# Synge's Theorem

1. Nov 17, 2003

### Mike2

Can anyone give me a phoenetic spelling of how to pronounce this name?

It seems Sygne's Theorem relates the differential concept of curvatrue to the globle concept of genus (how many holes in the surface). Is an integration necessary to derive this genus from the curvature? Or is there more of a direct formula from curvature to genus?

Thanks.

2. Nov 17, 2003

### marcus

Always heard it pronounced "sing"

Believe it's spelled Synge and pronounced "sing"
Irish mathematician/physicist 1897-1995
Stuff about Synge's theorem is available by google on web.

Last edited: Nov 17, 2003
3. Nov 18, 2003

### lethe

Re: Always heard it pronounced "sing"

is it? i got nothing by google, except for some course syllabi. wholly uninformative.

i also didn t see it anywhere in my geometry book, neither do Carmo, nor Petersen.

i would be interested to know what it is too, if anyone knows.

as far as Mike2 s description goes, it sounds a lot like the Gauss-bonnet theorem, which i know quite well. it states that the integral of the Gaussian curvature over a hyperspace gives you the euler character of the manifold, which is essentially the genus.

4. Nov 18, 2003

Staff Emeritus
I don't know about his "theorem" , but here is an account of Synge the man. Not nineteenth century at all, but twentieth, and the author of a classic textbook on relativity.

5. Nov 18, 2003

### HallsofIvy

Staff Emeritus
"Not nineteenth century at all, but twentieth"

What made you mention the "nineteenth century"? The only reference to time was marcus' "Irish mathematician/physicist 1897-1995".
Synge was borne in the 19th century but I doubt that he published much before he was 3!

(And he only missed the twenty-first century by 5 years! He lived to be 93!)

6. Nov 18, 2003

### Mike2

Re: Re: Always heard it pronounced "sing"

See , Geometry of Physics, by Theodore Frankel, chapter 12.

Would anyone be interested in how I'm thinking such a Theorem would be relevant to String theory?

If so, should I post here or in the "diff Eq on String" thread, since it would be an extention of those ideas?

Thanks.

7. Nov 19, 2003

### marcus

Re: Re: Re: Always heard it pronounced "sing"

I have to confess that I am curious. And since you have started discussing it here why dont you at least state the theorem (and perhaps some relevant definitions to get it into context) right here and give us a clue.

Gauss-Bonnet is a beautiful theorem and maybe its relevance to stringy business has been explained already (Lethe could help us on that) and who knows, maybe Synge's is too.

I cant discuss stringtheory with you but I would certainly like to hear what you have to say.

8. Nov 19, 2003

### lethe

from Frankel:
Synge's Theorem

Let $$M^{2n}$$ be an even-dimensional, orientable manifold with positive sectional curvatures, $$K(\mathbf{X}\wedge\mathbf{Y}) > 0$$. Then any closed geodesic is unstable, that is, can be shortened by a variation.

Corollary:

A compact, orientable, even-dimensional manifold with positive sectional curvatures is simply connected.

to remind you, sectional curvature is the Gaussian curvature determined by two tangent vectors X and Y

Last edited: Nov 19, 2003
9. Nov 19, 2003

Staff Emeritus
And relative to that see this new paper, wherein is calculated the rational homotopy and cohomology groups of any compact semisimple simply connected structure group of a principle bundle over a compact manifold. She does the conections too. Sounds like a great cite for LQG work.

10. Nov 19, 2003

### lethe

Re: Re: Re: Re: Always heard it pronounced "sing"

agreed! the Gauss-Bonnet theorem is really beautiful, and deep, and the starting point for much mathematics! in fact, it is my favorite theorem in all mathematics, just beating out stokes theorem.

it is really quite an exciting theorem.

heres one: string theory is basically GR in 2 dimensions. there is a field term in the action, and an einstein-hilbert term.

however, the einstein-hilbert term in 2 dimensions is stationary, by the Gauss-Bonnet theorem, and so does not contribute to the dynamics in the classical theory.

however, in the path-integral formulation of the quantum theory, you must sum over worldsheets, meaning that there is an integration over metrics for a possible topology, and a sum over possible topologies. each topology carries a coupling constant that looks like $$e^\chi$$, since the action goes in the exponential. thus higher genus worldsheets are higher order quantum corrections to the classical theory.

Last edited: Nov 19, 2003
11. Nov 19, 2003

### lethe

Re: Re: Re: Always heard it pronounced "sing"

looking at frankel, i see that you mistook the Gauss-Bonnet theorem for Synge s theorem.

whoops.

12. Nov 19, 2003

### marcus

Hey people, Lethe is showing us here something I didnt know about PF. One can use tex.

I have rewritten "tex" with spaces so it would not take effect, in the above quote.

13. Nov 19, 2003

### lethe

Marcus-

this is a new feature at PF, and a really cool one, in my opinion.

gone are the days of you reposting my HTML math codes to make them visible. its LaTeX all the way, from here on out.

see this thread for more details.

