Problem 1 (yau-contest 2021)
Section: Geometry and Topology

(a) Show that $P^{2n}$ can not be the boundary of a compact manifold.

(b) Show that $P^3$ is the boundary of some compact manifold.

Problem 2 (yau-contest 2021)
Section: Geometry and Topology

Suppose $M$ is a noncompact, complete $n$-dimensional manifold, and suppose there is an open subset $U \subset M$ and an open set $V \subset \mathbf{R}^n$ such that $M\setminus U$ is isomorphic to $\mathbf{R}^n\setminus V$. If $\mathrm{Ric}_M \ge 0$, show that $M$ is isometric to $\mathbf{R}^n$.

Problem 3 (yau-contest 2021)
Section: Geometry and Topology

Compute all the homotopy groups of the $n$-torus $T^n = S^1 \times S^1 \times \cdots \times S^1$, $n \ge 2$.

Problem 4 (yau-contest 2021)
Section: Geometry and Topology

Consider the upper half space $\mathbf{H}^3 = \{(x, y, z) \mid z > 0\}$ equipped with hyperbolic metric $$g = \frac{dx^2 + dy^2 + dz^2}{z^2}.$$ Let $P$ be the surface defined by $\{ z = x \tan \alpha,\ z > 0 \}$ for some $\alpha \in \bigl(0, \tfrac{\pi}{2}\bigr)$. Compute the mean curvature of $P$.

Problem 5 (yau-contest 2021)
Section: Geometry and Topology

Suppose $M$ is a compact 2-dimensional Riemannian manifold without boundary, with positive sectional curvature. Show that any two compact closed geodesics on $M$ must intersect with each other.

Problem 6 (yau-contest 2021)
Section: Geometry and Topology

Suppose $\Sigma$ is a smooth compact embedded hypersurface (i.e. a codimension 1 submanifold) without boundary in $\mathbf{R}^n$ for $n \ge 3$. Show that $\Sigma$ is orientable.