The Cotton tensor of a smooth connection #
On a Riemannian manifold, the Cotton tensor is the antisymmetrized covariant derivative of the
Schouten tensor P:
C(X, Y, Z) = (∇_Z P)(X, Y) - (∇_Y P)(X, Z).
It measures the failure of P to satisfy the Codazzi equation. Classically, in dimension 3,
where the Weyl tensor vanishes identically, the Cotton tensor is the obstruction to local conformal
flatness, and in higher dimensions it is a multiple of the divergence of the Weyl tensor; this file
develops the definition and its first properties, not these characterizations.
The covariant derivative ∇P is the covariant derivative of a field of bilinear forms,
CovariantDerivative.derivBilinearTensor, applied to the Schouten tensor. As for
CovariantDerivative.schoutenTensor, the definitions are made for any smooth connection on a
Riemannian tangent bundle, and the same connection differentiates P. For a smooth metric, P
pairs differentiable vector fields to differentiable functions, so ∇P and C are given on
vector fields by the usual formulas (CovariantDerivative.derivSchoutenTensor_apply and
CovariantDerivative.cottonTensor_apply_of_mdifferentiableAt).
The Cotton tensor is antisymmetric in its last two arguments by definition. For a
metric-compatible torsion-free connection, P is symmetric, hence so is ∇P in its first two
arguments, and the cyclic sum of the Cotton tensor vanishes. If P is a constant multiple of the
metric, metric compatibility gives ∇P = 0, so the Cotton tensor vanishes; in particular it
vanishes on every space of constant sectional curvature of dimension at least 3. The Cotton
tensor at a point also vanishes when the curvature vanishes near that point, since then P
vanishes near it.
Main declarations #
CovariantDerivative.derivSchoutenTensor: the covariant derivative∇Pof the Schouten tensor, withCovariantDerivative.derivSchoutenTensor_applyits value on differentiable vector fields.CovariantDerivative.IsMetricCompatible.derivSchoutenTensor_eq:∇P = (∇Ric - d Scal ⊗ g / (2 (n - 1))) / (n - 2).CovariantDerivative.cottonTensor: the Cotton tensor, withCovariantDerivative.cottonTensor_apply_of_mdifferentiableAtits value on differentiable vector fields.CovariantDerivative.cottonTensor_antisymm_last:C(X, Y, Z) = -C(X, Z, Y).CovariantDerivative.IsMetricCompatible.cottonTensor_cyclic_eq_zero: the cyclic sumC(X, Y, Z) + C(Y, Z, X) + C(Z, X, Y)vanishes for a metric-compatible torsion-free connection.CovariantDerivative.cottonTensor_eq_zero_of_eventually_curvatureTensor_eq_zero: the Cotton tensor vanishes at a point near which the curvature vanishes.CovariantDerivative.IsMetricCompatible.cottonTensor_eq_zero_of_schoutenTensor_eq_smul_inner: the Cotton tensor vanishes whenPis a constant multiple of the metric.IsMetricCompatible.cottonTensor_eq_zero_of_curvatureTensor_eq_smul_inner_sub, in theCovariantDerivativenamespace: the Cotton tensor vanishes in constant curvature, in dimension at least3.
References #
- É. Cotton, Sur les variétés à trois dimensions, Ann. Fac. Sci. Toulouse (2) 1 (1899), 385–438.
- J. M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer GTM 176 (2018), Chapter 7 (the Schouten, Weyl and Cotton tensors).
The covariant derivative ∇P of the Schouten tensor P of a smooth connection, evaluated as
cov.derivSchoutenTensor x u v w = (∇_w P)(u, v). This formula holds for a smooth metric
(IsContMDiffRiemannianBundle I ∞, see CovariantDerivative.derivSchoutenTensor_apply); as for
CovariantDerivative.derivBilinearTensor, the value is 0 when P does not pair differentiable
vector fields to differentiable functions.
Instances For
∇P is the covariant derivative of the Schouten tensor as a field of bilinear forms, so the
CovariantDerivative.derivBilinearTensor API applies to it.
The Cotton tensor C(u, v, w) = (∇_w P)(u, v) - (∇_v P)(u, w) of a smooth connection, where
P is the Schouten tensor and ∇P is CovariantDerivative.derivSchoutenTensor. The usual
formula on vector fields holds for a smooth metric (see
CovariantDerivative.cottonTensor_apply_of_mdifferentiableAt); otherwise ∇P, and hence the
Cotton tensor, may be 0.
Instances For
Evaluation of the Cotton tensor: C(u, v, w) = (∇_w P)(u, v) - (∇_v P)(u, w).
The Cotton tensor is antisymmetric in its last two arguments.
The covariant derivative of the Schouten tensor on vector fields differentiable at x:
(∇_v P)(σ, τ) = v (P(σ, τ)) - P(∇_v σ, τ) - P(σ, ∇_v τ).
The Cotton tensor on vector fields differentiable at x:
C(σ, τ, ρ) = ρ (P(σ, τ)) - τ (P(σ, ρ)) - P(∇_ρ σ, τ) - P(σ, ∇_ρ τ) + P(∇_τ σ, ρ) + P(σ, ∇_τ ρ).
For a metric-compatible connection in dimension n, the covariant derivative of the Schouten
tensor is (∇_w P)(u, v) = ((∇_w Ric)(u, v) - d Scal(w) ⟨u, v⟩ / (2 (n - 1))) / (n - 2).
The covariant derivative of the Schouten tensor vanishes at a point near which the curvature vanishes, since the Schouten tensor then vanishes near that point.
The Cotton tensor vanishes at a point near which the curvature vanishes.
For a metric-compatible torsion-free connection, ∇P is symmetric in its first two
arguments: (∇_w P)(u, v) = (∇_w P)(v, u).
For a metric-compatible torsion-free connection, the cyclic sum of the Cotton tensor
vanishes: C(u, v, w) + C(v, w, u) + C(w, u, v) = 0.
For a metric-compatible connection whose Schouten tensor is a constant multiple c g of the
metric, the covariant derivative of the Schouten tensor vanishes.
For a metric-compatible connection whose Schouten tensor is a constant multiple c g of the
metric, the Cotton tensor vanishes.
A metric-compatible connection of constant curvature R(w,u)v = κ (⟨u,v⟩ w - ⟨w,v⟩ u), with
κ the same at every point, has vanishing Cotton tensor in dimension at least 3.