@TedShifrin .... your input and suggestions have always been extremely helpful! Could you help me with my current problem about isometric embeddings (I have been stuck for days)? I would be super grateful for your support: chat.stackexchange.com/transcript/message/52623421#52623421
@eigenvalue: Ugh. You should post stuff actually typed up. All right ... I have the handwritten thing. So what are your specific questions? Where does eqn (1) come from? Is that your first question?
There are presumably initial conditions being imposed which explain the next stuff.
What's the original question? You're given a spacelike surface with the metric $dx^2+t\,dt^2$ and want to isometrically locally imbed in Minkowski?
When I check the literature, they only show how to set up the embedding functions and the equations. But nowhere they show how to set up initial conditions or how to actually calculate the solutions.
In the literature they only set up the embedding maps and show how to get the equations. But I have never seen an exact calculation so that I can understand it ... is there a good source about isometric embeddings with some examples?
I've done research work with singularities in various geometric settings, but not this one.
@eigenvalue: There's nothing relevant about isometric embeddings here. In general, that's a very hard overdetermined PDE question. Here they're reducing to ODE by having the $x$ variable disappear. So it's just about solving that ODE.
This one is underdetermined so you just have to try to think of functions so that you can get $\xi'^2 = t + \nu'^2$. This means that we need $\nu'^2 \ge -t$.
This is obviously no issue when $t>0$, but we need to think of it for $t<0$.
If $X$ is a locally compact metric space must every bounded subset be contained in a compact one? This is obvious for proper metric spaces, and a locally compact one is something like "locally proper"
If $\mathcal{F}_\Gamma$ is a fundamental domain for the action of $\Gamma$ on $\frak h$ where $\Gamma \subset \operatorname{SL}_2(\Bbb Z)$ is a congruence subgroup then
well.. if $A_1, \dots, A_m$ are representatives for $\Gamma \setminus \operatorname{SL}_2(\Bbb Z)$ then $\mathcal{F}_\Gamma = \bigcup_{\mu = 1}^m A_\mu \mathcal{F}$
I'm just saying that change of variables would give you $M\cdot\mathcal F$, and you have to sort out why that should be a fundamental domain of the conjugate.
@Ted actually, I don't think I need to do that at all; the integral is independent of the choice of fundamental domain I think, so I just need to show that $M \cdot \mathcal{F}_\Gamma$ is a fundamental domain for the action of $\Gamma$ on the upper half plane and then it's fine
I htink
think*
(this is a theorem)
anyway the point is that I'm showing that the Petersson inner product on the space of weight $2k$ cusp forms is somehow invariant under the action of the Petersson slash operator
Suppose I look at the polynomial $x^n - \sqrt{2}$. I can argue this is irreducible over $K = \Bbb Q(\sqrt{2})$ by noting that $K$ is the fraction field of the UFD $A = \Bbb Z[\sqrt{2}]$ so that by Gauss lemma it suffices to show it's irreducible over $A$ but $\sqrt{2}$ is an irreducible element by a norm argument. Then I just apply Eisenstein's criterion.
But what if I have something nasty like $x^n - \sqrt{5}$?
I have vaguely heard these things from some seniors taking algebraic number theory. Dunno much about it. What's the smallest example of a real quadratic field with ring of integers not a PID?
K is called real quadratic if d > 0. K has class number 1 for the following values of d (sequence A003172 in the OEIS): 2*, 3, 5*, 6, 7, 11, 13*, 14, 17*, 19, 21, 22, 23, 29*, 31, 33, 37*, 38, 41*, 43, 46, 47, 53*, 57, 59, 61*, 62, 67, 69, 71, 73*, 77, 83, 86, 89*, 93, 94, 97*, ...
K has class number 1 exactly for the following negative values of d: −1, −2, −3, −7, −11, −19, −43, −67, −163
Ah, OK, just take an element of order $8$ in $\Bbb F_{p^2}$ by Lagrange. Then it has to satisfy $x^8 - 1$ and cannot satisfy $x^4 - 1$, so is a root of $x^4 + 1$
