Geometry - August 2025

Directions

Do Problems 1–4, and then do two of the remaining Problems 5–7.

Make clear which problem you do NOT want graded. All manifolds, functions, vector fields, etc. are assumed to be smooth.

Problem 1.

Let $S$ be the subset of ${\mathbb{R}}^4$ defined by the equations

1. Prove that $S$ is a submanifold of ${\mathbb{R}}^4$.
2. What is $\dim S$? Is $S$ compact?

Problem 2.

Let $T\subset {\mathbb{R}}^3$ be the solid tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0) and (0, 0, 1). The orientation form $dx\wedge dy\wedge dz$ on $T$ induces “outward normal” orientations on each of its faces. Using these orientations, evaluate

where $S=S_1\cup S_2\cup S_3$ is the union of

$S_1=$ face in the xy-plane

$S_2=$ face in the xz-plane

$S_3=$ face in the yz-plane

Problem 3.

Let ${\Omega }^p\left(M\right)$ denote the set of smooth $p$-forms on a smooth manifold $M$.

1. The exterior derivative $d$ is a linear map $d:{\Omega }^p\left(M\right)\to {\Omega }^{p+1}\left(M\right)$ for each $p\ge 0$ such that:

   Fill in the blanks for (i)–(iii)

   1. $d^2=$ _____
   2. For $f\in C^{\infty }\left(M\right)$, the 1-form $df$ is defined by $df\left(X\right)=$ _____ for all vector fields $X$.
   3. $d(\omega \wedge \eta )=$ ________________ $\forall \omega \in {\Omega }^p\left(M\right),\eta \in {\Omega }^q\left(M\right)$.
   4. $d$ is local: for each $p\in M$, the value of $d\omega$ at $p$ depends only on the restriction of $\omega$ to an arbitrarily small neighborhood of $p$.
2. Show that Properties (i)–(iv) determine $d\omega$ for a 1-form $\omega$. Hint: write $\omega$ in local coordinates.
3. Use induction to prove that properties (i)–(iv) uniquely determine $d\omega$ for all $p$-forms $\omega$.

Problem 4.

This problem is about the definition of vector fields as derivations. Complete the definitions:

1. A vector field is a derivation, i.e. linear map $X:C^{\infty }\left(M\right)\to C^{\infty }\left(M\right)$ such that _____
2. The bracket $[X,Y]$ of vector fields $X$ and $Y$ is defined by _____
3. Prove that your answer to (b) is a vector field.
4. For a diffeomorphism $F:M\to M$, the pushforward of $X$ is the vector field $F_{*}X$ defined by _____

   One can show (you don’t have to) that

   $$F_{*}[X,Y]=[F_{*}X,F_{*}Y]\text{(1)}$$
5. Let $F_t$ be the flow of a vector field $Z$. Replace $F$ by $F_t$ in (1) and differentiate with respect to $t$ at $t=0$ to obtain a formula involving Lie derivatives.
6. Relate your formula in (e) to the Jacobi Identity.

Problem 5.

Let $f:M\to N$ be a submersion whose image is all of $N$. Prove that the pullback map $f^{*}:{\Omega }^p\left(N\right)\to {\Omega }^p\left(M\right)$ is an injection.

Problem 6.

Let $f:S^2\to M$ be a smooth map, where $S^2$ is regarded as the boundary of the unit ball $B^3$ in ${\mathbb{R}}^3$. Suppose that there is a DeRham cohomology class $\left[\omega \right]\in H^2\left(M\right)$ with

Prove that $f$ does not extend to a map $f:B^3\to M$.

You may use the fact that integration of 2-forms defines an isomorphism

Problem 7.

Let $M$ be an $n$-dimensional manifold with boundary, and $f:\partial M\to \mathbb{R}$ a function on the boundary. Show that $f$ extends to a smooth function on $M$, as follows.

1. (Local extension). Fix $p\in \partial M$. Show that there is a neighborhood $U_p$ of $p$ in $M$ and a map $F_p:U_p\to \mathbb{R}$ with $F_{|\partial M\cap U_p}=f$.
2. (Global extension). Use (a) and a partition of unity to show that there is a smooth map $F:M\to \mathbb{R}$ such that $F_{|\partial M}=f$.