Problem 1 (yau-contest 2012)
Section: Geometry and Topology

Prove that the real projective space $RP^n$ is a differentiable manifold of dimension $n$.

Problem 2 (yau-contest 2012)
Section: Geometry and Topology

Let $M, N$ be $n$-dimensional smooth, compact, connected manifolds, and $f : M \to N$ a smooth map with rank equals to $n$ everywhere. Show that $f$ is a covering map.

Problem 3 (yau-contest 2012)
Section: Geometry and Topology

Given any Riemannian manifold $(M^n, g)$, show that there exists a unique Riemannian connection on $M^n$.

Problem 4 (yau-contest 2012)
Section: Geometry and Topology

Let $S^n$ be the unit sphere in $\mathbb{R}^{n+1}$ and $f : S^n \to S^n$ a continuous map. Assume that the degree of $f$ is an odd integer. Show that there exists $x_0 \in S^n$ such that $f(-x_0) = -f(x_0)$.

Problem 5 (yau-contest 2012)
Section: Geometry and Topology

State and prove the Stokes theorem for oriented compact manifolds.

Problem 6 (yau-contest 2012)
Section: Geometry and Topology

Let $M$ be a surface in $\mathbb{R}^3$. Let $D$ be a simply-connected domain in $M$ such that the boundary $\partial D$ is compact and consists of a finite number of smooth curves. Prove the Gauss-Bonnet Formula:

$$\int_{\partial D} k_g\, ds + \sum_j (\pi - \alpha_j) + \iint_D K\, dA = 2\pi,$$

where $k_g$ is the geodesic curvature of the boundary curve. Each $\alpha_j$ is the interior angle at a vertex of the boundary, $K$ is the Gaussian curvature of $M$, and the 2-form $dA$ is the area element of $M$.