^c_{c]d})

where $\Gamma^a_{bc}$ are the Christoffel symbols of the metric, and $\Gamma _{{jk,i}}^{{l}}$ is the partial derivative of $\Gamma _{{jk}}^{{l}}$ in the $i$ -th coordinate direction.

Unlike the Riemann curvature tensor or the Ricci tensor, which both can be naturally defined for any affine connection, the scalar curvature requires a metric of some kind. The metric can be pseudo-Riemannian instead of Riemannian. Indeed, such a generalization is vital to relativity theory. More generally, the Ricci tensor can be defined in broader class of metric geometries (by means of the direct geometric interpretation, below) that includes Finsler geometry.

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 $\epsilon$ ; their hypersurface measures ("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).

Special cases

Surfaces

In two dimensions, scalar curvature is exactly twice the Gaussian curvature. For an embedded surface in Euclidean space, this means that

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/r².

The 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].

Space forms

A space form is by definition a Riemannian manifold with constant sectional curvature. Space forms are locally isometric to one of the following types:

Euclidean space: The Riemann tensor of an n-dimensional Euclidean space vanishes identically, so the scalar curvature does as well.
n-spheres: The sectional curvature of an n-sphere of radius r is K = 1/r². Hence the scalar curvature is S = n(n−1)/r².
Hyperbolic spaces: By the hyperboloid model, an n dimensional hyperbolic space can be identified with the subset of (n+1)-dimensional Minkowski space

x_0^2-x_1^2-\cdots-x_n^2 = r^2,\quad x_0>0.

The parameter r is a geometrical invariant of the hyperbolic space, and the sectional curvature is K = −1/r². The scalar curvature is thus S = −n(n−1)/r².

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: $R_{ijk}^l$ or $R_{abcd}$
the Ricci tensor: $R_{ij}$
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. Alternatively, in a coordinate-free notation one may use Riem for the Riemann tensor, Ric for the Ricci tensor and R for the curvature scalar.

References

Besse, A.L. (1987), Einstein manifolds, Springer .
Eisenhart, L.P. (1949), Riemannian geometry, Princeton Univ. Press .
Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Vol. 2, Wiley-Interscience, ISBN 978-0-471-15732-8 .
Jürgen Jost (1995), Riemannian Geometry and Geometric Analysis, Springer .
Ricci, G. (1903–1904), "Direzioni e invarianti principali in una varietà qualunque", Atti R. Inst. Veneto, 63 (2): 1233–1239 .
L.A. Sidorov (2001), "Ricci tensor", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Various notions of curvature defined in differential geometry

Differential geometry of curves	Curvature Torsion of a curve Frenet–Serret formulas Radius of curvature (applications) Affine curvature Total curvature Total absolute curvature

Differential geometry of surfaces	Principal curvatures Gaussian curvature Mean curvature Darboux frame Gauss–Codazzi equations First fundamental form Second fundamental form Third fundamental form

Riemannian geometry	Curvature of Riemannian manifolds Riemann curvature tensor Ricci curvature Scalar curvature Sectional curvature

Curvature of connections	Curvature form Torsion tensor Cocurvature Holonomy

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