Math 829F: Riemann Surfaces

On this page you will find your weekly assignments.

Here is a sample LaTeX template for you to use on assignments, if you'd like.

Week 3 Assignment
Due by 1:00 pm on Tuesday, 28 June.

1. [JJ,Ex 2.3.A.1] Show that the exponential map \(\exp_p :T_pS^2 \to S^2\) is not holomorphic, where \(S^2\) is the Riemann sphere with the conformal Riemannian metric induced from the Euclidean metric on \(\mathbb{R}^3\) and \(p\) is any point in \(S^2\).

2. [JJ, Ex 2.4.1] Let \(\Gamma < PSL(2,\mathbb{R})\) be a discrete subgroup of isometries that acts properly discontinuously and freely on \((H,g_H)\), where \(g_H\) is the hyperbolic metric on the upper half plane \(H\), such that \(\Gamma\setminus H\) (or \(H/\Gamma\) in [JJ]) is a compact Riemann surface. Show that each nontrivial abelian subgroup of \(\Gamma\) is infinite cyclic (i.e., \( \cong \mathbb{Z}\)).

3. Let \(S\) be a \(G\)-manifold; that is, there exists an action \(\tau :G \times S \to S\) of the group \(G\) on the manifold \(S\). Fix \(p \in S\) and consider the isotropy group \[G_p := \{g \in G \mid \tau_g(p) = p\}\] of the action at \(p\). Show that for any \(g \in G\), \(G_{\tau_g(p)} = gG_pg^{-1}\).

Week 2 Assignment
Due by 1:00 pm on Tuesday, 21 June.

1. Let \(A\) be an \(\mathbb{R}\)-algebra, and let \(\text{Der}(A)\) denote the space of derivations of \(A\). Show that (\(a\)) if \(D \in \text{Der}(A)\) and \(k \in \mathbb{R}\), then \(D(k)=0\); and (\(b\)) if \(D_1, D_2 \in \text{Der}(A)\), then their commutator \([D_1,D_2] = D_1D_2 - D_2D_1\) is also in \(\text{Der}(A)\).

Remark. The result of 1(b) means that \(\text{Der}(A)\) is a Lie algebra. In particular (and specific to our interests in this course), this means that the space \(\mathfrak{X}(S)\) of (smooth) vector fields on a Riemann surface forms a Lie algebra, and we can make use of this additional structure whenever it is convenient.

2. Let \(g_\lambda = \lambda^2(z)dz\,d\bar{z}\) be a conformal Riemannian metric on a Riemann surface \(S\). Show that the angle between two vectors \(x,y \in T_pS\) as measured with respect to \(g_\lambda\) is exactly the angle between \(x\) and \(y\) determined by the dot product on \(T_pS\) regarded as \(\mathbb{R}^2\). How are the distances \(d_{E,p}\) and \(d_{\lambda,p}\) related? (Here \(d_{E,p}\) is the Euclidean distance \(d(x,y)_p := \sqrt{(x-y)\cdot(x-y)}\) in the vector space \(T_pS\), and \(d_{\lambda,p}\) is defined analogously with respect to the inner product \(g_{\lambda,p}\).)

3. Consider the torus \(\mathbb{T}^2\) as constructed in Chapter 1. Regard \(\pi :\mathbb{C} \to \mathbb{T}^2\) as the universal covering with covering transformation group \(H_\pi \cong \mathbb{Z}^2\). Suppose \(g_\lambda = \lambda^2 (z)dz\,d\bar{z}\) is a conformal Riemannian metric on \(\mathbb{C}\) for which each \(\varphi \in H_\pi\) is an isometry. Then \(g_\lambda\) defines a conformal Riemannian metric on \(\mathbb{T}^2\). Show that the curvature \(K_\lambda([z]) = -\Delta\log\lambda([z])\) satisfies \[\int_{\mathbb{T}^2} K_\lambda = 0.\] This justifies calling \(\mathbb{T}^2\) (as constructed) a flat torus.

Solutions to Assignment 2.

Week 1 Assignment
Here are the exercises for week 1 of the course. These are due by 1.00 pm on Tuesday, 14 June. Each student should submit a PDF file with their unique solutions via a direct message to me in Slack.

The file should be called [last-name]_[assignment-#].pdf. For example, my solutions for Week 1 would be called "ryan_1.pdf".

Please let me know if you have any questions.

1. Construct a 3-dimensional torus \(\mathbb{T}^3\) by defining an appropriate equivalence relation on \(\mathbb{R}^3\), following the construction of \( \mathbb{T}^2\) in the book. Be sure to prove that the equivalence relation you define is indeed an equivalence relation, and to provide the charts. Give an example of what a transition function \(\varphi_{\alpha \beta} :\mathbb{R}^3 \to \mathbb{R}^3\) would look like. You do not need to prove that the transition functions are smooth (although it should be obvious).

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

3. Prove Corollary 1.3.1.

Solutions to Assignment 1.