Problem 1 (yau-contest 2010)
Section: Geometry and Topology

Let $D^* = \{(x,y) \in \mathbb{R}^2 \mid 0 < x^2 + y^2 < 1\}$ be the punctured unit disc in the Euclidean plane. Let $g$ be the complete Riemannian metric on $D^*$ with constant curvature $-1$. Find the distance under the metric between the points $(e^{-2\pi}, 0)$ and $(-e^{-\pi}, 0)$.

Problem 2 (yau-contest 2010)
Section: Geometry and Topology

Show that every closed hypersurface in $\mathbb{R}^n$ has a point at which the second fundamental form is positive definite.

Problem 3 (yau-contest 2010)
Section: Geometry and Topology

Prove that the real projective space $\mathbb{RP}^n$ is orientable if and only if $n$ is odd.

Problem 4 (yau-contest 2010)
Section: Geometry and Topology

Suppose $\pi : M_1 \longrightarrow M_2$ is a $C^\infty$ map of one connected differentiable manifold to another. And suppose for each $p \in M_1$, the differential $\pi_* : T_p M_1 \longrightarrow T_{\pi(p)} M_2$ is a vector space isomorphism.

(a) Show that if $M_1$ is connected, then $\pi$ is a covering space projection.

(b) Give an example where $M_2$ is compact but $\pi : M_1 \longrightarrow M_2$ is not a covering space (but has the $\pi_*$ isomorphism property).

Problem 5 (yau-contest 2010)
Section: Geometry and Topology

Let $\Sigma_g$ be the closed orientable surface of genus $g$. Show that if $g>1$, then $\Sigma_g$ is a covering space of $\Sigma_2$.

Problem 6 (yau-contest 2010)
Section: Geometry and Topology

Let $M$ be a smooth 4-dimensional manifold. A symplectic form is a closed 2-form $\omega$ on $M$ such that $\omega \wedge \omega$ is a nowhere vanishing 4-form.

(a) Construct a symplectic form on $\mathbb{R}^4$.

(b) Show that there are no symplectic forms on $S^4$.