Timeline for Reference for the Banach Manifold structure of $C^k(M,N)$
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Nov 2, 2015 at 21:31 | comment | added | James Mracek | You've probably already got an answer by now, but in case you haven't you can find it here: mat.univie.ac.at/~michor/apbookh-ams.pdf theorem 42.1 | |
| Jul 30, 2015 at 9:50 | comment | added | uro | Thanks again for your help. Maybe I can ask one last question. In the notes you linked me, the $C^\infty$ structure on $C^k(M,N)$ is consctructed like that: Pick $f\in C^\infty(M,N)$, then use the exponential map to construct a chart $x_f:U_f\rightarrow V_f$ where $U_f$ is a neighborhood of $f$ in $C^k(M,N)$. Then it is shown that for another $g\in C^\infty (M,N)$, the transition maps $x_f\circ x_g^{-1}$ are smooth. Other sources I found seem to indicate that this works aswell for $f,g \in C^k(M,N)$, i.e. for $f,g \in C^k(M,N)$ the transtion map $x_f\circ x_g^{-1}$ is smooth. Is that true? | |
| Jul 30, 2015 at 3:53 | comment | added | Andy Sanders | See this paper for the formal details arxiv.org/pdf/1403.3111v2.pdf. Then, just note that if $f_t:M\rightarrow N$ is a variation of $f,$ then the first variation is a section of $f^{*}TN.$ | |
| Jul 29, 2015 at 13:29 | comment | added | uro | Thanks, that helped a lot! I have a follow-up question, maybe you (or someone else) can help me with it: Can the tangent bundle of the smooth banach manifold $C^k(M,N)$ be realised as $\bigcup_{f\in C^k(M,N)}C^k(f^*TN)\rightarrow C^k(M,N)$ where a $C^k$-section $X\in C^k(f^*TN)$ is mapped to $f$? If so, what's the idea to prove that? | |
| Jul 28, 2015 at 16:23 | comment | added | Andy Sanders | Let me refine that by saying that I don't think there is any mention of the compact-open topology there, just so I don't send you on a wild goose chase. Nonetheless, you'll find the proof that it is a smooth Banach manifold. | |
| Jul 28, 2015 at 16:10 | comment | added | Andy Sanders | You can find proofs of these statements in the notes of John Moore found at the following link: math.ucsb.edu/~moore/globalanalysisshort.pdf. | |
| Jul 28, 2015 at 15:35 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
Slight reformatting.
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| Jul 28, 2015 at 15:26 | review | First posts | |||
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| Jul 28, 2015 at 15:23 | history | asked | uro | CC BY-SA 3.0 |