Geometry - August 2024

Aug. 22, 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 $GL(n,\mathbb R)=\{A:A\text{ is a real invertible }n\times n\text{ matrix}\}$ and let $O(n)=\{A:A\text{ is a real }n\times n\text{ matrix satisfying }AA^t=I\}$

1. Show that $GL(n,\mathbb R)$ is a smooth manifold of dimension $n^2$.
2. Show that $O(n)$ is a smooth submanifold of $GL(n,\mathbb R)$. What is its dimension? Prove your claim.
4. Find the tangent space at the identity of $GL(n,\mathbb R)$ and $O(n)$.

Problem 2.

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

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 $f(x,y,z)=y$ on $\Sigma$.

Problem 3.

Consider $\mathbb R^4$ with coordinates $(x,y,z,w)$. Let $f:\mathbb R^2\to\mathbb R^4$ be defined by $f(u,v)=(v,u,v^3,u^2v)$. Define a 2-form on $\mathbb R^4$ by

$$\alpha=w^2\,dy\wedge dz+y^3\,dy\wedge dw-2yw\,dz\wedge dw.$$

1. compute $f^*\alpha$.
2. show $d\alpha=0$.
3. Find a 1-form $\beta$ on $\mathbb R^4$ such that $\alpha=d\beta$.

Problem 4.

1. What is an orientation on a finite dimensional real vector space $V$? If $\dim V=3$ and $\{e_1,e_2,e_3\}$ is a basis, determine if $[e_1,e_2,e_3]$ and $[e_3,-e_1,-e_2]$ determine the same orientation or not.
2. What does it mean for a manifold to be orientable? Prove that $\mathbb{RP}^2$ is not orientable.
3. Suppose $f$ is a smooth function on $\mathbb R^{n+1}$ with 0 as a regular value. Show that the zero set of $f$ is an orientable submanifold of $\mathbb R^{n+1}$.

Problem 5.

Let $X$ be the vector field on $\mathbb R^3$ such that its flow $\Phi_t$ is the rotation which fixes x-axis and rotates in the yz-plane counterclockwise by $t$ in radians. Similarly, $Y$ is the vector field that generates rotations around the y-axis.

1. Find $X,Y$ explicitly.
2. Compute $[X,Y]$.
3. Determine the flow generated by $[X,Y]$.

Problem 6.

On $\mathbb R^3-\{0\}$ consider the differential form

$$\omega=\frac{x\,dy\wedge dz+y\,dz\wedge dx+z\,dx\wedge dy}{(x^2+y^2+z^2)^{3/2}}.$$

1. Show that $\omega$ is closed.
2. Let $\iota:S^2\to\mathbb R^3\setminus\{0\}$ be the inclusion map. Show that $\iota^*\omega$ is an orientation form on $S^2$, i.e. it is nowhere zero.
3. Let $\Omega\subset\mathbb R^3$ be a bounded domain with smooth boundary s.t. 0 is contained in the interior of $\Omega$. Show that $\int_{\partial\Omega}\omega=4\pi$.