MATH 230A: Differential Geometry

Hi everybody,

I'll start keeping a list of clarifications/corrections for the final here.

• 0. Asterisked problems are optional, but you may assume their truth throughout the exam.
• 1.
• 2.
• 3.
  • a. In this problem, of course the notation $[\alpha \wedge \beta ]\in H^{\ast }(X\times Y)$ doesn't make sense. If $p_X:X\times Y\to X$ and $p_Y:X\times Y\to Y$ are the projection maps, then the formula should actually read
  • $[\alpha ]\otimes [\beta ]\mapsto [p_X^{\ast }\alpha \wedge p_Y^{\ast }\beta ]$
  • b. You may assume the Five Lemma. The Five Lemma takes as a hypothesis the following:
    • Exact sequences $A_1\to A_2\to A_3\to A_4\to A_5$ and $A_1^{\prime }\to A_2^{\prime }\to A_3^{\prime }\to A_4^{\prime }\to A_5^{\prime }$
      whose maps we'll name $f_i:A_i\to A_{i+1}$ and $f_i^{\prime }:A_i^{\prime }\to A_{i+1}^{\prime }$, 
    • morphisms $h_i:A_i\to A_i^{\prime }$ such that $h_{i+1}\circ f_i=f_i^{\prime }\circ h_i$,
    • further, $h_1,h_2,h_4,$ and $h_5$ are known to be isomorphisms.
  • Then the Five Lemma concludes that $h_3$ must also be an isomorphism.
• 4. c Assume your manifold is connected.
• 5.
  • d. Assume all the manifolds are compact for this problem.
  • e. Assume M has finite-dimensional cohomology.
• 6.
  • I don't want you to spend too much time in rigorously proving the smoothness of everything, but here are some ways of thinking about the smooth structure on the Grassmannian:
    • The Grassmannian of k-planes in n-space is the quotient of $G{L}_n$ (which acts transitively on the space of k-dimensional subspaces) by a particular closed Lie subgroup (which stabilizes, say, ${\mathbb{R}}^k\subset {\mathbb{R}}^n$). The orbit-stabilizer theorem, souped up for Lie groups, tells you that the Grassmannian is then a smooth manifold.
    • You can use part d of this problem.
    • Closely related: You can also consider the description of the Grassmannian as primitives of ${\mathrm{\Lambda }}^k{\mathbb{R}}^n$, modulo scalars. (These give two different descriptions, and require a little bit of work to prove that the equations cutting out the Grassmannian give rise to a smooth manifold.) This shows that not only is the Grassmannian smooth, it can actually be cut out using polynomial equations (as opposed to, say, transcendental functions).
    • As I stated in class, the easiest way to think of the Grasssmanian is as a quotient of $k\times n$ matrices with no null space, modulo the action of invertible $k\times k$ matrices. The idea is that a matrix with no null space are injective maps from ${\mathbb{R}}^k$ into ${\mathbb{R}}^n$, and two of these have the same image if and only if they can be linearly re-parametrized---i.e., if they are related by an invertible $k\times k$ matrix.
  • 6.a. You should construct a map $F\to {\mathbb{R}}^N$. There are, actually, other ways to construct this without using partitions of unity.
  • 6.d. should say $G{r}_k({\mathbb{R}}^N)$, not $G_k$.
• 7.c. The angle function should be only locally defined. That is, for any curve $\gamma :[a,b]\to U\setminus \{p\}$, there is some small $ϵ>0$ so that for any $t\in [a,a+ϵ]$, the indicated equation holds. Also, the expression for angle was missing a ${\cos }^{-1}$. It should read ${\cos }^{-1}\frac{⟨u,\overline{u}⟩}{|u||\overline{u}|}$. Keep in mind that ${\cos }^{-1}$ is a bad "function," in that one should think of it as a local lift of the map $\mathbb{R}\to S^1$, rather than an actual function on ${\mathbb{R}}^2-\{0\}$. So taking this formula too literally may indeed hurt you.
• 8.

Dear class,

I hope you've spent a nice Thanksgiving. This note is to alert you that I've posted the final exam.

