Scalar curvature of a smooth connection #
The scalar curvature of a connection on a Riemannian tangent bundle is the metric trace of its
Ricci tensor. The definition is basis-free; in an orthonormal basis it is the sum of the diagonal
Ricci curvatures. In the chart-local frame of a tangent trivialization it is the classical
contraction gⁱʲ Rᵢⱼ of the Ricci coefficients against the inverse Gram matrix. Applying it to
the Levi-Civita connection gives the scalar curvature of a Riemannian manifold; for a smooth
metric g, Bundle.ContMDiffRiemannianMetric.scalarCurvature records it as a function of g, so
that several metrics on one manifold can be compared. For a metric f g rescaled by a positive
function, Bundle.ContMDiffRiemannianMetric.scalarCurvature_rescale_eq_sum computes its scalar
curvature in an orthonormal basis for g.
As for CovariantDerivative.ricciTensor, the connection need not preserve the metric or be torsion
free. The pointwise formulas do not need the metric to vary smoothly; for a smooth metric, the
chart formula shows that scalar curvature is a smooth function. Compactness and absence of
boundary are not required.
For the convention
R(w,u)v = κ (⟨u,v⟩ w - ⟨w,v⟩ u), the scalar curvature is n (n - 1) κ in
real dimension n. Consequently, in dimension at least two, a function κ for which the curvature
has this form near a point is differentiable there
(CovariantDerivative.mdifferentiableAt_of_eventually_curvatureTensor_eq_smul_inner_sub).
The definition and convention follow J. M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer GTM 176 (2018), Chapter 7.
The scalar curvature of a smooth connection on the tangent bundle, defined as the metric trace of its Ricci tensor.
Instances For
Scalar curvature is the metric trace of Ricci curvature.
In an orthonormal basis, scalar curvature is the sum of the diagonal Ricci curvatures.
In the chart-local frame eᵢ at a, scalar curvature is the contraction gⁱʲ Ric(eᵢ, eⱼ)
of the Ricci coefficients against the inverse Gram matrix of the frame. This holds at every point
of the chart source; the frame need not be orthonormal.
The scalar curvature of a smooth connection on the tangent bundle is smooth when the metric is smooth. Neither metric compatibility nor vanishing torsion is required, so this applies in particular to the Levi-Civita connection of a smooth metric.
The scalar curvature of a smooth connection on the tangent bundle is continuous when the metric is smooth.
In an orthonormal basis, scalar curvature is the double contraction of the curvature tensor. The inner sum contracts the curvature output against its first argument; the outer sum contracts the two Ricci arguments.
Vanishing Ricci curvature implies vanishing scalar curvature.
A point with zero curvature tensor has zero scalar curvature.
If Ricci curvature is a scalar multiple of the metric, scalar curvature is that scalar times the real dimension.
Contracting curvature of the form R(w,u)v = B(u,v)w - B(w,v)u gives scalar curvature
(dim - 1) trace(B).
The scalar curvature of the constant-curvature model
R(w,u)v = κ (⟨u,v⟩ w - ⟨w,v⟩ u) is n (n - 1) κ.
In dimension at least two, a function κ for which the curvature tensor has the pointwise
constant-curvature form R(w, u)v = κ(y) (⟨u, v⟩ w - ⟨w, v⟩ u) at every point y near x is
differentiable at x: near x it is the smooth scalar curvature divided by n (n - 1).
The scalar curvature of a smooth Riemannian metric g: the scalar curvature of the
Levi-Civita connection of g, with g as the fibre inner product.
Instances For
The scalar curvature of g is that of its Levi-Civita connection, with g installed as the
fibre inner product.
If the installed fibre metric is g, the scalar curvature of g is that of the Levi-Civita
connection. This transports scalar curvature along an equality of fibre metrics, such as
toRiemannianMetric_ofLE, which cannot be rewritten inside the instance-dependent arguments.
The scalar curvature of a rescaled metric f g at x, computed in an orthonormal basis b of
T_x M for g: it is (f x)⁻¹ ∑ⱼ Ric^(f g)_x (b j) (b j), since dividing b by √(f x) gives a
basis orthonormal for f g.
The scalar curvature of a smooth Riemannian metric is a smooth function.
The scalar curvature of a smooth Riemannian metric is continuous.