Problem 1 (yau-contest 2019)
Section: Geometry and Topology

Is $TS^2$ diffeomorphic to $S^2 \times \mathbb{R}^2$? Verify your answer. Here $TS^2$ is the total space of the tangent bundle of $S^2$.

Problem 2 (yau-contest 2019)
Section: Geometry and Topology

Solve the problem which Russell Crowe assigns to his students in the movie "A beautiful mind" (2001):

$$V = \{ F : \mathbb{R}^3 \setminus X \to \mathbb{R}^3 \text{ s.t. } \nabla \times F = 0 \},$$
$$W = \{ F = \nabla g \},$$
$$\dim(V/W) = ?$$

First give the general answer for any closed $X \subset \mathbb{R}^3$, and then specialize it to
(a) $X = \{ x = y = z = 0 \}$,
(b) $X = \{ x = y = 0 \}$ and
(c) $X = \{ x = 0 \}$.

Problem 3 (yau-contest 2019)
Section: Geometry and Topology

Let $T^2 = S^1 \times S^1$ be the 2-torus with the standard orientation, and let $F : T^2 \to T^2$ be a smooth map of degree $1$ such that $F \circ F = \mathrm{Id}$ and $F$ has no fixed points. Prove that the induced map $F^* : H^1(T^2) \to H^1(T^2)$ is the identity.

Problem 4 (yau-contest 2019)
Section: Geometry and Topology

Let $U(n)$ be the group of $n \times n$ unitary matrices, and $O(n)$ be the group of $n \times n$ orthogonal matrices. Let $SU(n) = \{ A \in U(n) \mid \det A = 1 \}$ be the special unitary group and $SO(n) = \{ A \in O(n) \mid \det A = 1 \}$ be the special orthogonal group. All $U(n)$, $SU(n)$, $O(n)$, $SO(n)$ are Lie groups with natural manifold structures.

(a) Compute the dimensions of $SU(n)$ and $SO(n)$.

(b) Compute the fundamental groups of $SU(n)$ and $SO(n)$ ($n \ge 2$).

Problem 5 (yau-contest 2019)
Section: Geometry and Topology

Let $(M, g)$ be a compact Riemannian manifold and $R$ be its Riemannian curvature tensor. $(M, g)$ will be called weakly negative if for any point $p \in M$ and for any nonzero vector field $X \in T_p M$, there exists a nonzero vector field $Y \in T_p M$ such that $R(X,Y,Y,X) < 0$.

(a) Let $X$ be a Killing vector field and $f = \tfrac{1}{2} g(X,X) = \tfrac{1}{2} |X|^2$. Show that for any vector field $V$

$$ (\mathrm{Hess} f)(V,V) = g(\nabla_V X, \nabla_V X) - R(V,X,X,V). $$

Here the Hessian of $f$ is $(\mathrm{Hess} f)(Y,Z) := g(\nabla_Y \mathrm{grad}(f), Z)$ for any vector fields $Y,Z$, where $\mathrm{grad}(f)$ is the gradient vector of $f$.

(b) Prove that if $(M, g)$ is weakly negative, then there are no nontrivial Killing vector fields.