Curvature in a coordinate frame #
The l-th coefficient of R(∂ᵢ, ∂ⱼ)∂ₖ is
∂ᵢΓˡⱼₖ - ∂ⱼΓˡᵢₖ + ∑ₘ (Γᵐⱼₖ Γˡᵢₘ - Γᵐᵢₖ Γˡⱼₘ).
The first two arguments are the curvature directions, the third is the vector being
differentiated, and the last index is the output coefficient.
Both the vector-valued expansion and its scalar coefficients are provided. They use the
existing tangent trivialization and TauCeti.Manifold.christoffelSymbol, so an explicit
metric's Christoffel calculation can be substituted directly to compute its curvature.
Only C¹ regularity of the connection on the chosen chart is required. The manifold is
C³ over the real or complex numbers, and analytic over other complete normed fields,
as prescribed by minSmoothness. No metric or torsion hypothesis is needed. The formulas
hold on chart sources, including boundary points.
For a smooth connection on a smooth real manifold, the pointwise curvature tensor has the
same expansion in terms of the model-space Christoffel map Γ of the chart: on vectors with
coordinates a, c and w, the curvature R(a, c)w has coordinates
dΓ(a)(w, c) - dΓ(c)(w, a) + Γ(Γ(w, c), a) - Γ(Γ(w, a), c),
where Γ(v, u) is the coordinate vector of ∇_u applied to the field with constant
coordinates v (CovariantDerivative.curvatureTensor_symm_trivializationAt). Covariant
derivatives along curves and parametrized surfaces are computed with Γ as a bilinear map
rather than through its scalar symbols, so this is the form in which their commutators are
compared with the curvature.
The sign convention follows J. M. Lee, Introduction to Riemannian Manifolds, 2nd ed.,
Springer GTM 176 (2018), Chapter 7, pp. 196–198:
R(X,Y)Z = ∇_X ∇_Y Z - ∇_Y ∇_X Z - ∇_[X,Y] Z.
The coordinate expansion of R(∂ᵢ, ∂ⱼ)∂ₖ. Its coefficients are the difference of
the directional derivatives of the Christoffel symbols plus their quadratic contraction.
This is valid for any C¹ connection, without a metric or a torsion hypothesis.
The scalar coordinate formula for R(∂ᵢ, ∂ⱼ)∂ₖ, in the output component l.
The order of the indices agrees with christoffelSymbol: Γ i k l is the l-th
component of ∇_{∂ᵢ} ∂ₖ.
The curvature tensor in a chart, in terms of the model-space Christoffel map
Γ = christoffelMap b ∇ of the chart at x₀. On tangent vectors with coordinates a, c and
w, the curvature R(a, c)w has coordinates
dΓ(a)(w, c) - dΓ(c)(w, a) + Γ(Γ(w, c), a) - Γ(Γ(w, a), c).