Geometry - January 2024

Jan 4, 2024

Instructions

Solve 5 out of the 6 problems. You must justify all your claims either by direct arguments or by referring to well known and basic theorems. Indicate clearly which 5 problems you would like us to grade.

Problem 1.

Let $M$ be a nonempty smooth $n$-manifold, and suppose $n\ge1$.

1. Show that the vector space $C^\infty(M)$ is infinite-dimensional.
2. Suppose $A,B\subset M$ are disjoint closed subsets. Prove that there exists $f\in C^\infty(M)$ s.t. $f|_A\equiv0$ and $f|_B\equiv1$.
3. Suppose $N$ is another smooth manifold and $F:M\to N$ is a continuous map. Prove that $F$ is smooth if $F^*(C^\infty(N))\subset C^\infty(M)$.

Problem 2.

Let $F:\mathbb R^3\to\mathbb R$ be defined by $F(x,y,z)=x^2+xy-2y^2+z^4$.

1. Prove that the level set $\Sigma:=F^{-1}\{1\}$ is a submanifold of $\mathbb R^3$.
2. Is $\Sigma$ compact? Prove your claim.
3. Find the critical points of the function $f(x,y,z)=x$ on $\Sigma$.

Problem 3.

Let $M$ be a smooth $n$-manifold.

1. What is a smooth vector filed on $M$?
2. Suppose $(U,x)$ and $(V,y)$ are two charts on $M$ with $U\cap V\ne\emptyset$. A vector field $X$ on $M$ has the local expression $\sum_i a^i\frac{\partial}{\partial x^i}$ on $U$ and $\sum_i b^i\frac{\partial}{\partial y^i}$ on $V$. Find a formula for $a^i$ in terms of $b^j$ on $U\cap V$.
3. On the local chart $(S^2\setminus\{N\},x)$ by the stereographic projection from the north pole (recall $x=\frac{(\xi_1,\xi_2)}{1-\xi_3}$), $\frac{\partial}{\partial x^1}$ defines a vector field on $S^2\setminus\{N\}$. Is it the local expression of a smooth vector field $X$ on $S^2$? If yes, what is the value of $X$ at the north pole? Justify your answers.

Problem 4.

Let $X$ be a smooth vector field on a smooth manifold $M$.

1. If $c:I\to M$ is a nonconstant integral curve, show that it is an immersion.
2. If an integral curve $c:\mathbb R\to M$ is not injective, show that it is periodic, i.e. there exists $T>0$ s.t. $c(t+T)=c(t)$, for all $t\in\mathbb R$.
3. For $X=-x^2\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$ on $\mathbb R^2$, find the maximal integral curve starting at the point $(1,0)$.

Problem 5.

Let $M$ be a smooth $n$-manifold.

1. What is a Riemannian metric on $M$?

2. Suppose $(U,x)$ and $(V,y)$ are two charts on $M$ with $U\cap V\ne\emptyset$. A Riemannian metric $g$ on $M$ has the local expression $\sum_{i,j}g_{ij}\,dx^i\otimes dx^j$ on $U$ and $\sum_{i,j}\widetilde g_{ij}\,dy^i\otimes dy^j$ on $V$. Find a formula relating $[g_{ij}]$ and $[\widetilde g_{ij}]$ on $U\cap V$.

3. Suppose $M$ is oriented. Show that the $n$-form

   $$\Omega=\sqrt{\det[g_{ij}]}\,dx^1\wedge\cdots\wedge dx^n$$

   on each positively oriented local chart $(U,x)$ is globally defined on $M$.

Problem 6.

On $\mathbb R^n\setminus\{0\}$ consider the differential $(n-1)$-form

$$\omega=|x|^{-n}\sum_{i=1}^n(-1)^{i+1}x_i\,dx^1\wedge\cdots\wedge\widehat{dx^i}\wedge\cdots\wedge dx^n.$$

1. Show that $\omega$ is closed.
2. Let $\iota:S^{n-1}\to\mathbb R^n\setminus\{0\}$ be the inclusion map. Show that $\iota^*\omega$ is an orientation form on $S^{n-1}$, i.e. it is nowhere zero.
3. Is $\omega$ exact? Justify your claim.