Exercises in Differential Geometry

Exercises in Differential Geometry

Universit¨at Regensburg, Winter Term 2015/16

Prof. Dr. Bernd Ammann / Dipl.-Math. Manuel Streil

Exercise Sheet no. 3

1. Problem (4 points)

We define the hyperbolic plane as

H={x+iy∈C|x∈R, y ∈R>0} endowed with the metric g_x+iy^hyp = _y¹2g^eucl.

a) Compute the Christoffel symbols with respect to the chart given by the identity H→H⊂R².

b) For A = ^{a b}_{c d}

∈ SL(2;R) let Ψ_A(z) = ^az+b_cz+d be the associated M¨obius transformation. Show that Ψ_A is an isometry of H, g^hyp

.

Hint: You can use without proof that SL(2;R) is generated by matrices of the form ^a_{0 a}−1⁰

, ⁰_{1 0}⁻¹

, ¹_{0 1}^b

with a∈R\ {0} and b∈R.

2. Problem (4 points)

We consider R³ with the standard Euclidean scalar product g^eucl and introduce polar coordinates via

Ψ :R^>0×(−π, π)×(0, π) → R³

(r, ϕ, ϑ) 7→ (rsinϑcosϕ, rsinϑsinϕ, rcosϑ).

Compute the Christoffel symbols ofg^eucl with respect to the chart Ψ⁻¹. Hint: Consider the pullback metric Ψ^∗ g^eucl

(r,ϕ,ϑ). 3. Problem (4 points)

Let M, N be smooth manifolds and ∇ the Levi-Civita-connection on M. For f ∈C^∞(N, M) we denote as in the lecture by^f∇the induced covariant derivative for vector fields along f.

a) Let v ∈T_pN for somep∈N and Z,Z˜∈X(f). Show the following product rule:

∂_vg Z,Z˜

=g

f∇_vZ,Z˜ +g

Z,^f∇_vZ˜ .

b) If y:U →V is a chart ofN, then

f∇ ^∂

∂yi

df

∂

∂y^j

=^f∇ ^∂

∂yj

df

∂

∂yⁱ

.

4. Problem (4 points)

Let (M, g) be a Riemannian manifold with Levi-Civita connection,v ∈T_pM for some p ∈ M and X ∈ X(M). We choose a smooth curve c: (−ε, ε) → M such that c(0) =p and c⁰(0) =v and denote by

P_c,t :T_c(0)M →T_c(t)M the parallel transport along c. Prove that

∇_vX = d dt

t=0

P_c,t⁻¹(X(c(t))).

• Submission deadline: Since there is no lecture on Thursday 5.11.2015, you can either give your solution to Mrs Bonn, office 217, by 12:00 or to A.

Platzer if you attend his exercise class.

• Please write your name and the number of your exercise class on every sheet of your proposal for solution.

• Each participant should hand in his own solution. A joint solution of a working group is not allowed.