14. Nov 19, 2003

### Mike2

Re: Re: Re: Re: Always heard it pronounced "sing"

As I understand it, and I could very well be wrong, Gauss-Bonnet requires the integration over a closed surface. But Synge's theorem relates the curvature to the stability of closed geodesic. It seems to be more of a differential version of Gauss-Bonnet.

Why might that be important? As I've read, String theory so far is done with the integral equations of Feynman path integrals. And we run into problems with renormalizations, etc. There is no "closed form" of String theory which solves all problems more easily.

I wonder, and this is more of a question, wouldn't it be more advantageous if we had a differential equation from which to derive the state of a string and how it propagates? I'm considering whether or not I may have a start to such a differential equation as described in the "diff Eq on Strings, check out the math" thread in the Strings, Branes, & LQG forum. This diff eq relies on a background potential, or "existence function" if you prefer. I suppose you could think of it as only existing on the string itself. The "force" being differentiated is the gradient of some scalar field. So this equation is a second order diff eq in the background potential. This second order diff eq involves connection coefficients and has periodic boundary conditions. So we may get eigenvalues and eigenfunction. The connection coefficients, and the metric associated with them, may be determined by these eigenfunctions.

I wonder, if the eigenfunctions did determine the metric, connection coefficients and the curvature, then would each eigenfunction correspond to the genus of the surface being used in the integral form of String theory? I wonder if I'm even on the right path.

15. Nov 19, 2003

### lethe

Re: Re: Re: Re: Re: Always heard it pronounced "sing"

this is correct (actually, you can formulate the Gauss-Bonnet theorem to allow for surfaces that are not closed, but this is a minor point)

you have to know the curvature everywhere to use the Gauss-Bonnet theorem, or talk about the average over the whole surface.

i think you also need to know something about the curvature everywhere in the space to use the Synge theorem. you don t have to know the curvature itself, but you do stipulate that the curvature is positive everywhere.

so both the Gauss-Bonnet theorem and the Synge theorem put global stipulations on the curvature. neither is a local theorem.

indeed, how could it be? it is impossible to put constraints on global topology with only local conditions.

the path integral formulation is only one way to quantize a theory. another choice is canonical quantization.

there are two canonical approaches to quantization in theories with gauge symmetries: Geupta-Bleuler (covariant) quantization, and gauge fixed (noncovariant) quantization.

in string theory, the latter approach is call light cone gauge quantization. all methods are well known in, e.g. QED, and work well in string theory.

canonical quantization involves no integration over all states, but is formulated purely in terms of local differntial equations.

i don t know what you mean here.

we do. i m not sure why you think we don t.

16. Nov 20, 2003

### Mike2

Re: Re: Re: Re: Re: Re: Always heard it pronounced "sing"

Without going through all the detail, I wouldn't know how at this point, it is seems obvious that all you need to know to decide whether a closed curve was "stable" is the curvature of at each point of the closed curve. If the curvature is negative everywhere then there is no way to shorten it. But if the curvature is positive somewhere than you can shorten it in at least that region of the closed curves. So I think we're looking at the integration over the length of the closed curve of some sort of function of the coordinate curves to determine stability, right?

[QUOTE
i don t know what you mean here.
[/QUOTE]

We'd like a better answer than the perturbation method, which is only approximate.

17. Nov 20, 2003

### lethe

Re: Re: Re: Re: Re: Re: Re: Always heard it pronounced "sing"

so then we are in agreement. Synge's theorem is about nonlocal curvature.

it sounds to me like you would at least have to know the curvature in a neighborhood of each point of the curve. not just at each point of the curve.

but we don t have to argue about how much information we think Synge's theorem should require. synge's theorem is explicit: you have to know the curvature is positive everywhere.

so this is my point: even though there is no integration involved in Synge, neither it nor Gauss-Bonnet is a local theorem.

yeah. and i would like to have an exact formula for cos x, instead of just a taylor approximation.

i wish every system were exactly solvable.

but seriously, there are lots of nonperturbative methods in string theory. so i m not really sure what you re on about here.

Last edited: Nov 20, 2003
18. Nov 20, 2003

### Mike2

Re: Re: Re: Re: Re: Re: Re: Re: Always heard it pronounced "sing"

We don't have to know the second derivative at any other point to know that if the second dervative is increasing and the first derivative is zero at that point, then you have a local extreme.

What I guess I'm suggesting is the same concept. If the second partial along the curve is of one sign and the second partial transverse to the curve is of the opposite sign, then with the first transverse partial zero, you have a saddle point there. If this happens to be the case at every point on a closed string/loop/curve, then the length of the curve must be an extrema, right? Sounds pretty straight forward anyway. Though it is probably not Synge's theorem anymore.

There are local extreme. The closed curve length could be locally extreme without being the most minimum of all, right? So extrema do not require information about the whole, just locally.

19. Nov 20, 2003

### lethe

Re: Re: Re: Re: Re: Re: Re: Re: Re: Always heard it pronounced "sing"

probably not. also, you can t take the derivative of something if you only know its value at a point. you need to know its value on a neighborhood of a point.

like i said above: the neighborhood of every point on the curve is probably sufficient to make it an extremum.

this is not local.