Geometry - January 2025

January 2025

Directions

Do all 6 problems (there is a choice in Problem 4). All manifolds, functions, vector fields, etc. are assumed to be smooth.

Problem 1.

Consider the map $f:\mathbb R^4\to\mathbb R^2$ given by $f(x,y,z,w)=(x^4+y^3,zw)$. Let $M=f^{-1}(1,1)$.

1. Show that $M$ is a smooth manifold and give its dimension.
2. Describe the tangent space $T_pM$ as a subspace of $T_p\mathbb R^4$ for the point $p=(x,y,z,w)=(1,0,1,0)$.
3. For which values of $a$ and $b$ is $f^{-1}(a,b)$ not a regular submanifold of $\mathbb R^4$?

Problem 2.

Consider the surface $S=\{(x,y,z)\mid y=x^2+z^2,y\le4\}$, oriented by the orientation form $dz\wedge dx$. Evaluate:

$$\int_S z\,dx\wedge dy$$

It may help to know that $\int_0^{2\pi}\sin^2\theta\,d\theta=\int_0^{2\pi}\cos^2\theta\,d\theta=\pi$.

Problem 3.

Let $I$ denote the identity element in the Lie group $GL(n,\mathbb R)=\{\text{invertible }n\times n\text{ real matrices}\}$. Consider the map $f:GL(n,\mathbb R)\to GL(n,\mathbb R)$ defined by $f(A)=A^2$.

1. Explain why $GL(n,\mathbb R)$ is a manifold.
2. Compute the differential $(df)_A$ at a point $A\in GL(n,\mathbb R)$.
3. Prove that there are neighborhoods $U,V$ of $I$ in $GL(n,\mathbb R)$ such that for every matrix $B\in V$ there is a unique matrix $A\in U$ with $A^2=B$.

Problem 4.

Do ONE of the following two (somewhat shorter) problems.

4A.

Let $f:M\to N$ be a smooth map from a manifold $M$ without boundary ($\partial M=\emptyset$) to a manifold $N$ with $\partial N\ne\emptyset$. Show that if $p\in M$ is a point such that $df_p$ is surjective, then $f(p)\in\operatorname{int}N$.

4B.

Let $\omega$ be a $p$-form on a manifold $M$. Suppose that there is a smooth positive function $f$ on $M$ such that $f\omega$ is exact.

1. Prove that there is a 1-form $\theta$ such that $d\omega=\theta\wedge\omega$.
2. In the case where $\omega$ is a 1-form, show that $d\omega\wedge\omega=0$.

Problem 5.

Let $X$ and $Y$ be a vector fields on a manifold $M$, and let $\omega\in\Omega^*(M)$ be a differential form on $M$.

1. State the definition of the Lie derivative $\mathcal L_X\omega$ in terms of flows.

2. Use your definition to prove that $\mathcal L_X$ is a derivation on $\Omega^*(M)$.

3. Use your definition to prove that $\mathcal L_X$ commutes with $d$.

4. In the case where $\omega$ is a 1-form, compute $\mathcal L_X(\omega(Y))$. Then use Cartan’s formula to show that

   $$(d\omega)(X,Y)=X\cdot\omega(Y)-Y\cdot\omega(X)-\omega([X,Y]).$$

Problem 6.

A $p$-form $\omega$ on a manifold $M$ has compact support if $\{x\in M\mid\omega(x)\ne0\}$ is compact. Let $\Omega^p_{\mathrm{cpt}}(M)$ denote the set of such forms. The deRham cohomology groups with compact support are defined by

$$H^p_{\mathrm{cpt}}(M)=\frac{\{\omega\in\Omega^p_{\mathrm{cpt}}(M)\mid d\omega=0\}}{\{d\eta\mid\eta\in\Omega^{p-1}_{\mathrm{cpt}}(M)\}}.$$

Consider the case $M=\mathbb R^n$.

1. Show that $H^0_{\mathrm{cpt}}(\mathbb R^n)=0$.
2. Construct a closed $n$-form $\omega$ on $\mathbb R^n$ with compact support and $\int_{\mathbb R^n}\omega\ne0$.
3. Use (b) to show that $H^n_{\mathrm{cpt}}(\mathbb R^n)\ne0$.