Geometry - January 2023

Fall 2022

Instructions

Each problem is worth 20 points. Your grade will be determined by your 5 best solutions. You may submit more than 5, but are not required to do so.

Problem (1)

1. Give an example of a smooth map $f:\mathbb R^2\to\mathbb R^2$ for which there exists $p\in\mathbb R^2$ with nonzero Jacobian matrix $J_f(p)=\left[\frac{\partial f^i}{\partial x^j}(p)\right]$ satisfying $\det J_f(p)=0$.

2. Suppose $f:\mathbb R^2\to\mathbb R^2$ is a smooth map satisfying the Cauchy-Riemann equations:

   $$\frac{\partial f^1}{\partial x^1}=\frac{\partial f^2}{\partial x^2}\qquad\text{and}\qquad\frac{\partial f^1}{\partial x^2}=-\frac{\partial f^2}{\partial x^1}.$$

   Prove that for $p\in\mathbb R^2$, $J_f(p)=0$ if and only if $\det J_f(p)=0$.

3. Suppose $f:\mathbb R^2\to\mathbb R^2$ satisfies the Cauchy-Riemann equations. Prove that at points $p\in\mathbb R^2$ where $f$ is a local-diffeomorphism, the local inverse of $f$ also satisfies the Cauchy-Riemann equations.

Problem (2)

Let $M=\{(x,y,z)\in\mathbb R^3\mid z-x^2-y^2=1\}$.

1. Prove $M$ is a regular submanifold that is diffeomorphic to $\mathbb R^2$.
2. Prove that $\omega=z\,dx\wedge dy\in\Omega^2(\mathbb R^3)$ is not an exact two form on $\mathbb R^3$.
3. Prove the restriction $\omega|_M\in\Omega^2(M)$ is an exact two form on $M$.

Problem (3)

Let $X=-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}\in\chi(\mathbb R^2)$.

1. Define $f:\mathbb R^2\to\mathbb R$ by $f(x,y)=e^{x^2+y^2}$. Prove if $g:\mathbb R^2\to\mathbb R$ is smooth, then

   $$X(g)=0\iff X(fg)=0.$$

2. Suppose $Y\in\chi(\mathbb R^2)$ is a smooth vector field having the same restriction as $X$ to the unit circle. Show there exists $p\in\mathbb R^2$ such that $Y(p)=0$.

Problem (4)

Let $V,W\in\chi(\mathbb R^3)$ be smooth vector fields defined by

$$V=\frac{\partial}{\partial x}+y\frac{\partial}{\partial z}$$

$$W=\frac{\partial}{\partial y}.$$

1. Determine the smooth functions $f_i:\mathbb R^3\to\mathbb R$, $i\in\{1,2,3\}$, defined by

   $$[V,W]=f_1\frac{\partial}{\partial x}+f_2\frac{\partial}{\partial y}+f_3\frac{\partial}{\partial z}.$$

2. Show for a smooth function $f:\mathbb R^3\to\mathbb R$, $V(f)=W(f)=0$ if and only if $f$ is a constant function.

Problem (5)

For each $\theta\in\mathbb R$ and $\lambda\in(0,\infty)$ define diffeomorphisms

$$R_\theta,D_\lambda:\mathbb R^2\setminus\{0\}\to\mathbb R^2\setminus\{0\}$$

by

$$R_\theta(x,y)=(\cos(\theta)x+\sin(\theta)y,-\sin(\theta)x+\cos(\theta)y)$$

and

$$D_\lambda(x,y)=(\lambda x,\lambda y).$$

Determine the smooth functions $f:\mathbb R^2\setminus\{0\}\to\mathbb R$ such that the smooth 2-form $\omega=f\,dx\wedge dy$ has the property that for every $(\theta,\lambda)\in\mathbb R\times(0,\infty)$, $R_\theta^*(\omega)=\omega$ and $D_\lambda^*(\omega)=\omega$.

Problem (6)

Let $X\in\chi(\mathbb R^2)$ be the smooth vector field defined by

$$X=x\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}.$$

1. Let $f:\mathbb R^2\to\mathbb R$ be the smooth function defined by $f(x,y)=xy$. Show $X(f)=0$.
2. Suppose that $c(t)$ is an integral curve. Explain why part (a) implies $f(c(t))$ is constant as a function of $t$.
3. Let $p_0=(x_0,y_0)\in\mathbb R^2$. Find the maximal integral curve $c(t)$ of $X$ with $c(0)=p_0$. Verify that $f(c(t))$ is constant in $t$ directly using your formula for the integral curve $c(t)$.

Problem (7)

Let $\lambda\in\mathbb R$ and let $F_\lambda:\mathbb R^2\to\mathbb R^2$ be defined by $F(x,y)=(xe^y-\lambda y,xe^y+y)$.

1. Determine those $\lambda\in\mathbb R$ for which $F_\lambda$ is a diffeomorphism of $\mathbb R^2$.
2. Determine those $\lambda\in\mathbb R$ for which $F_\lambda$ is an orientation preserving diffeomorphism of $\mathbb R^2$.

Problem (8)

Let $M$ be a smooth manifold. Let $\omega\in\Omega^1(M)$ and let $f:M\to\mathbb R$ be a smooth and positive function. Show that if $d(f\omega)=0$, then $\omega\wedge d\omega=0$.