Problem 1 (yau-contest 2015)
Section: Geometry and Topology

Let $n, m$ be positive integers. Show that the product of spheres $S^n \times S^m$ has trivial tangent bundle if and only if $n$ or $m$ is odd.

Problem 2 (yau-contest 2015)
Section: Geometry and Topology

Show that there does not exist a compact three-dimensional manifold $M$ whose boundary is the real projective space $\mathbb{RP}^2$.

Problem 3 (yau-contest 2015)
Section: Geometry and Topology

Let $M^n$ be a smooth manifold without boundary and $X$ a smooth vector field on $M$. If $X$ does not vanish at $p \in M$, show that there exists a local coordinate chart $(U; x_1, \ldots, x_n)$ centered at $p$ such that in $U$ the vector field $X$ takes the form $X = \frac{\partial}{\partial x_1}$.

Problem 4 (yau-contest 2015)
Section: Geometry and Topology

Let $M \to \mathbb{R}^3$ be a compact simply-connected closed surface. Prove that if $M$ has constant mean curvature, then $M$ is a standard sphere.

Problem 5 (yau-contest 2015)
Section: Geometry and Topology

Let $M$ be an $n$-dimensional compact Riemannian manifold with diameter $\pi/c$ and Ricci curvature $\ge (n-1)c^2 > 0$. Show that $M$ is isometric to the standard $n$-sphere in $\mathbb{R}^{n+1}$ with radius $1/c$.

Problem 6 (yau-contest 2015)
Section: Geometry and Topology

Suppose $(M, g)$ is a Riemannian manifold and $p \in M$. Show that the second-order Taylor series of $g$ in normal coordinates centered at $p$ is $$g_{ij}(x) = \delta_{ij} - \frac{1}{3} \sum_{k,l} R_{iklj} x_k x_l + O(|x|^3).$$