Problem 1 (yau-contest 2015)
Section: Geometry and Topology

Let $SO(3)$ be the set of all $3\times 3$ real matrices $A$ with determinant $1$ and satisfying $^{t}A\,A=I$, where $I$ is the identity matrix and $^{t}A$ is the transpose of $A$. Show that $SO(3)$ is a smooth manifold, and find its fundamental group. You need to prove your claims.

Problem 2 (yau-contest 2015)
Section: Geometry and Topology

Let $X$ be a topological space. The suspension $S(X)$ of $X$ is the space obtained from $X\times [0,1]$ by contracting $X\times\{0\}$ to a point and contracting $X\times\{1\}$ to another point. Describe the relation between the homology groups of $X$ and $S(X)$.

Problem 3 (yau-contest 2015)
Section: Geometry and Topology

Let $F:M\to N$ be a smooth map between two manifolds. Let $X_1, X_2$ be smooth vector fields on $M$ and let $Y_1, Y_2$ be smooth vector fields on $N$. Prove that if $Y_1 = F_{\ast}X_1$ and $Y_2 = F_{\ast}X_2$, then $F_{\ast}[X_1, X_2] = [Y_1, Y_2]$, where $[\ ,\ ]$ is the Lie bracket.

Problem 4 (yau-contest 2015)
Section: Geometry and Topology

Let $M_1$ and $M_2$ be two compact convex closed surfaces in $\mathbb{R}^3$, and $f : M_1 \to M_2$ a diffeomorphism such that $M_1$ and $M_2$ have the same inner normal vectors and Gauss curvatures at the corresponding points. Prove that $f$ is a translation.

Problem 5 (yau-contest 2015)
Section: Geometry and Topology

Prove the second Bianchi identity:

$$R_{ijkl,h} + R_{ijhl,k} + R_{ijhk,l} = 0$$

Problem 6 (yau-contest 2015)
Section: Geometry and Topology

Let $M_1, M_2$ be two complete $n$-dimensional Riemannian manifolds and $\gamma_i : [0,a] \to M_i$ are two arc length parametrized geodesics. Let $\rho_i$ be the distance function to $\gamma_i(0)$ on $M_i$. Assume that $\gamma_i(a)$ is within the cut locus of $\gamma_i(0)$ and for any $0 \le t \le a$ we have the inequality of sectional curvatures

$$K_1\!\left(X_1, \frac{\partial}{\partial \dot{\gamma}_1}\right) \ge K_2\!\left(X_2, \frac{\partial}{\partial \dot{\gamma}_2}\right),$$

where $X_i \in T_{\gamma_i(t)}M_i$ is any unit vector orthogonal to the tangent $\frac{\partial}{\partial \dot{\gamma}_i}$. Then

$$\mathrm{Hess}(\rho_1)(\tilde{X}_1, \tilde{X}_1) \le \mathrm{Hess}(\rho_2)(\tilde{X}_2, \tilde{X}_2),$$

where $\tilde{X}_i \in T_{\gamma_i(a)}M_i$ is any unit vector orthogonal to the tangent $\frac{\partial}{\partial \dot{\gamma}_i}(a).$