Problem 1 (yau-contest 2013)
Section: Geometry and Topology

Find the homology and fundamental group of the space $X = S^1 \times S^1/\{p, q\}$ obtained from the torus by identifying two distinct points $p, q$ to one point.

Problem 2 (yau-contest 2013)
Section: Geometry and Topology

Suppose $(X,d)$ is a compact metric space and $f : X \to X$ is a map so that $d(f(x), f(y)) = d(x, y)$ for all $x, y \in X$. Show that $f$ is an onto map.

Problem 3 (yau-contest 2013)
Section: Geometry and Topology

Let $M^2$ ba a complete regular surface and $K$ be the Gaussian curvature. Suppose $\sigma : [0,\infty) \to M$ is a geodesic such that $K(\sigma(t)) \le f(t)$, where $f$ is a differentiable function on $[0,\infty)$. Prove that any solution $u(t)$ of the equation $$u''(t) + f(t)u(t) = 0$$ has a zero on $[0, t_0]$, where $\sigma(t_0)$ is the first conjugate point to $\sigma(0)$ along $\sigma$.

Problem 4 (yau-contest 2013)
Section: Geometry and Topology

Let $g_1, g_2$ be Riemannian metrics on a differentiable manifold $M$, and denote by $R_1$ and $R_2$ their respective Riemannian curvature tensor. Suppose that $R_1(X, Y, X, X) = R_2(X, Y, Y, X)$ holds for any tangent vectors $X, Y \in T_pM$. Show that $R_1(X, Y, Z, W) = R_2(X, Y, Z, W)$ for any $X, Y, Z, W \in T_pM$.

Problem 5 (yau-contest 2013)
Section: Geometry and Topology

Let $M^n$ be an even dimensional, orientable Riemannian manifold with positive sectional curvature. Let $\sigma : [0, l] \to M$ be a closed geodesic, namely, $\sigma$ is a geodesic with $\sigma(0) = \sigma(l)$ and $\sigma'(0) = \sigma'(l)$. Show that there exist an $\epsilon > 0$ and a smooth map $F : [0, l] \times (-\epsilon, \epsilon) \to M$, such that $F(t, 0) = \sigma(t)$, and for any fixed $s \ne 0$ in $(-\epsilon, \epsilon)$, $\sigma_s(t) = F(t, s)$ is a closed smooth curve with length less than that of $\sigma$.

Problem 6 (yau-contest 2013)
Section: Geometry and Topology

Let $(M^2, ds^2)$ be a minimal surface in $\mathbb{R}^3$, where $ds^2$ is the restriction of the Euclidean metric. Assume that the Gaussian curvature $K$ of $(M^2, ds^2)$ is negative. Denote by $\tilde{K}$ the Gaussian curvature of the metric $d\tilde{s}^2 = -K\, ds^2$. Show that $\tilde{K} = 1$.