Problem 1 (yau-contest 2014)
Section: Geometry and Topology

Let $X$ be the quotient space of $S^2$ under the identifications $x \sim -x$ for $x$ in the equator $S^1$. Compute the homology groups $H_n(X)$. Do the same for $S^3$ with antipodal points of the equator $S^2 \subset S^3$ identified.

Problem 2 (yau-contest 2014)
Section: Geometry and Topology

Let $M \to \mathbb{R}^3$ be a graph defined by $z = f(u,v)$ where $\{u,v,z\}$ is a Descartes coordinate system in $\mathbb{R}^3$. Suppose that $M$ is a minimal surface. Prove that:

(a) The Gauss curvature $K$ of $M$ can be expressed as
$$
K = \Delta \log\!\left(1 + \frac{1}{W}\right), \qquad W := \sqrt{1 + \left(\frac{\partial f}{\partial u}\right)^{\!2} + \left(\frac{\partial f}{\partial v}\right)^{\!2}}\, ,
$$
where $\Delta$ denotes the Laplacian with respect to the induce metric on $M$ (i.e., the first fundamental form of $M$).

(b) If $f$ is defined on the whole $uv$-plane, then $f$ is a linear function (Bernstein theorem).

Problem 3 (yau-contest 2014)
Section: Geometry and Topology

Let $M = \mathbb{R}^2/\mathbb{Z}^2$ be the two dimensional torus, $L$ the line $3x = 7y$ in $\mathbb{R}^2$, and $S = \pi(L) \subset M$ where $\pi : \mathbb{R}^2 \to M$ is the projection map. Find a differential form on $M$ which represents the Poincaré dual of $S$.

Problem 4 (yau-contest 2014)
Section: Geometry and Topology

Let $p : (\tilde{M}, \tilde{g}) \to (M, g)$ be a Riemannian submersion. This is a submersion $p : M \to M$ such that for each $x \in \tilde{M}$,
$$
Dp : \ker^{\perp}(Dp) \to T_{p(x)}(M)
$$
is a linear isometry.

(a) Show that $p$ shortens distances.

(b) If $(\tilde{M}, \tilde{g})$ is complete, so is $(M, g)$.

(c) Show by example that if $(M, g)$ is complete, $(\tilde{M}, \tilde{g})$ may not be complete.

Problem 5 (yau-contest 2014)
Section: Geometry and Topology

Let $\Psi : M \to \mathbb{R}^3$ be an isometric immersion of a compact surface $M$ into $\mathbb{R}^3$. Prove that
$$
\int_M H^2 \, d\sigma \ge 4\pi,
$$
where $H$ is the mean curvature of $M$ and $d\sigma$ is the area element of $M$.

Problem 6 (yau-contest 2014)
Section: Geometry and Topology

The unit tangent bundle of $S^2$ is the subset
$$
T^1(S^2) = \{(p, v) \in \mathbb{R}^3 \mid \|p\| = 1,\; (p, v) = 0 \text{ and } \|v\| = 1\}.
$$
Show that it is a smooth submanifold of the tangent bundle $T(S^2)$ of $S^2$ and $T^1(S^2)$ is diffeomorphic to $\mathbb{RP}^3$.