Scalar curvature

In the context of tensor index notation, it is common to use the letter R to represent three different things:[7]

1. the Riemann curvature tensor: R_ijk^l or R_ijkl
2. the Ricci tensor: R_ij
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. Other notations used for scalar curvature include scal,[8] κ,[9] K,[10] r,[11] s or S,[12] and τ.[13]

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 scalar curvature.

Some authors instead define Ricci curvature and scalar curvature with a normalization factor, so that[10]

${\displaystyle R_{ij}={\frac {1}{n-1}}g^{kl}R_{kijl}{\text{ and }}R={\frac {1}{n}}g^{ij}R_{ij}.}$

The purpose of such a choice is that the Ricci and scalar curvatures become average values (rather than sums) of sectional curvatures.[14]

As a direct consequence of the Bianchi identities, any (pseudo-)Riemannian metric has the property that[5]

${\displaystyle {\frac {1}{2}}\nabla _{i}R=g^{jk}\nabla _{j}R_{ki}.}$

This identity is called the contracted Bianchi identity. It has, as an almost immediate consequence, the Schur lemma stating that if the Ricci tensor is pointwise a multiple of the metric, then the metric must be Einstein (unless the dimension is two). Moreover, this says that (except in two dimensions) a metric is Einstein if and only if the Ricci tensor and scalar curvature are related by

${\displaystyle R_{ij}={\frac {1}{n}}Rg_{ij},}$

where n denotes the dimension.[18] The contracted Bianchi identity is also fundamental in the mathematics of general relativity, since it identifies the Einstein tensor as a fundamental quantity.[19]

Given a (pseudo-)Riemannian metric g on a space of dimension n, the scalar curvature part of the Riemann curvature tensor is the (0,4)-tensor field

${\displaystyle {\frac {1}{n(n-1)}}R(g_{il}g_{jk}-g_{ik}g_{jl}).}$

(This follows the convention that R_ijkl = g_lp∂_iΓ_jk^p − ....) This tensor is significant as part of the Ricci decomposition; it is orthogonal to the difference between the Riemann tensor and itself. The other two parts of the Ricci decomposition correspond to the components of the Ricci curvature which do not contribute to scalar curvature, and to the Weyl tensor, which is the part of the Riemann tensor which does not contribute to the Ricci curvature. Put differently, the above tensor field is the only part of the Riemann curvature tensor which contributes to the scalar curvature; the other parts are orthogonal to it and make no such contribution.[20] There is also a Ricci decomposition for the curvature of a Kähler metric.[21]

The scalar curvature of a conformally changed metric can be computed:[22]

${\displaystyle R(e^{2f}g)=e^{-2f}{\Big (}R(g)-2(n-1)\Delta ^{g}f-(n-2)(n-1)g(df,df){\Big )},}$

using the convention Δ = g^ij ∇_i∇_j for the Laplace–Beltrami operator. Alternatively,[22]

${\displaystyle R(\psi ^{4/(n-2)}g)=-{\frac {4{\frac {n-1}{n-2}}\Delta ^{g}\psi -R(g)\psi }{\psi ^{\frac {n+2}{n-2}}}}.}$

Under an infinitesimal change of the underlying metric, one has[23]

${\displaystyle {\frac {\partial R}{\partial t}}=-\Delta ^{g}\left(g^{ij}{\frac {\partial g_{ij}}{\partial t}}\right)+\left(\nabla _{k}\nabla _{l}{\frac {\partial g_{ij}}{\partial t}}-R_{kl}{\frac {\partial g_{ij}}{\partial t}}\right)g^{ik}g^{jl}.}$

This shows in particular that the principal symbol of the differential operator which sends a metric to its scalar curvature is given by

${\displaystyle (\xi _{i},h_{ij})\mapsto -g(\xi ,\xi )g^{ij}h_{ij}+h_{ij}\xi ^{i}\xi ^{j}.}$

Furthermore the adjoint of the linearized scalar curvature operator is

${\displaystyle f\mapsto \nabla _{i}\nabla _{j}f-(\Delta f)g_{ij}-fR_{ij},}$

and it is an overdetermined elliptic operator in the case of a Riemannian metric. It is a straightforward consequence of the first variation formulas that, to first order, a Ricci-flat Riemannian metric on a closed manifold cannot be deformed so as to have either positive or negative scalar curvature. Also to first order, an Einstein metric on a closed manifold cannot be deformed under a volume normalization so as to increase or decrease scalar curvature.[23]

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 ${\displaystyle \rho _{1},\,\rho _{2}}$ 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

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

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

The scalar curvature is also constant when given a Kähler metric of constant holomorphic sectional curvature.[21]

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.

