Problem 1 (yau-contest 2020)
Section: Geometry and Topology

Let $S^n$ be the unit sphere in $\mathbb{R}^{n+1}$.

(a) Find a 6-form $\alpha$ on $\mathbb{R}^7 \setminus \{0\}$ such that $d\alpha = 0$, and $\int_{S^6} \alpha = 1$.

(b) For any smooth map $f : S^{11} \to S^6$, show that there exists a 5-form $\varphi$ on $S^{11}$ such that $f^*\alpha = d\varphi$. (1)

(c) Let $H(f) = \int_{S^{11}} \varphi \wedge d\varphi$. Show that $H(f)$ is independent of the choice of $\varphi$ satisfying (1).

(d) Show that $H(f)$ is an integer, for any smooth map $f : S^{11} \to S^6$.

Problem 2 (yau-contest 2020)
Section: Geometry and Topology

For any $h \in C^{\infty}(\mathbb{R}^2)$ and $h > 0$ on $\mathbb{R}^2$, define the Ricci curvature $\operatorname{Ric}(h)$ associated with $h$ by
$$\operatorname{Ric}(h) = \frac{1}{h}\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) \log h,$$
where $(x,y)$ are the standard Cartesian coordinates in $\mathbb{R}^2$. Either construct a positive smooth function $h_1$ such that $\operatorname{Ric}(h_1) = 1$, or show that no such function $h_1$ exists.

Problem 3 (yau-contest 2020)
Section: Geometry and Topology

Let $M$ be an $n$-dimensional Riemannian manifold, and $p \in M$. Let $\{e_1,\ldots,e_n\}$ be an orthonormal basis of the tangent space $T_pM$, and let $\{x^1,\ldots,x^n\}$ be a coordinate system of $M$ centered at $p$ such that
$$\exp_p^{-1}(q) = \sum_{j=1}^{n} x^j(q)\, e_j,$$
where $\exp_p$ denotes the exponential map. Let $\gamma(t) = \exp_p(te_1)$, $0 \le t \le \delta$, where $\delta$ is a positive constant less than $1$.

(a) For $2 \le \alpha \le n$, which one of the following,
$$t\, \frac{\partial}{\partial x^\alpha}\Big|_{\gamma(t)}, \quad \frac{\partial}{\partial x^\alpha}\Big|_{\gamma(t)},$$
is a Jacobi field along $\gamma(t)$? Prove your assertion.

(b) Denote $g_{ij} = \left\langle \frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j} \right\rangle$, $1 \le i,j \le n$. Compute $\frac{\partial^2 g_{22}}{\partial x^1\, \partial x^1}$ at the point $p$.

(c) Show that
$$\max_{0\le t \le \delta} \left| \frac{\partial g_{22}}{\partial x^1}(\gamma(t)) \right| \le C\, \delta\, A,$$
where $C>0$ is a constant depending only on $n$, and $A$ is the $C^0$-bound of the curvature tensor of $M$ along $\gamma(t)$, for $0 \le t \le \delta$.

Problem 4 (yau-contest 2020)
Section: Geometry and Topology

Let $\mathrm{SO}(n)$ be the set of $n\times n$ orthogonal real matrices with determinant equal to $1$. Endow $\mathrm{SO}(n)$ with the relative topology as a subspace of Euclidean space $\mathbb{R}^{n^2}$.

(a) Show that $\mathrm{SO}(n)$ is compact.

(b) Is $\mathrm{SO}(3)$ homeomorphic to the real projective space $\mathbb{RP}^3$? Prove your assertion.

(c) Compute the fundamental group of $\mathrm{SO}(2020)$.

Problem 5 (yau-contest 2020)
Section: Geometry and Topology

Let $X$ be a topological space and $\pi : \mathbb{R}^2 \to X$ a covering map. Let $K$ be a compact subset of $X$ and $B$ the closed unit ball centered at the origin in $\mathbb{R}^2$.

(a) Suppose $\pi : \mathbb{R}^2 \setminus B \to X \setminus K$ is a homeomorphism. Show that $\pi : \mathbb{R}^2 \to X$ is a homeomorphism.

(b) Suppose $\mathbb{R}^2 \setminus B$ is homeomorphic to $X \setminus K$, where the homeomorphism may not be $\pi$. Is $X$ necessarily homeomorphic to $\mathbb{R}^2$? Prove your assertion.

Problem 6 (yau-contest 2020)
Section: Geometry and Topology

Let $F_n$ be the free group of rank $n$.

(a) Give an example of a finite connected graph such that its fundamental group is $F_2$.

(b) Does $F_2$ contain a proper subgroup isomorphic to $F_2$?

(c) Does $F_2$ contain a proper finite index subgroup isomorphic to $F_2$?