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 small geodesic ball in a 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 of an n-manifold is defined as the trace of the Ricci tensor, and it can be defined as n(n − 1) times the average of the sectional curvatures at a point.

At first sight, the scalar curvature in dimension at least 3 seems to be a weak invariant with little influence on the global geometry of a manifold, but in fact some deep theorems show the power of scalar curvature. One such result is the positive mass theorem of Schoen, Yau and Witten. Related results give an almost complete understanding of which manifolds have a Riemannian metric with positive scalar curvature.

Definition

The scalar curvature S (commonly also R, or Sc) is defined as the trace of the Ricci curvature tensor with respect to the metric:

\( {\displaystyle S=\operatorname {tr} _{g}\operatorname {Ric} .} \)

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 Rij 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:

\( {\displaystyle S=g^{ab}\left({\Gamma ^{c}}_{ab,c}-{\Gamma ^{c}}_{ac,b}+{\Gamma ^{d}}_{ab}{\Gamma ^{c}}_{cd}-{\Gamma ^{d}}_{ac}{\Gamma ^{c}}_{bd}\right)=2g^{ab}\left({\Gamma ^{c}}_{a[b,c]}+{\Gamma ^{d}}_{a[b}{\Gamma ^{c}}_{c]d}\right)} \)

where \( {\displaystyle {\Gamma ^{a}}_{bc}} \) are the Christoffel symbols of the metric, and \( {\displaystyle {\Gamma ^{l}}_{jk,i}} \) is the partial derivative of\( {\displaystyle {\Gamma ^{l}}_{jk}} \) in the ith coordinate direction.

Unlike the Riemann curvature tensor or the Ricci tensor, both of which can be 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 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 ) {\displaystyle (M,g)} (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

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

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 of radius ε {\displaystyle \varepsilon } \varepsilon ; their hypersurface measures ("areas") satisfy the following equation:

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

Special cases
Surfaces

In two dimensions, scalar curvature is exactly twice the Gaussian curvature. For an embedded surface in Euclidean space R3, this means that

\( {\displaystyle S={\frac {2}{\rho _{1}\rho _{2}}}\,} \)

where \( \rho_1,\,\rho_2 \) are the principal radii of the surface. For example, the scalar curvature of the 2-sphere of radius r is equal to 2/r2.

The 2-dimensional Riemann curvature tensor has only one independent component, and it can be expressed in terms of the scalar curvature and metric area form. Namely, in any coordinate system, one has

\( {\displaystyle 2R_{1212}\,=S\det(g_{ij})=S\left[g_{11}g_{22}-(g_{12})^{2}\right].} \)

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 space: By the hyperboloid model, an n-dimensional hyperbolic space can be identified with the subset of (n + 1)-dimensional Minkowski space

\( {\displaystyle x_{0}^{2}-x_{1}^{2}-\cdots -x_{n}^{2}=r^{2},\quad x_{0}>0.} \)

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.

Products

The scalar curvature of a product M × N of Riemannian manifolds is the sum of the scalar curvatures of M and N. For example, for any smooth closed manifold M, M × S2 has a metric of positive scalar curvature, simply by taking the 2-sphere to be small compared to M (so that its curvature is large). This example might suggest that scalar curvature has little relation to the global geometry of a manifold. In fact, it does have some global significance, as discussed below.
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.

Yamabe problem
Main article: Yamabe problem

The Yamabe problem was solved by Trudinger, Aubin, and Schoen. Namely, every Riemannian metric on a closed manifold can be multiplied by some smooth positive function to obtain a metric with constant scalar curvature. In other words, every metric on a closed manifold is conformal to one with constant scalar curvature.
Positive scalar curvature

For a closed Riemannian 2-manifold M, the scalar curvature has a clear relation to the topology of M, expressed by the Gauss–Bonnet theorem: the total scalar curvature of M is equal to 4π times the Euler characteristic of M. For example, the only closed surfaces with metrics of positive scalar curvature are those with positive Euler characteristic: the sphere S2 and RP2. Also, those two surfaces have no metrics with scalar curvature ≤ 0.

The sign of the scalar curvature has a weaker relation to topology in higher dimensions. Given a smooth closed manifold M of dimension at least 3, Kazdan and Warner solved the prescribed scalar curvature problem, describing which smooth functions on M arise as the scalar curvature of some Riemannian metric on M. Namely, M must be of exactly one of the following three types:[1]

Every function on M is the scalar curvature of some metric on M.
A function on M is the scalar curvature of some metric on M if and only if it is either identically zero or negative somewhere.
A function on M is the scalar curvature of some metric on M if and only if it is negative somewhere.

Thus every manifold of dimension at least 3 has a metric with negative scalar curvature, in fact of constant negative scalar curvature. Kazdan–Warner's result focuses attention on the question of which manifolds have a metric with positive scalar curvature, that being equivalent to property (1). The borderline case (2) can be described as the class of manifolds with a strongly scalar-flat metric, meaning a metric with scalar curvature zero such that M has no metric with positive scalar curvature.

A great deal is known about which smooth closed manifolds have metrics with positive scalar curvature. In particular, by Gromov and Lawson, every simply connected manifold of dimension at least 5 which is not spin has a metric with positive scalar curvature.[2] By contrast, Lichnerowicz showed that a spin manifold with positive scalar curvature must have Â genus equal to zero. Hitchin showed that a more refined version of the Â genus, the α-invariant, also vanishes for spin manifolds with positive scalar curvature.[3] This is only nontrivial in some dimensions, because the α-invariant of an n-manifold takes values in the group KOn, listed here:

n (mod 8) 0 1 2 3 4 5 6 7
KOn Z Z/2 Z/2 0 Z 0 0 0

Conversely, Stolz showed that every simply connected spin manifold of dimension at least 5 with α-invariant zero has a metric with positive scalar curvature.[4]

Lichnerowicz's argument using the Dirac operator has been extended to give many restrictions on non-simply connected manifolds with positive scalar curvature, via the K-theory of C*-algebras. For example, Gromov and Lawson showed that a closed manifold that admits a metric with sectional curvature ≤ 0, such as a torus, has no metric with positive scalar curvature.[5] More generally, the injectivity part of the Baum–Connes conjecture for a group G, which is known in many cases, would imply that a closed aspherical manifold with fundamental group G has no metric with positive scalar curvature.[6]

There are special results in dimensions 3 and 4. After work of Schoen, Yau, Gromov, and Lawson, Perelman's proof of the geometrization theorem led to a complete answer in dimension 3: a closed orientable 3-manifold has a metric with positive scalar curvature if and only if it is a connected sum of spherical 3-manifolds and copies of S2 × S1.[7] In dimension 4, positive scalar curvature has stronger implications than in higher dimensions (even for simply connected manifolds), using the Seiberg–Witten invariants. For example, if X is a compact Kähler manifold of complex dimension 2 which is not rational or ruled, then X (as a smooth 4-manifold) has no Riemannian metric with positive scalar curvature.[8]

Finally, Akito Futaki showed that strongly scalar-flat metrics (as defined above) are extremely special. For a simply connected Riemannian manifold M of dimension at least 5 which is strongly scalar-flat, M must be a product of Riemannian manifolds with holonomy group SU(n) (Calabi–Yau manifolds), Sp(n) (hyperkähler manifolds), or Spin(7).[9] In particular, these metrics are Ricci-flat, not just scalar-flat. Conversely, there are examples of manifolds with these holonomy groups, such as the K3 surface, which are spin and have nonzero α-invariant, hence are strongly scalar-flat.