In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.
The definition of scalar curvature via partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature of a Lorentzian metric is one of the key terms in the Einstein field equations. Furthermore, this scalar curvature is the Lagrangian density for the Einstein–Hilbert action, the Euler–Lagrange equations of which are the Einstein field equations in vacuum.
The geometry of Riemannian metrics with positive scalar curvature has been widely studied. On noncompact spaces, this is the context of the positive mass theorem proved by Richard Schoen and Shing-Tung Yau in the 1970s, and reproved soon after by Edward Witten with different techniques. Schoen and Yau, and independently Mikhael Gromov and Blaine Lawson, developed a number of fundamental results on the topology of closed manifolds supporting metrics of positive scalar curvature. In combination with their results, Grigori Perelman's construction of Ricci flow with surgery in 2003 provided a complete characterization of these topologies in the three-dimensional case.
Definition Given a Riemannian metric , the scalar curvature Scal is defined as the trace of the Ricci curvature tensor with respect to the metric:
The scalar curvature cannot be computed directly from the Ricci curvature since the latter is a (0,2)-tensor field; the metric must be used to raise an index to obtain a (1,1)-tensor field in order to take the trace. In terms of local coordinates one can write, using the Einstein notation convention, that:
where are the components of the Ricci tensor in the coordinate basis, and where are the inverse metric components, i.e. the components of the inverse of the matrix of metric components . Based upon the Ricci curvature being a sum of sectional curvatures, it is possible to also express the scalar curvature as
where denotes the sectional curvature and is any orthonormal frame at . By similar reasoning, the scalar curvature is twice the trace of the curvature operator. Alternatively, given the coordinate-based definition of Ricci curvature in terms of the Christoffel symbols, it is possible to express scalar curvature as
where are the Christoffel symbols of the metric, and is the partial derivative of in the σ-coordinate direction.
The above definitions are equally valid for a pseudo-Riemannian metric. The special case of Lorentzian metrics is significant in the mathematical theory of general relativity, where the scalar curvature and Ricci curvature are the fundamental terms in the Einstein field equation.
However, unlike the Riemann curvature tensor or the Ricci tensor, the scalar curvature cannot be defined for an arbitrary affine connection, for the reason that the trace of a (0,2)-tensor field is ill-defined. However, there are other generalizations of scalar curvature, including in Finsler geometry.
Traditional notation In the context of tensor index notation, it is common to use the letter to represent three different things: the Riemann curvature tensor: or 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. Other notations used for scalar curvature include , , , , or , and .
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
The purpose of such a choice is that the Ricci and scalar curvatures become average values (rather than sums) of sectional curvatures.
Basic properties It is a fundamental fact that the scalar curvature is invariant under isometries. To be precise, if is a diffeomorphism from a space to a space , the latter being equipped with a (pseudo-)Riemannian metric , then the scalar curvature of the pullback metric on equals the composition of the scalar curvature of with the map . This amounts to the assertion that the scalar curvature is geometrically well-defined, independent of any choice of coordinate chart or local frame. More generally, as may be phrased in the language of homotheties, the effect of scaling the metric by a constant factor is to scale the scalar curvature by the inverse factor .
Furthermore, the scalar curvature is (up to an arbitrary choice of normalization factor) the only coordinate-independent function of the metric which, as evaluated at the center of a normal coordinate chart, is a polynomial in derivatives of the metric and has the above scaling property. This is one formulation of the Vermeil theorem.
Bianchi identity As a direct consequence of the Bianchi identities, any (pseudo-)Riemannian metric has the property that
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
where denotes the dimension. The contracted Bianchi identity is also fundamental in the mathematics of general relativity, since it identifies the Einstein tensor as a fundamental quantity.
Ricci decomposition Given a (pseudo-)Riemannian metric on a space of dimension , the scalar curvature part of the Riemann curvature tensor is the (0,4)-tensor field
(This follows the convention that .) 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. There is also a Ricci decomposition for the curvature of a Kähler metric.
Basic formulas The scalar curvature of a conformally changed metric can be computed:
using the convention for the Laplace–Beltrami operator. Alternatively,
Under an infinitesimal change of the underlying metric, one has
This shows in particular that the principal symbol of the differential operator which sends a metric to its scalar curvature is given by
Furthermore the adjoint of the linearized scalar curvature operator is
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.
Relation between volume and Riemannian scalar curvature When the scalar curvature is positive at a point, the volume of a small geodesic 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 . 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
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 ; their hypersurface measures ("areas") satisfy the following equation:
These expansions generalize certain characterizations of Gaussian curvature from dimension two to higher dimensions.
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
where 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
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:
The scalar curvature is also constant when given a Kähler metric of constant holomorphic sectional curvature.
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.
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
on , where is a constant-curvature Riemannian metric on a three-dimensional manifold . The scalar curvature of the Robertson–Walker metric is given by
where is the constant curvature of .
Scalar-flat spaces 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 .
Yamabe problem
The Yamabe problem was resolved in 1984 by the combination of results found by Hidehiko Yamabe, Neil Trudinger, Thierry Aubin, and Richard Schoen. They proved that every smooth 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 Riemannian metric on a closed manifold is conformal to one with constant scalar curvature.
Riemannian metrics of positive scalar curvature