In both mathematics and general relativity, warped product metrics are an important source of examples. For example, the general Robertson–Walker spacetime, important to cosmology, is the Lorentzian metric

${\displaystyle -dt^{2}+f(t)^{2}g}$

on (a, b) × M, where g is a constant-curvature Riemannian metric on a three-dimensional manifold M. The scalar curvature of the Robertson–Walker metric is given by

${\displaystyle 6{\frac {f'(t)^{2}+f(t)f''(t)+6k}{f(t)^{2}}},}$

where k is the constant curvature of g.[26]

It is automatic that any Ricci-flat manifold has zero scalar curvature; the best-known spaces in this class are the Calabi–Yau manifolds. In the pseudo-Riemannian context, this also includes the Schwarzschild spacetime and Kerr spacetime.

There are metrics with zero scalar curvature but nonvanishing Ricci curvature. For example, there is a complete Riemannian metric on the tautological line bundle over real projective space, constructed as a warped product metric, which has zero scalar curvature but nonzero Ricci curvature. This may also be viewed as a rotationally symmetric Riemannian metric of zero scalar curvature on the cylinder R × Sⁿ.[27]

In the 1960s, André Lichnerowicz found that on a spin manifold, the difference between the square of the Dirac operator and the tensor Laplacian (as defined on spinor fields) is given exactly by one-quarter of the scalar curvature. This is a fundamental example of a Weitzenböck formula. As a consequence, if a Riemannian metric on a closed manifold has positive scalar curvature, then there can exist no harmonic spinors. It is then a consequence of the Atiyah–Singer index theorem that, for any closed spin manifold with dimension divisible by four and of positive scalar curvature, the  genus must vanish. This is a purely topological obstruction to the existence of Riemannian metrics with positive scalar curvature.[29]

Lichnerowicz's argument using the Dirac operator can be "twisted" by an auxiliary vector bundle, with the effect of only introducing one extra term into the Lichnerowicz formula.[30] Then, following the same analysis as above except using the families version of the index theorem and a refined version of the  genus known as the α-genus, Nigel Hitchin proved that in certain dimensions there are exotic spheres which do not have any Riemannian metrics of positive scalar curvature. Gromov and Lawson later extensively employed these variants of Lichnerowicz's work. One of their resulting theorems introduces the homotopy-theoretic notion of enlargeability and says that an enlargeable spin manifold cannot have a Riemannian metric of positive scalar curvature. As a corollary, a closed manifold with a Riemannian metric of nonpositive curvature, such as a torus, has no metric with positive scalar curvature. Gromov and Lawson's various results on nonexistence of Riemannian metrics with positive scalar curvature support a conjecture on the vanishing of a wide variety of topological invariants of any closed spin manifold with positive scalar curvature. This (in a precise formulation) in turn would be a special case of the strong Novikov conjecture for the fundamental group, which deals with the K-theory of C*-algebras.[31] This in turn is a special case of the Baum–Connes conjecture for the fundamental group.[32]

In the special case of four-dimensional manifolds, the Seiberg–Witten equations have been usefully applied to the study of scalar curvature. Similarly to Lichnerowicz's analysis, the key is an application of the maximum principle to prove that solutions to the Seiberg–Witten equations must be trivial when scalar curvature is positive. Also in analogy to Lichnerowicz's work, index theorems can guarantee the existence of nontrivial solutions of the equations. Such analysis provides new criteria for nonexistence of metrics of positive scalar curvature. Claude LeBrun pursued such ideas in a number of papers.[33]

By contrast to the above nonexistence results, Lawson and Yau constructed Riemannian metrics of positive scalar curvature from a wide class of nonabelian effective group actions.[30]

Later, Schoen–Yau and Gromov–Lawson (using different techniques) proved the fundamental result that existence of Riemannian metrics of positive scalar curvature is preserved by topological surgery in codimension at least three, and in particular is preserved by the connected sum. This establishes the existence of such metrics on a wide variety of manifolds. For example, it immediately shows that the connected sum of an arbitrary number of copies of spherical space forms and generalized cylinders S^m × Sⁿ has a Riemannian metric of positive scalar curvature. Grigori Perelman's construction of Ricci flow with surgery has, as an immediate corollary, the converse in the three-dimensional case: a closed orientable 3-manifold with a Riemannian metric of positive scalar curvature must be such a connected sum.[34]

Based upon the surgery allowed by the Gromov–Lawson and Schoen–Yau construction, Gromov and Lawson observed that the h-cobordism theorem and analysis of the cobordism ring can be directly applied. They proved that, in dimension greater than four, any non-spin simply connected closed manifold has a Riemannian metric of positive scalar curvature.[35] Stephan Stolz completed the existence theory for simply-connected closed manifolds in dimension greater than four, showing that as long as the α-genus is zero, then there is a Riemannian metric of positive scalar curvature.[36]

According to these results, for closed manifolds, the existence of Riemannian metrics of positive scalar curvature is completely settled in the three-dimensional case and in the case of simply-connected manifolds of dimension greater than four.