Per university policy, the official due date of this exam is Thursday, December 10th, 11:59 PM. (The end of reading period.) However, there will be no penalty for exams handed in by Thursday, December 17th, at noon. Plan out your weeks accordingly; I hope this long window will allow you to reduce stress, and spend the time you need on this Final.

Unless your situation is exceptional, I will not accept your exam after noon on December 17th.

You may hand in exams electronically via e-mail, or leave them in my mailbox in the math department (3rd floor, right by the main math office).

You can access the exam here:
http://www.piazza.com/class_profile/get_resource/idkbto5s954ch/ihjzhycr1e27o6

You can also view it on the course page: https://piazza.com/harvard/fall2015/math230a/resources

We have Tuesday and Thursday's lectures left, and I hope this final will cap off what's been a fun semester!

Hiro

No, not really.

I've posted online a non-exhaustive collection of papers you might find interesting. They're in the "Resources" page of the Piazza site, under "Papers to read."

I've posted mainly ICM papers or ICM addresses from the past, and I figured you might enjoy perusing through some of the things that some people are talking about, or have been talking about, in differential geometry. For reasons of scope, there are many many papers not included on this list--there is a bias for topics we could cover in this class. Differential geometry is more than just the study of curvature.

Regardless, now that your final P Set is nearing its end, your one final assignment before I give out the final next week is to relax and maybe enjoy some fun reading.

Hiro

Dear class,

There is a non-obvious trick required to show that the two exponential maps for Lie groups (with a bi-invariant metric) agree, so let me tell you what it is.

First, fix a left-invariant vector field $X$ with flow ${\mathrm{\Phi }}_t$. Show that for any group element $g$, we have

$L_g\circ {\mathrm{\Phi }}_t={\mathrm{\Phi }}_t\circ L_g$

where as usual, $L_g:G\to G$ is left multiplication by $g$, and $R_g$ is right multiplication.

Now, consider the operation

$R_{{\mathrm{\Phi }}_t(e)}{L}_{{\mathrm{\Phi }}_t(e{)}^{-1}}:G\to G$

otherwise known as conjugating by the element ${\mathrm{\Phi }}_t(e)$, where $e\in G$ is the identity. Show that

$R_{{\mathrm{\Phi }}_t(e)}={\mathrm{\Phi }}_t$

and thereby compute the derivative, with respect to $t$, of

$⟨D(R_{{\mathrm{\Phi }}_t(e)}{L}_{{\mathrm{\Phi }}_t(e{)}^{-1}}))Y,D\left(R_{{\mathrm{\Phi }}_t(e)}{L}_{{\mathrm{\Phi }}_t(e{)}^{-1}}\right)Z⟩.$

(Use the definition of Lie derivative!) Note that whatever you get, you know it equals zero by the metric being bi-invariant.

Using the previous two steps, show that for any three left-invariant vector fields, one has

$⟨[X,Y],Z⟩=⟨X,[Y,Z]⟩.$

This should help with the problem.

Hi Class,

You have partial control over what we might cover in the last we weeks. Have your voice heard:

https://docs.google.com/forms/d/1ZC10whhDzQgssemLtZQeSwMgh0WE6i6Lo_dthZ0fkKg/viewform

Hiro

So what I said in class is very wrong, about $A$ being recovered from its values on an orthonormal collection of $v$. Here's a more correct statement, which may seem mysterious:

If $A:E\otimes E\to \mathbb{R}$ is a symmetric map, and if $⟨-,-⟩$ is a non-degenerate inner product on $E$ (unrelated to $A$) then knowledge of the map

$v\mapsto A(v,v)/⟨v,v⟩$

completely recovers $A$ itself. This is obvious since I'm taking $v\ne 0$ to be arbitrary.

Anyway, we'll see why this has anything to do with anything next time.

I'm out of town tomorrow, so I won't be able to hold my normal office hours.  If you'd like to discuss a previous problem and/or something on the current pset, I'll be available on Friday.  Most times between 10 and 3 should work, as long as you let me know at least an hour beforehand.

-Phil

Dear class,

Please take some time to fill out this survey to let me know how the class is going for you.

Hiro