Ricci curvature of a smooth connection #
The Ricci tensor of a connection on the tangent bundle is the bilinear form
Ric(u,v) = trace (w ↦ R(w,u)v). The trace makes this definition independent of a choice
of basis. Applying it to the Levi-Civita connection gives Riemannian Ricci curvature.
A general connection need not have symmetric Ricci curvature.
We work with a RiemannianBundle on the tangent bundle for the orthonormal-basis formulas.
The connection need not preserve this metric or be torsion free,
and the metric need not vary smoothly. Compactness and absence of boundary are not required.
We give the coordinate formula in any basis and the inner-product formula in an
orthonormal basis, including contraction in factor bases on a product manifold.
The constant-curvature calculation fixes the sign convention:
R(w,u)v = κ (⟪u,v⟫ w - ⟪w,v⟫ u) gives Ric = (dim - 1) κ g.
The calculation is stated for any bilinear form in place of κ g, and then specialised
to κ g.
The Ricci coefficients of a smooth connection in a chart-local frame are smooth on the chart, so the Ricci tensor pairs vector fields differentiable (respectively smooth) at a point to a function differentiable (respectively smooth) there.
The convention follows J. M. Lee, Introduction to Riemannian Manifolds, 2nd ed.,
Springer GTM 176 (2018), Chapter 7. The curvature tensor being contracted is
CovariantDerivative.curvatureTensor, with convention
R(X,Y)Z = ∇_X ∇_Y Z - ∇_Y ∇_X Z - ∇_[X,Y] Z.
The Ricci tensor of a smooth connection on the tangent bundle, defined by
Ric(u,v) = trace (w ↦ R(w,u)v). Metric compatibility and vanishing torsion are not
assumed; symmetry is not asserted for an arbitrary connection.
Instances For
The defining trace formula for Ricci curvature. The two flips leave the first curvature argument free, so it is this argument that is traced against the output.
Ricci curvature in an arbitrary basis: sum the diagonal coefficients of
w ↦ R(w,u)v. The basis need not be orthogonal.
In an orthonormal basis, Ricci curvature is the sum of the corresponding inner products for the given fibre metric.
The Ricci tensor of a metric-compatible torsion-free connection is symmetric.
The Ricci tensor of a metric-compatible torsion-free connection is a symmetric bilinear form.
A point with zero curvature tensor has zero Ricci tensor.
A smooth connection on a manifold of dimension at most one has zero Ricci tensor.
Contracting curvature of the form R(w,u)v = B(u,v)w - B(w,v)u gives
Ric = (dim - 1) B. In particular, the constant-curvature model with B = κ g
has Ricci tensor (dim - 1) κ g, with the sign fixed by the curvature convention.
The Ricci tensor of the constant-curvature model R(w,u)v = κ (⟪u,v⟫ w - ⟪w,v⟫ u) is
(dim - 1) κ g.
The Ricci coefficients Ric(eᵢ, eⱼ) of a smooth connection on the tangent bundle in the
chart-local frame at a are smooth on the chart source. No metric is involved.
The Ricci tensor of a smooth connection pairs two vector fields differentiable at x to a
function differentiable at x. No metric is involved.
The Ricci tensor of a smooth connection pairs two vector fields that are C^∞ at x to a
function that is C^∞ at x. No metric is involved.
Contract the curvature of any smooth connection on a product manifold in the standard orthonormal bases of the factors. Ricci is an ordinary trace, so no metric on the product, metric compatibility, or torsion assumption is needed.