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∇_{x}s
and∇e_{x}s.

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

Exercise 3. Letsbe a smooth section of the vector bundleE →M and letx_{0}∈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,k6n}of 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 isH^{2} :={(x, y) ∈R^{2} |y > 0}

endowed with the metric g_{(x,y)} := _{y}^{1}2(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 D^{2} ⊂ R^{2} endowed with the metric
g_{(x,y)}:= _{(1−x}2^{4}−y^{2})^{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 D^{2}\ {0}.

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