The first variation of the Levi-Civita connection #
Let g_s be a one-parameter family of smooth nondegenerate metrics on M, of any signature,
jointly C² in (s, x) as a map into the bundle of bilinear forms on TM, and let ∇ˢ be a
Levi-Civita connection of g_s for each s. Write h = ∂ₛ g_s for the parameter derivative at
s = t, a symmetric field of bilinear forms. This file proves that for a vector field Y which is
differentiable at x, the curve s ↦ ∇ˢ_v Y in T_x M is differentiable at t, and that its
derivative (∂ₜ∇)_v Y satisfies the first variation formula
2 g_t((∂ₜ∇)_v Y, w) = (∇ᵗ_v h)(Y, w) + (∇ᵗ_Y h)(v, w) - (∇ᵗ_w h)(v, Y),
where ∇ᵗh is the covariant derivative CovariantDerivative.derivBilinearTensor of h. In index
notation this is the classical variation ∂ₜΓ^k_ij = ½ g^kl (∇_i h_jl + ∇_j h_il - ∇_l h_ij) of the
Christoffel symbols.
For a smooth family g of nondegenerate metrics over a parameter manifold P, as bundled by
TauCeti.Manifold.ContMDiffNondegenerateMetricFamily, the formula holds along every parameter
curve c which is C² at t, with h the first variation δ_{c'(t)} g of the family in the
direction of the velocity of c. Consequently the derivative at t of s ↦ ∇^{c s}_v Y depends
only on c t and c'(t).
The proof differentiates the Koszul formula 2 g_s(∇ˢ_X Y, Z) = K_{g_s}(X, Y, Z) in s. The
parameter derivative of the Koszul expression is the Koszul expression of h
(TauCeti.Manifold.hasDerivAt_koszulBilinear, in
TauCeti.Geometry.Manifold.VectorBundle.CovariantDerivative.LeviCivita.KoszulVariation).
Differentiability of s ↦ ∇ˢ_v Y comes from writing it as g_s⁻¹ applied to half the Koszul
covector, and the identity
TauCeti.Manifold.koszulBilinear_sub_two_mul_apply_eq_derivBilinearTensor turns
K_h(v, Y, w) - 2 h(∇ᵗ_v Y, w) into the right-hand side. No positivity is used. Since the
difference ∇ˢ - ∇ᵗ is tensorial, s ↦ ∇ˢ_v Y differs by a constant from the same curve for any
smooth extension of Y x, so only differentiability of Y at x is needed.
Main results #
TauCeti.Manifold.differentiableAt_covariantDerivative_apply_of_isLeviCivitaConnection: a family of Levi-Civita connections of a jointlyC²family of nondegenerate metrics varies differentiably.TauCeti.Manifold.two_mul_apply_deriv_eq_derivBilinearTensor_of_isLeviCivitaConnection: the first variation formula for the Levi-Civita connection.
In the namespace TauCeti.Manifold.ContMDiffNondegenerateMetricFamily, for a smooth family over a
parameter manifold along a parameter curve:
differentiableAt_covariantDerivative_apply_of_isLeviCivitaConnectionandtwo_mul_apply_deriv_eq_derivBilinearTensor_of_isLeviCivitaConnection: the two results above, withhthe first variation of the family in the direction of the velocity of the curve.deriv_covariantDerivative_apply_eq_of_mfderiv_eq: the variation of the connection depends only on the position and velocity of the curve.
References #
- A. L. Besse, Einstein Manifolds, Springer (1987), Section 1.K, Theorem 1.174.
- B. Chow and D. Knopf, The Ricci Flow: An Introduction, AMS (2004), Chapter 3, Lemma 3.2.
- B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983, Ch. 3, Thm. 11: the Koszul formula for nondegenerate metrics.
Families of Levi-Civita connections #
Levi-Civita connections vary differentiably with the metric. Let g_s be a family of
smooth nondegenerate metrics, jointly C² in (s, y) at (t, x), and let ∇ˢ be a Levi-Civita
connection of g_s for every s. Then for every vector field Y which is differentiable at x
and every v ∈ T_x M, the curve s ↦ ∇ˢ_v Y in T_x M is differentiable at t.
The first variation of the Levi-Civita connection. Let g_s be a family of smooth
nondegenerate metrics, jointly C² in (s, y) at (t, x), with parameter derivative
h = ∂ₛ g_s at s = t, and let ∇ˢ be a Levi-Civita connection of g_s for every s. Then for
every vector field Y which is differentiable at x and all v, w ∈ T_x M,
2 g_t((∂ₜ∇)_v Y, w) = (∇ᵗ_v h)(Y, w) + (∇ᵗ_Y h)(v, w) - (∇ᵗ_w h)(v, Y),
where (∂ₜ∇)_v Y is the derivative at t of s ↦ ∇ˢ_v Y and ∇ᵗh is the covariant derivative
CovariantDerivative.derivBilinearTensor of h along ∇ᵗ.
Families of metrics over a parameter manifold #
Levi-Civita connections vary differentiably along a parameter curve. Let g be a smooth
family of nondegenerate metrics parametrized by P, let c be a parameter curve which is C²
at t, and let ∇ˢ be a Levi-Civita connection of g_{c s} for every s. Then for every vector
field Y which is differentiable at x and every v ∈ T_x M, the curve s ↦ ∇ˢ_v Y in T_x M
is differentiable at t.
The first variation of the Levi-Civita connection along a parameter curve. Let g be a
smooth family of nondegenerate metrics parametrized by P, let c be a parameter curve which is
C² at t, and let ∇ˢ be a Levi-Civita connection of g_{c s} for every s. Write
h = δ_{c'(t)} g for the first variation of g at c t in the direction of the velocity of c,
which is the parameter derivative ∂ₜ g_{c t}. Then for every vector field Y which is
differentiable at x and all v, w ∈ T_x M,
2 g_{c t}((∂ₜ∇)_v Y, w) = (∇ᵗ_v h)(Y, w) + (∇ᵗ_Y h)(v, w) - (∇ᵗ_w h)(v, Y),
where (∂ₜ∇)_v Y is the derivative at t of s ↦ ∇ˢ_v Y and ∇ᵗh is the covariant derivative
CovariantDerivative.derivBilinearTensor of h along ∇ᵗ.
The variation of the Levi-Civita connection depends only on the position and velocity of
the parameter curve. Let g be a smooth family of nondegenerate metrics parametrized by P,
and let ∇ᵖ be a
Levi-Civita connection of g_p for every p. If two parameter curves c₁ and c₂, both C²
at t, have the same position and velocity at t, then for every vector field Y
differentiable at x the curves s ↦ ∇^{c₁ s}_v Y and s ↦ ∇^{c₂ s}_v Y have the same
derivative at t.