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. 4

1. Problem (4 points)

We consider again the hyperbolic plane (H, g^hyp).

a) Compute explicitly the parallel transport P_c,t :T_(0,1)H →T_(t,1)H along the curve c: [0,1]→H with c(s) = (st,1).

b) Let x0 ∈ R and a ∈ R\ {0}. Show that R 3 t 7→ (x0, e^at) is a geodesic of (H, g^hyp).

2. Problem (4 points)

Let (M, g) be a semi-Riemannian manifold with associated Levi-Civita connec- tion ∇. For a diffeomorphism f : M → M and a vector field X ∈ X(M) we define f∗X ∈ X(M) by (f∗X)_p = d_f⁻¹_(p)f(X_f⁻¹_(p)) and f^∗X ∈ X(M) by f^∗X = (f⁻¹)∗X.

a) Let f ∈Isom(M, g).We define ˜∇:X(M)×X(M)→X(M) by

∇˜_XY =f^∗(∇_f∗X f_∗Y). Show that ˜∇=∇.

b) Let f ∈ Isom(M, g) and c: (−ε, ε)→M be geodesic. Use a) to show that f ◦c: (−ε, ε)→M is also a geodesic.

c) Show that every isometryf of R^{k+`</sup>, g<sup>(k,`)}

is affine, i.e there exists a linear map A:R^{k+`</sup> →R<sup>k+`} and a constantb ∈R^{k+`</sup> such thatf(x) =Ax+b for all x∈R<sup>k+`}.

3. Problem (4 points)

Let (M, g) be a semi-Riemannian manifold. For a map ψ : M → M we denote by Fix(ψ) the set of fixed points of ψ, i.e. Fix(ψ) ={p∈M |ψ(p) = p}.

a) Let ψ ∈ Isom(M, g). Prove that for p ∈ Fix(ψ) and ξ ∈ TpM such that d_pψ(ξ) =ξ the geodesic c: (−ε, ε)→M with c(0) =p and c⁰(0) = ξ takes all its values in Fix(ψ).

b) Use a) to the determine the geodesics of

Hⁿ={X ∈Rⁿ⁺¹ |g^(n,1)(X, X) =−1}.

4. Problem (4 points)

We continue discussing SU(n) =

A ∈C^n×n | A^∗A = Id and detA= 1 . Recall that the tangential space at X ∈SU(n) is given byL_Xsu(n) with

su(n) =

A∈C^n×n|A+A^∗ = 0 and Tr(A) = 0 and L_X :C^n×n →C^n×n, Y 7→XY. For V, W ∈T_XSU(n) we define

g_X(V, W) =−1

2Tr (((d_XL_X⁻¹)V)◦((d_XL_X⁻¹)W)), where◦ denotes matrix multiplication.

a) Show that g :SU(n)3X 7→gX is a Riemannian metric on SU(n).

b) Prove that SU(2) = a −¯b b a¯

a, b∈C with |a|²+|b|² = 1

. c) Use b) to show that S³ =

x∈R⁴ | kxk_eucl= 1 , g^eucl

and (SU(2), g) are isometric.

• Submission deadline: Thursday 12.11.2015 at the beginning of the lecture

• 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.