Problem 1 (yau-contest 2023)
Section: Geometry and Topology

(a) Let $G$ be a Lie group, $g$ be its Lie algebra. The Maurer-Cartan form on $G$ is the unique left-invariant $g$-valued 1-form such that $\omega_e : T_e G \to g$ is the identity map, where $e$ is the identity in $G$. Show that the Maurer-Cartan form $\omega$ satisfies the Maurer-Cartan equation $d\omega + \tfrac{1}{2}[\omega, \omega] = 0$.

(b) Let $G$ be a matrix group $GL(n, \mathbb{R})$, give detailed computation to find the Maurer-Cartan form $\omega$.

(c) Let $SE(3)$ be the special Euclidean group, i.e. it contains all transformations of $\mathbb{R}^3$ (as 3-dim Euclidean space) of the form $x \mapsto t + Ax$, where $t \in \mathbb{R}^3$ and $A \in SO(3)$. Find an expression of the Maurer Cartan form $\omega$ of $SE(3)$, also check it satisfies the standard Cartan structure equations in Euclidean space.

Problem 2 (yau-contest 2023)
Section: Geometry and Topology

Let $M_1, M_2$ be closed, connected, oriented 3-manifolds with the first fundamental groups
$$\pi_1(M_1) = \mathbb{Z}_3 \oplus \mathbb{Z}^2, \quad \pi_1(M_2) = \mathbb{Z}_6 \oplus \mathbb{Z}^3,$$
(a) Find all homology groups $H_n(M_1, \mathbb{Z})$ and $H_n(M_2, \mathbb{Z})$.

(b) Find all homology groups $H_n(M_1 \times M_2, \mathbb{Q})$.

(c) Does there exist a closed connected oriented 3-manifold $M$ with
$$\pi_1(M) = \mathbb{Z}_3 \oplus \mathbb{Z}^2 \quad \text{or} \quad \pi_1(M) = \mathbb{Z}_6 \oplus \mathbb{Z}^3?$$

Problem 3 (yau-contest 2023)
Section: Geometry and Topology

(a) Let $f$ be a diffeomorphism group of a circle $S^1$, assume $f$ has no fixed point and it is generated by a smooth vector field, show that $f$ must be conjugate to a rotation.

(b) Show that there is a diffeomorphism $f : S^1 \to S^1$, such that $f$ can not be generated by a smooth vector field but it is arbitrarily closed the identity map $i : S^1 \to S^1$ in $C^{\infty}$-topology.

Problem 4 (yau-contest 2023)
Section: Geometry and Topology

(a) State the Leray-Hirsh theorem.

(b) Let
$$Fl_k(\mathbb{C}^n) = \{ \text{all } k\text{-flags in } \mathbb{C}^n \} = \{(F_0, \cdots, F_k) \mid F_i \text{ is an } i\text{-dim subspace of } \mathbb{C}^n,\ \text{s.t. } F_0 \subset F_1 \subset \cdots \subset F_k \subset \mathbb{C}^n\}.$$
Let $\Phi : Fl_k(\mathbb{C}^n) \to Fl_{k-1}(\mathbb{C}^n)$ be the projection map sending a $k$-flag $(F_0, \cdots, F_k)$ to a $(k-1)$-flag $(F_0, \cdots, F_{k-1})$, it is known this is a fiber bundle. What is the fiber of $\Phi$?

(c) Compute the Euler Characteristic $\chi(Fl_n(\mathbb{C}^n))$.

Problem 5 (yau-contest 2023)
Section: Geometry and Topology

On the Euclidean space $\mathbb{R}^n$, we consider an $n-1$ form $\alpha$, which is of class $C^1$, such that both $\alpha$ and $d\alpha$ are in $L^1$. Show that $\int_{\mathbb{R}^n} d\alpha = 0$.

Problem 6 (yau-contest 2023)
Section: Geometry and Topology

A complete Riemannian metric $g_{ij}$ on a smooth manifold $M^n$ is called a gradient expanding Ricci soliton if there exists a smooth function $f$ on $M^n$ such that the Ricci tensor $Ric$ of the metric $g$ is given by
$$Ric + Hess f = \lambda g,$$
for some negative constant $\lambda < 0$. Show that if $M$ is compact, then a gradient expanding Ricci soliton must be an Einstein metric.