Problem 1 (yau-contest 2017)
Section: Geometry and Topology

Let $M$ be a smooth, compact, oriented $n$-dimensional manifold. Suppose that the Euler characteristic of $M$ is zero. Show that $M$ admits a nowhere vanishing vector field.

Problem 2 (yau-contest 2017)
Section: Geometry and Topology

Let $S^2 \xrightarrow{q_1} S^2 \vee S^2 \xrightarrow{q_2} S^2$ be the maps that crush out one of the two summands. Let $f : S^2 \to S^2 \vee S^2$ be a map such that $q_i \circ f : S^2 \to S^2$ is a map of degree $d_i$. Compute the integral homology groups of $(S^2 \vee S^2) \cup_f D^3$. Here $D^3$ is the unit solid ball with boundary $S^2$.

Problem 3 (yau-contest 2017)
Section: Geometry and Topology

Let $X$ and $Y$ be smooth vector fields on a smooth manifold. Prove that the Lie derivative satisfies the identity $$L_X Y = [X, Y].$$

Problem 4 (yau-contest 2017)
Section: Geometry and Topology

State and prove the Liouville formula for the geodesic curvature $\kappa_g$ along a regular curve on a smooth surface in $\mathbb{R}^3$.

Problem 5 (yau-contest 2017)
Section: Geometry and Topology

On a Riemannian manifold, let $F$ be the set of smooth functions $f$ on $M$ with $|\mathrm{grad} f| \le 1$. For any $x, y$ in the manifold, show that $$d(x, y) = \sup\{|f(x) - f(y)| : f \in F\}.$$

Problem 6 (yau-contest 2017)
Section: Geometry and Topology

Let $M$ be an $n$-dimensional oriented closed minimal submanifold in an $(n+p)$-dimensional unit sphere $S^{n+p}$. Denote by $K_M$ the sectional curvature of $M$. Prove that if $$K_M > \frac{p-1}{2p-1},$$ then $M$ is the great sphere $S^n$.