Math 829F: Riemann Surfaces

On this page you will find the recommended exercises for each part of the course. These will neither be collected nor graded, but you are strongly encouraged to at least attempt them all. Exercises marked with an asterisk (*) should be regarded as "advanced." These starred exercises may be quite difficult for beginning graduate students, but should be reasonable for students with at least 2 years experience as a grad student.

Part I.2: Differential Geometry of Riemann Surfaces

1. Let \(S\) be a Riemann surface and \(z = x + iy\) local conformal coordinates at \(p \in S\). Let \(T^\mathbb{C} S = TS \otimes \mathbb{C}\) be the complexified tangent bundle over \(S\), and \((T^\mathbb{C})^*S\) the complexified cotangent bundle. Let \(dz := dx + idy\), \(d\bar{z} := dx - idy\), \(\frac{\partial}{\partial z} := \frac{1}{2}\left(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right)\), and \(\frac{\partial}{\partial \bar{z}} := \frac{1}{2}\left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right)\). Compute \(dz(\frac{\partial}{\partial z})\), \(dz(\frac{\partial}{\partial \bar{z}})\), \(d\bar{z}(\frac{\partial}{\partial z})\), \(d\bar{z}(\frac{\partial}{\partial \bar{z}})\). Show that \(\frac{\partial}{\partial z} \frac{\partial} {\partial\bar{z}} = \frac{1}{4} \Delta\), where \(\Delta\) is the Laplacian on \(\mathbb{R}^2\).

2. Show that \(\frac{\partial}{\partial z}(\bar{z}) = 0\) and \(\frac{\partial}{\partial\bar{z}}(z) = 0\).

From [JJ] Chapter 2:

3. Show that \(\text{Iso}(S) = \{f \in \text{Diff}(S) \mid f \text{ is an isometry}\}\) is a group under pointwise operations.

4. Fill in the details of the proof of Lemma 2.3.5 that \(\text{Iso}(H) = PSL(2,\mathbb{R})\).

5. Let \(S^2 \subset \mathbb{R}^3\) be the sphere with the induced metric \(g = dx^2 + dy^2 + dz^2\) from \(\mathbb{R}^3\). Show that under stereographic projection, \[(x,y,z) \mapsto w = \frac{x + iy}{1 -z}\] this metric takes the form \[\frac{4}{(1 + \vert w \vert^2)^2}\,dw\,d\bar{w}.\]

6. The curvature of \(S^2\) with the above metric is \(K \equiv 1\).

7. \(\text{Area}(S^2) = 4\pi\) with respect to the metric of exercise 5.

8*. \(\text{Iso}(S^2) = \{f \in \text{Möb} \mid f(w) = \frac{aw - \bar{c}}{cw - \bar{a}}, \vert a\vert^2 + \vert c\vert^2 = 1\}\) with respect to the metric of exercise 5.

9. Verify that the geodesic equation (2.3.8) is invariant under (conformal) coordinate transformations.

10. Work through the details in the book (page 33) to verify that transition functions between exponential coordinate charts are smooth.

11. Compute the Christoffel symbols for... (TBA), and use them to write down the 2 by 1 system of ODE determined by the geodesic equation (2.3.A.9).

12. Fill in the "elementary geometric argument" alluded to in the proof of Theorem 2.5.1.

13. Prove Corollary 2.5.3.

14. [~Ex 2.5.1] Restate Theorem 2.5.1 for spherical triangles, and adjust the proof accordingly.

15. Prove Lemma 2.5.1.

16. Prove Corollary 2.5.7.

Part I.1: Topological Foundations

From [JJ] Chapter 1:

1. [Ex 1.1.1] Show that the dimension of a smooth manifold is uniquely determined. Hint: Prove that if \(M_1\) and \(M_2\) are smooth manifolds and \(f :M_1 \to M_2\) is a diffeomorphism, then dim(\(M_1\)) = dim(\(M_2\)).

2. [~Ex 1.1.2] Generalize the construction in Example 1.1.2 to define a 3-dimensional real torus through an appropriate equivalence relation on \(\mathbb{R}^3\).

3. Verify the claim in the proof of Lemma 1.2.1 that the map \(\kappa_\gamma :\pi_1(M,p_1) \to \pi_1(M,p_0) :g \mapsto \gamma^{-1}g\gamma\) is an isomorphism of groups.

4. Prove Lemma 1.2.2.

5. [~Ex 1.2.1 & Ex 1.2.2] Prove that \(S^n\) is simply connected for \(n \geq 2\). Compute \(\pi_1(S^1)\) to show that it is not simply connected.

6. [Ex 1.2.3] Use your computation of \(\pi_1(S^1)\) to help compute \(\pi_1(\mathbb{T}^2)\), where \(\mathbb{T}^2\) is the torus defined in Example 1.1.2.

7. Prove the claim in the proof of Lemma 1.3.1 that the set of points in \(M\) with precisely \(n\) inverse images is both open and closed in \(M\).

8. Prove that the lift \(f' :S \to M'\) of Theorem 1.3.1 is continuous.

9. Prove Corollary 1.3.1.

10. Prove the claim in Definition 1.3.6 that the deck (covering) transformations form a group \(H_\pi\).

11. Work through all of the details of Example 1.3.2. Be sure that you understand every argument and prove any bald claims.

12*. [Ex 1.3.2] Construct a manifold \(M\) with a (nontrivial) covering map \(\pi :S^3 \to M\).

13*. [Ex 1.3.3] Consider the upper half plane \[ H := \left\{ z = x + iy \in \mathbb{C} \mid y > 0 \right\},\] and maps of the form \[ \gamma :z \to \frac{az + b}{cz + d}, \] where \(a,b,c,d \in \mathbb{Z}\), \(ad-bc = 1\), and \[ \left(\begin{matrix}a & b \\ c & d \end{matrix}\right) \equiv \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right)\text{mod}(3).\] The space of all such maps \(\gamma\) forms a group \(\Gamma\) which acts on \(H\).

Show that if \(\gamma \in \Gamma\) is different from the identity map, then \(\gamma\) has no fixed points in \(H\). Interpret \(\Gamma\) as the group of deck (covering) transformations associated with a manifold \(M := H/\Gamma\) and a covering \(\pi :H \to M\). Construct different coverings of \(M\) associated with conjugacy classes of subgroups of \(\Gamma\).

From [AS] Chapter 1:

14*. [1.10B-G] Compute the fundamental group of the punctured plane, \(\pi_1(\mathbb{R}^2\setminus\{0\})\).