More connections

Differential Geometry 2017–2018 ENS de Lyon

More connections

Exercise 1(Derivatives at a vanishing point). LetE→M be a vector bundle and let∇and

∇e be two connections onE. Let s∈Γ(E) and x∈M be such that s(x) = 0. Compare∇_xs and∇e_xs.

Exercise 2. LetE →M be a vector bundle eqquiped with a connection ∇. Let x₀ ∈M and y∈Ex0, is there a local (resp. global) section s∈Γ(E) such thats(x0) =y and ∇_x0s= 0?

Exercise 3. Letsbe a smooth section of the vector bundleE →M and letx₀∈M. Is there a connection∇ onE such that∇svanishes on some neighborhood ofx0 inM?

Exercise 4 (Christoffel symbols). 1. Let(M, g)be a Riemannian manifold. We denote by (x1, . . . , xn) local coordinates on a open subetU ofM and byG= (gij)the matrix of g in these coordinates. Let(g^kl)denote the coefficients ofG⁻¹. Check that the Christoffel symbols(Γ^k_ij)_16i,j,k6nof the Levi–Civita connection∇of(M, g)are symmetric in (i, j).

Prove that for any i, j, k ∈ {1, . . . , n}we have:

Γ^k_ij = 1 2

n

X

l=1

g^kl ∂g_il

∂x_j +∂gjl

∂x_i −∂gij

∂x_l

.

2. Recall that the half-plane model of the hyperbolic plane isH² :={(x, y) ∈R² |y > 0}

endowed with the metric g_(x,y) := _y¹2(dx ⊗dx + dy ⊗dy). Compute the covariant derivatives of _∂x^∂ and _∂y^∂ for the Levi–Civita connection.

3. Recall that the Poincaré disc is the unit open disc D² ⊂ R² endowed with the metric g_(x,y):= _(1−x2⁴−y²)²(dx⊗dx+ dy⊗dy). Compute the covariant derivatives for the Levi–

Civita connection of the vector fields _∂r^∂ and _∂θ^∂ associated with the polar coordinates on D²\ {0}.

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