Problem 1 (yau-contest 2011)
Section: Geometry and Topology

Suppose $K$ is a finite connected simplicial complex. True or false:

a) If $\pi_1(K)$ is finite, then the universal cover of $K$ is compact.

b) If the universal cover of $K$ is compact then $\pi_1(K)$ is finite.

Problem 2 (yau-contest 2011)
Section: Geometry and Topology

Compute all homology groups of the the $m$-skeleton of an $n$-simplex, $0 \le m \le n$.

Problem 3 (yau-contest 2011)
Section: Geometry and Topology

Let $M$ be an $n$-dimensional compact oriented Riemannian manifold with boundary and $X$ a smooth vector field on $M$. If $n$ is the inward unit normal vector of the boundary, show that
$$\int_M \mathrm{div}(X)\, dV_M = \int_{\partial M} X \cdot n \, dV_{\partial M}.$$

Problem 4 (yau-contest 2011)
Section: Geometry and Topology

Let $F^k(M)$ be the space of all $C^\infty$ $k$-forms on a differentiable manifold $M$. Suppose $U$ and $V$ are open subsets of $M$.

a) Explain carefully how the usual exact sequence
$$0 \longrightarrow F(U \cup V) \longrightarrow F(U) \oplus F(V) \longrightarrow F(U \cap V) \longrightarrow 0$$
arises.

b) Write down the “long exact sequence” in de Rham cohomology associated to the short exact sequence in part (a) and describe explicitly how the map
$$H^{k}_{deR}(U \cap V) \longrightarrow H^{k+1}_{deR}(U \cup V)$$
arises.

Problem 5 (yau-contest 2011)
Section: Geometry and Topology

Let $M$ be a Riemannian $n$-manifold. Show that the scalar curvature $R(p)$ at $p \in M$ is given by
$$R(p) = \frac{1}{\operatorname{vol}(S^{n-1})} \int_{S^{n-1}} \mathrm{Ric}_p(x)\, dS^{n-1},$$
where $\mathrm{Ric}_p(x)$ is the Ricci curvature in direction $x \in S^{n-1} \subset T_pM$, $\operatorname{vol}(S^{n-1})$ is the volume of $S^{n-1}$ and $dS^{n-1}$ is the volume element of $S^{n-1}$.

Problem 6 (yau-contest 2011)
Section: Geometry and Topology

Prove the Schur’s Lemma: If on a Riemannian manifold of dimension at least three, the Ricci curvature depends only on the base point but not on the tangent direction, then the Ricci curvature must be constant everywhere, i.e., the manifold is Einstein.