Problem 1 (qiuzhen-qualify 2023-fall)

(10 pts) On a Riemannian manifold $(M, g)$, suppose that $f$ is a smooth function such that $|\mathrm{grad}\, f| = 1$. Show that the integral curves of $\mathrm{grad}\, f$ are geodesics.

Problem 2 (qiuzhen-qualify 2023-fall)

(10 pts) Let $M$ be the quotient space

$([0,1] \times \mathbb{CP}^2)/(0, [z_0, z_1, z_2]) \sim (1, [\bar{z}_0, \bar{z}_1, \bar{z}_2]).$

Compute the homology group $H_k(M; \mathbb{Z})$ for all $k \ge 0$.

Problem 3 (qiuzhen-qualify 2023-fall)

(10 pts) Consider a map $f : T^2 \to \mathbb{RP}^2$ from the torus to the projective plane. Suppose the induced map $f_* : \pi_1(T^2) \to \pi_1(\mathbb{RP}^2)$ is trivial. Is $f$ always null-homotopic? Prove your conclusion.

Problem 4 (qiuzhen-qualify 2023-fall)

(10 pts) Let $F : (M, g) \to (M, g)$ be an isometry that fixes $p \in M$. Show that $dF_p = -\mathrm{id}$ on $T_p M$ if and only if $F^2 = \mathrm{id}_M$ and $p$ is an isolated fixed point.

Problem 5 (qiuzhen-qualify 2023-fall)

(15 pts) Let $M$ be a closed smooth manifold. Show that $M$ is a smooth fiber bundle over $S^1$ if and only if there exists a closed, nowhere vanishing differential $1$-form on $M$. (A differential form $\alpha$ is nowhere vanishing if for any $x \in M$, there exists $v \in T_x M$ such that $\alpha(v) \ne 0$.)

Problem 6 (qiuzhen-qualify 2023-fall)

(15 pts) Consider an Einstein metric $(N^{n-1}, g_N)$ with $\mathrm{Ric} = \frac{n-2}{n-1}\, \lambda\, g_N$, $\lambda < 0$. Find a $\rho : \mathbb{R} \to (0, \infty)$ such that $(M^n, g) = (\mathbb{R} \times N, dr^2 + \rho^2(r) g_N)$ becomes an Einstein metric with $\mathrm{Ric} = \lambda g$.

Problem 7 (qiuzhen-qualify 2023-fall)

(15 pts) (1) Let $M$ be a compact, orientable $(2n+1)$-dimensional manifold with boundary $N$. Consider the map

$i^* : H^n(M; \mathbb{R}) \to H^n(N; \mathbb{R})$

induced by the inclusion $i : N \to M$. Show that

dimension of the image of $i^* = \frac{1}{2}$ dimension of $H^n(N; \mathbb{R})$.

(2) Show that for any $n \ge 1$, $\#_n \mathbb{CP}^2$ is not the boundary of a compact, orientable $5$-dimensional topological manifold. Here $\#_n \mathbb{CP}^2$ denotes the connected sum of $n$ copies of complex projective space $\mathbb{CP}^2$.

Problem 8 (qiuzhen-qualify 2023-fall)

(15 pts) Suppose $(M, g)$ is a Riemannian manifold. There is a unique bundle endomorphism

$\mathcal{R} : \Lambda^2 TM \to \Lambda^2 TM$

called the curvature operator of $g$, that satisfies

$g(\mathcal{R}(W \wedge X), Y \wedge Z) = -R(W, X, Y, Z)$

for all tangent vectors $W, X, Y, Z$ at a point of $M$. A Riemannian metric is said to have positive curvature operator if $\mathcal{R}$ is positive definite.

(a) Show that positive curvature operator implies positive sectional curvature.

(b) Let $g$ be the Fubini-Study metric on $\mathbb{CP}^2$. Do it have positive sectional curvature? Does $g$ have positive curvature operator? Give details to justify your conclusions.