Problem 1 (yau-contest 2011)
Section: Geometry and Topology

Suppose $M$ is a closed smooth $n$-manifold.

a) Does there always exist a smooth map $f:M\to S^n$ from $M$ into the $n$-sphere, such that $f$ is essential (i.e. $f$ is not homotopic to a constant map)? Justify your answer.

b) Same question, replacing $S^n$ by the $n$-torus $T^n$.

Problem 2 (yau-contest 2011)
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 2011)
Section: Geometry and Topology

Let $C_1, C_2$ be two linked circles in $\mathbb{R}^3$. Show that $C_1$ cannot be homotopic to a point in $\mathbb{R}^3\setminus C_2$.

Problem 4 (yau-contest 2011)
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 5 (yau-contest 2011)
Section: Geometry and Topology

A regular curve $C$ in $\mathbb{R}^3$ is called a Bertrand Curve, if there exists a diffeomorphism $f:C\to D$ from $C$ onto a different regular curve $D$ in $\mathbb{R}^3$ such that $N_x C = N_{f(x)} D$ for any $x\in C$. Here $N_x C$ denotes the principal normal line of the curve $C$ passing through $x$, and $T_x C$ will denote the tangent line of $C$ at $x$. Prove that:

a) The distance $|x-f(x)|$ is constant for $x\in C$; and the angle made between the directions of the two tangent lines $T_x C$ and $T_{f(x)}D$ is also constant.

b) If the curvature $k$ and torsion $\tau$ of $C$ are nowhere zero, then there must be constants $\lambda$ and $\mu$ such that $\lambda k+\mu\tau=1$.

Problem 6 (yau-contest 2011)
Section: Geometry and Topology

Let $M$ be the closed surface generated by carrying a small circle with radius $r$ around a closed curve $C$ embedded in $\mathbb{R}^3$ such that the center moves along $C$ and the circle is in the normal plane to $C$ at each point. Prove that
$$\int_M H^2\, d\sigma \ge 2\pi^2,$$
and the equality holds if and only if $C$ is a circle with radius $\sqrt{2}\,r$. Here $H$ is the mean curvature of $M$ and $d\sigma$ is the area element of $M$.