Timeline for Is there an Error on pg. 17 of Tromba's "Teichmuller Theory in Riemannian Geometry"?
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| Jan 17, 2011 at 23:50 | comment | added | Willie Wong | In other words: I have a nagging doubt that transverse-intersection and a suitable notion of transitivity of submanifolds are probably equivalent notions given some sort of implicit function theorem. So I don't see what avoiding transversality will gain you. | |
| Jan 17, 2011 at 23:48 | comment | added | Willie Wong | I suspect the final argument will actually be the same. The proof of the transverse intersection in finite dimensions have a lot in common with the implicit/inverse function theorems (what you outlined certainly works in finite dimensions); and I think the same is true for the infinite dimensional case. An argument that gives you that certain "submanifolds" are transitive will most likely also give you the transverse intersection theorem. Anyway, I don't have any handy references for infinite dimensional manifolds. Perhaps you should check Lang's book on geometry? | |
| Jan 17, 2011 at 21:55 | comment | added | BrainDead | @Willie Wong - Thank you for your answer. But what I was asking was whether the transverse-intersection theorem was necessary at all. I made an edit to my post to clarify what exactly I meant by 2). Please take a look. | |
| Jan 17, 2011 at 17:20 | comment | added | Willie Wong | In particular: the finite dimensional analogue gives if $M,N$ two smooth submanifolds of $X$ such that along $x\in M\cap N$ you have $T_xM$ and $T_xN$ span $T_xX$, then $M\cap N$ is a smooth submanifold of $X$ with codimension $\mathop{codim}(M) + \mathop{codim}(N)$. | |
| Jan 17, 2011 at 17:15 | comment | added | Willie Wong | @BrainDead: I don't have my copy of Tromba handy, so I cannot check the exact wording he used. If you are sure about your reading, then maybe you should send him an e-mail about the misprint. On question (2) I don't remember off hand whether any additional conditions are needed for the infinite dimensional transverse-intersection theorem. But the finite dimensional analogue of your statement is true: transverse intersection of two smooth submanifolds form a smooth submanifold of codimension the sum of the original codimensions. | |
| Jan 17, 2011 at 16:12 | comment | added | BrainDead | What about my question 2)? $\mathcal{A}^s$ is a $C^\infty$ submanifold of $\mathcal{N}$, and thus is a $C^\infty$ submanifold of $\mathcal{H}^s$? | |
| Jan 17, 2011 at 16:10 | comment | added | BrainDead | Thank you for the answer. Yes, that should've been $\det^{-1}(1)$. I made the correction just now. But there is definitely a typo on pg. 17, since he says: "[after using $J^2=-id$ to get $Ddet(J)H = -tr(JH)$]...It remains to show that $H\mapsto -\tr JH$ is surjective. Then the implicit function theorem shows that $\mc{M}$ is a submanifold with tangent sapce $\ker(D\det(J))=\\{H|\tr JH =0 \\}.$ Thanks for the clarification. I thought perhaps I was missing some kind of a known theorem or some feature of the Implicit Function Theorem that allows me to only look at those with $J^2=-id$. | |
| Jan 17, 2011 at 15:59 | vote | accept | BrainDead | ||
| Jan 17, 2011 at 15:04 | history | answered | Willie Wong | CC BY-SA 2.5 |