After playing for few time with the method of moving frames for surface in $\mathbb{R}^3$, I decided to try to apply it to study geometric surfaces (topological surfaces with an inner product). I have no trouble in doing calculations: with the first fundamental form I can recover $\omega_1$ and $\omega_2$, then $dA$, then $\omega_{12}$, $K$… but I have a theoretical problem. Following the method, I have $\{{e}_1, e_2\}$ ON positive basis for $T_pS$ for every p in S and I want to write $dP, de_1, de_2$ with respect of this basis. I would write
$dP = \omega_1e_1 + \omega_2e_2\\ de_1 = \omega_{12}e_2 *\\ de_2 = -\omega_{12}e_1 *$
But then, if I take the integrability condition (exterior derivative) I get
$d\omega_1 = \omega_{12}\wedge\omega_2 \\ d\omega_2 = -\omega_{12}\wedge\omega_1 \\ d\omega_{12} = 0$
Clearly, the last equation is not correct. So I suspect that the (*) equations are not correct too. I think that the problem is that $e_i$ changes also outside the tangent plane (as in $\mathbb{R}^3$). So I tried to get more formal writing where the functions lives:
$P :S \to S \\
\ \ \ \ \ \ p \to p
\\ e_i: S \to TS \\
\ \ \ \ \ \ p \to e_i(p) \in T_pS$
and also their derivatives **
$(dP)_p : T_pS \to T_pS \\
(de_i)_p: T_pS \to T_{e_i(p)}TS$
So I think that is correct to express $(dP)_p$ in terms of $e_i(p)$, because $(dP)_p$ lives in $T_pS$. Instead $(de_i)_p$ lives in a 4-dimensional space, $T_{e_i(p)}TS$, where, I suppose, $e_1$ and $e_2$ induce only half of the base. Is this true?
I'm really confused and I don't know how to google these things.
Thanks in advance
** I don't anything about TS, the tangent bundle, other than that is the union of the tangent planes and it is a 2n dimensional manifold.
Best Answer
When you did stuff in $\Bbb R^3$, you could think of differentiating vector-valued functions to make sense of equations like $de_1 = \omega_{12}e_2 + \omega_{13}e_3$. Now, you cannot really write $de_1$, as $e_1$ is a vector bundle valued function. So this is really the covariant derivative $\nabla e_1 = \omega_{12} e_2$. But one usually skips this and just defines $\omega_{12}$ uniquely, as you suggested, by the two equations $d\omega_1 = \omega_{12}\wedge\omega_2$ and $d\omega_2 = -\omega_{12}\wedge\omega_1$. [Note that if $\omega_{12} = P\omega_1+Q\omega_2$, then the first equation determines $P$ and the second determines $Q$.] Now, the next structure equation becomes $d\omega_{12} = \Omega_{12} = -K\omega_1\wedge\omega_2$ [not $0$].
Let me suggest a nice new book on moving frames. Perhaps you can find it in your library. For more advanced things, the master of 20th century moving frames, S.S. Chern, wrote this book.
I'll be glad to help within reason, too. :)