Examples of "scalar_curvature"
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.
In differential geometry, a constant scalar curvature Kähler metric (cscK metric), is (as the name suggests) a Kähler metric on a complex manifold whose scalar curvature is constant.
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 4 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 "S" and RP. Also, those two surfaces have no metrics with scalar curvature ≤ 0.
The scalar curvature of de Sitter space is given by
This curvature is one-half of the Ricci scalar curvature.
If formula_39, the scalar curvature equation might be more complicated.
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:
The scalar curvature, written in components, then expands to
The scalar curvature is the trace of the Ricci curvature,
It is the negative "L"-gradient flow of the (normalized) total scalar curvature, restricted to a given conformal class: it can be interpreted as deforming a Riemannian metric to a conformal metric of constant scalar curvature, when this flow converges.
if and only if formula_1 admits a metric of positive scalar curvature. The significance of this fact is that much is known about the topology of manifolds with metrics of positive scalar curvature.
where formula_16 is the determinant of a spacetime Lorentz metric and formula_17 is the scalar curvature.
Scalar curvature is a function on any Riemannian manifold, usually denoted by "Sc".
where "R" is the scalar curvature, a measure of the curvature of space.
The "gradient" of the scalar curvature follows from the Bianchi identity (proof):
The trace of the Ricci tensor, called the scalar curvature, is
where S(x) is scalar curvature, the trace of the Ricci curvature, on M.
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. 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. This is only nontrivial in some dimensions, because the α-invariant of an "n"-manifold takes values in the group "KO", listed here:
If the dimension of "M" is three or greater, then any smooth function "ƒ" which takes on a negative value somewhere is the scalar curvature of some Riemannian metric. The assumption that "ƒ" be negative somewhere is needed in general, since not all manifolds admit metrics which have strictly positive scalar curvature. (For example, the three-dimensional torus is such a manifold.) However, Kazdan and Warner proved that if "M" does admit some metric with strictly positive scalar curvature, then any smooth function "ƒ" is the scalar curvature of some Riemannian metric.
Given a coordinate system and a metric tensor, scalar curvature can be expressed as follows