I am simply going to jump into the proof of the Myers theorem and refer to yesterday’s post for background. In particular, to prove it we only need to prove the following lemma:

**Lemma 1** *If is a complete Riemannian manifold of dimension and Ricci curvature bounded below by , then a geodesic of length does not minimize energy if . *

If we choose a geodesic parametrized by unit length, we then will be done if we find an endpoint-preserving variation of with . In other words if we can find a vector field along with and

we will obtain a contradiction; cf. the initial discussion in yesterday’s post and the second variation formula.

**Construction of **

Choose parallel vector fields along which, together with , form an orthonormal basis for at each . To do this choose for , and then parallel translate. Then since is skew-symmetric in the last two variables and

In particular, we could use the sum of the to get the vector field satisfying the boxed inequality—except . So we define

By the product rule for covariant differentiation and the paralellism of ,

So using orthonormality and , we can get that

if minimizes energy, so .

**Where next?**

This ends my MaBloWriMo series on differential geometry; there’s one more day in November, of course, but I need to first learn more new material before I can go further. Besides, it is probably healthy both for this blog and for myself to cover some other topics.

The posting over the next few weeks will probably be less structured than these entries. I’m not yet quite sure what I want to discuss, but likely the topic will be one or two out of Riemann surfaces, Koszul complexes and depth, spectral sequences, and singular integrals. Feel free to suggest something.

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November 29, 2009 at 3:54 pm

I have a selfish reason for you to cover Riemann surfaces, which is that it would tie in extremely well with the series I’m currently still doing!

November 29, 2009 at 4:27 pm

With your series on commutative algebra?

November 29, 2009 at 10:04 pm

The commutative algebra is just background – I’m planning on getting to the relationship between rings of integers and algebraic curves, and of course Riemann surfaces and algebraic curves are very closely related.

November 30, 2009 at 6:09 pm

Hi,

What will you study using spectral sequences?

November 30, 2009 at 6:23 pm

I’m not sure. I remember I ran into trouble in the book by Altman and Kleiman “Introduction to Grothendieck Duality Theory” when they prove Serre duality using (I think) Yoneda products and some kind of spectral sequence. Then again I am starting to think it is better to wait on that subject until I can actually take a course in algebraic geometry, so my motivation would come from algebraic topology, except that’s another subject I haven’t been able to learn (beyond the absolute basics–the heavy algebra obscures the motivation, and I can’t (yet) visualize things in four dimensions). All I know is that Eisenbud seems to have a clearly written account of them in his book on commutative algebra. You probably have more experience here–any recommendations?

November 30, 2009 at 7:02 pm

I am not familiar with Grothendieck Duality. (Maybe another topic for you to talk about?) Serre Duality for curves requires only a couple easy lifting lemmas. 🙂

Eisenbud and Bott and Tu are great.

November 30, 2009 at 7:07 pm

Thanks for the recommendations! Serre duality for Riemann surfaces sounds like a good idea. I don’t know why Farkas and Kra leave it out.

The existence of residues becomes much easier when you have can integrate 1-forms.

November 30, 2009 at 7:26 pm

Integration of one-forms is essential. As you noted, they provide the residue map necessary for the duality pairing. Moreover it is only by integrating one-forms that you get the period matrix (http://en.wikipedia.org/wiki/Period_mapping, http://en.wikipedia.org/wiki/Abel%E2%80%93Jacobi_map, http://rigtriv.wordpress.com/2009/02/03/elliptic-curves-and-jacobians/). This construction, especially for elliptic curves, forms the basis for the study of jacobians and higher-dimensional abelian varieties.

You might ask whether the residue maps are the only maps on H^1(D)*; the construction of the pairing intrinsically assumes this. This is the necessary theorem that says we basically approximate any functional by residue maps:

If A is a divisor on an algebraic curve X and A and B are functionals H^1(D) -> C, there exists an effective divisor C and two invertible functions f1, f2 in L(C) such that [a rather large diagram, which I can draw if you want] commutes.

The diagram basically gives maps between the group of tail divisors of certain divisors (“lifting).

Serre Duality is proved using sheaf cohomology in the lectures by Gunning, and in an elementary way in Miranda.

As for spectral sequences, there are I suppose some standard things you can do with them. If you are interested in group cohomology, it is a good idea to understand the basic Leray-Serre spectral sequence in addition to the universal coefficients theorem for homology and cohomology, derived functors, free resolutions, cohomology of the Eilenberg-Maclane spaces, and a tinge of covering space theory, For example, a theorem of Leray says that if X is a free G CW-complex, there is a spectral sequence with E2(p,q) = Hp(G,Hq(X)) converging to H_{p + q}(X/G). This is a statement about group cohomology.

November 30, 2009 at 8:10 pm

I’d say I recognize about 2/3 of the terms you just mentioned! That is why I started this blog.

Do you know a good source for the last paragraph or so (in particular Eilenberg-Maclane spaces, Leray’s theorem, etc)? I saw the preview of Bott and Tu and it seemed to be primarily from the de Rham viewpoint.

November 30, 2009 at 9:17 pm

Yes, Bott and Tu heavily emphasizes the de Rham theory.

The material on cohomology can be found in chapter 3 of Hatcher’s Algebraic Topology book. Homotopy theory is the topic of chapter 4. I’ve seen Postnikov towers in Spanier, but it is probably in Hatcher as well. For spectra and such, I am familiar with “Infinite Loop Spaces” by Frank Adams.

Additionally, John Baez writes a lot about these kinds of things, e.g. http://math.ucr.edu/home/baez/calgary/BG.html. His expositions are fantastic. As you know, the nLab is also a great reference. Finally, May’s notes are good if you like abstract nonsense.