Problem 1 (qiuzhen-qualify 2025-spring)

Let $M,N$ be closed, connected manifolds of the same dimension. Suppose $f : M \to N$ is an immersion. Is the following true or false? If true, prove your claim. If false, give a counterexample and prove that your example is indeed a counterexample.

(a) $f$ is injective.

(b) $f$ is surjective.

(c) $f$ is a homeomorphism.

(d) $f$ is a covering map.

Problem 2 (qiuzhen-qualify 2025-spring)

Let $B^n(r) = \{ x \in \mathbb{R}^n \mid |x| < r \}$. Compute the sectional curvature of
$$g = \frac{r^4}{(r^2 - |x|^2)^2} \sum_{i=1}^n dx^i \otimes dx^i.$$

Problem 3 (qiuzhen-qualify 2025-spring)

Let $X$ denote a closed orientable surface of genus $2$. See the picture below with four curves labeled as $e_1, e_2, e_3, e_4$. Consider a group homomorphism
$$\rho : \pi_1(X) \to \mathbb{Z}/2\mathbb{Z}$$
such that $\rho(e_4)=1$ and $\rho(e_i)=0$ for $i=1,2,3$.

(a) Draw a picture of a covering space $Y$ that corresponds to the subgroup $\ker \rho \subset \pi_1(X)$.

(b) On the picture, label all curves on $Y$ which are preimages of $e_1, e_2, e_3, e_4$.

(c) The signature $\sigma(A)$ of a square matrix $A$ is the number of its positive eigenvalues minus the number of its negative eigenvalues. Let $T$ denote the nontrivial deck transformation of the covering $Y \to X$. Compute the signature $\sigma(T_*)$ of $T_* : H_1(Y) \to H_1(Y)$.

(d) Prove that there exists a homeomorphism $\phi : X \to X$ such that $\phi(e_1) = e_4$.

Problem 4 (qiuzhen-qualify 2025-spring)

Let $(M,g)$ be a Riemannian manifold and $X$ be a Killing field. If $\gamma$ is a geodesic, show that $J(t) = X \circ \gamma(t)$ is a Jacobi field along $\gamma$.

Problem 5 (qiuzhen-qualify 2025-spring)

Compute the cohomology ring of $H^*((S^2 \times S^8)\#(S^4 \times S^6); \mathbb{Z})$.

Problem 6 (qiuzhen-qualify 2025-spring)

A Riemannian metric $h$ on a Lie group $G$ is said to be left-invariant if $L_g^* h = h$ for all $g \in G$. A right-invariant metric is defined similarly. A metric that is both left- and right-invariant is said to be bi-invariant. Let $(M,g)$ be a Riemannian manifold with a bi-invariant Riemannian metric $g$.

(A) Show that for all left-invariant vector fields $X, Y, Z$,
$$h([X,Y], Z) = h(X, [Y,Z]).$$

(B) Show that for any left-invariant vector fields $X, Y$,
$$\nabla_X Y = \frac{1}{2}[X,Y],$$
where $\nabla$ is the Levi-Civita connection.

Problem 7 (qiuzhen-qualify 2025-spring)

Suppose that $M$ is a compact connected nonorientable 3-dimensional manifold. Prove that $\pi_1(M)$ is infinite.

Problem 8 (qiuzhen-qualify 2025-spring)

Let $(M,g)$ be a simply connected complete Riemannian manifold. If the differential of each exponential map is length increasing, i.e.
$$\left|(d\exp_p)_v\, \vec{v}\right| \ge \left|\vec{v}\right|$$
for all $p \in M$ and all $v, \vec{v} \in T_p M$, show that $(M,g)$ has non-positive sectional curvature.

Problem 9 (qiuzhen-qualify 2025-spring)

Let $\mathrm{SL}_3(\mathbb{C})$ be the space of $3\times3$ complex matrices with determinant $1$. Compute $\pi_3(\mathrm{SL}_3(\mathbb{C}))$.

Problem 10 (qiuzhen-qualify 2025-spring)

Let $(M,g)$ be an $n$-dimensional compact oriented Riemannian manifold with positive sectional curvature. Given an isometry $F : M \to M$ such that $F$ preserves the orientation when $n$ is even, $F$ changes the orientation when $n$ is odd. Show that $F$ has a fixed point.