Problem 1(A) (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

What's the fundamental group of $SO(3)$?

Problem 1(B) (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Does $SO(3)$ admit a smooth Riemannian metric with constant Ricci curvature?

Problem 2(A) (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Is it true that every continuous map $f : S^{2024} \to RP^{2024}$ is null-homotopic?

Problem 2(B) (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Is it true that every continuous map $g : S^{2024} \to CP^{1012}$ is null-homotopic?

Problem 3(A) (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Let $(M,g)$ be a connected Riemannian manifold of dimension $n$. Suppose that there exists some $f \in C\infty(M, \mathbb{R})$ such that

$$Ric(g) = (n-1)\, f\, g.$$

If $n=2$, is $f$ necessarily a constant?

Problem 3(B) (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Let $(M,g)$ be a connected Riemannian manifold of dimension $n$. Suppose that there exists some $f \in C\infty(M, \mathbb{R})$ such that

$$Ric(g) = (n-1)\, f\, g.$$

If $n=2024$, is $f$ necessarily a constant?

Problem 4 (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Consider the quotient space

$$X = ([0,1] \times S^1 \times S^1)/\sim,$$

where the equivalence relation $\sim$ is generated by

$$(0, x, y) \sim (0, z, w) \text{ if } xy = zw,$$

and

$$(1, x, y) \sim (1, z, w) \text{ if } x^2 y^6 = z^2 w^6.$$

Here we treat $S^1$ as the space of unit complex numbers. Compute $H_n(X; \mathbb{Z})$ for all $n$.

Problem 5 (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Recall a space $X$ is called an H-space if there exists a point $e \in X$ and a continuous map $\mu : X \times X \to X$ such that the map $X \to X$ defined by $x \mapsto \mu(e,x)$ and the map $X \to X$ defined by $x \mapsto \mu(x,e)$ are both homotopic to the identity map. Show that $CP^n$ is not an H-space for any $1 \le n < \infty$.

Problem 6 (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Let $(M,g)$ be a Cartan-Hadamard manifold. Given $p \in M$, let

$$f : M \to [0,+\infty)$$

be the function $f(x) = \tfrac{1}{2} d(x,p)^2$. Show that $f$ is strictly geodesically convex, i.e. for any (nontrivial) geodesic $\gamma : [0,1] \to M$, the following inequality holds for all $t \in (0,1)$:

$$f(\gamma(t)) < (1-t) f(\gamma(0)) + t f(\gamma(1)).$$

Problem 7 (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Let $M$ be an oriented, connected, closed manifold of dimension $n \ge 2$. Let $f : S^n \to M$ be a continuous map of mapping degree $\deg(f)=1$. Show that $f$ must be a homotopy equivalence.

Problem 8 (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Let $(M,g)$ be a smooth Riemannian manifold of dimension $n$ and $p \in M$. Show that when $r$ is small enough,

$$\operatorname{Vol}(B(p,r)) = \omega_n r^n \left( 1 - \frac{s(p)}{6(n+2)} r^2 + O(r^3) \right),$$

where $\omega_n$ is the volume of the unit ball in $\mathbb{R}^n$ and $s(p)$ is the scalar curvature of $(M,g)$ at point $p$.

Problem 9 (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Let $M$ be a smooth, closed manifold of dimension $\ge 1$. And let $f : M \to M$ be a smooth map such that $f \circ f(x) = x$ for any $x \in M$. Show that the set $\{ x \in M \mid f(x) = x \}$ can not be a single point.

Problem 10(A) (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Does $S^1 \times S^1$ admit a Riemannian metric with conjugate radius $= +\infty$?

Problem 10(B) (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Does $S^1 \times S^1$ admit a Riemannian metric with conjugate radius $< +\infty$?

Problem 10(C) (qiuzhen-qualify 2024-fall)
Section: Geometry & Topology

Does $S^2 \times S^1$ admit a Riemannian metric with conjugate radius $= +\infty$?