Timeline for Slowly increasing smooth mappings with values in a Lie group?
Current License: CC BY-SA 4.0
12 events
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| Jul 17 at 22:13 | comment | added | Isaac | @or I have not seen the term "jet" before. Could you explain more? I am more used to the notion of Frechet derivative, which of course should be understood locally (= on local charts) for manifolds. Well, I admit that it can be a little bit awkward for manifolds. | |
| Jul 17 at 17:36 | comment | added | o r | Yes, I mean that function. The argument can be applied to show the boundedness of the norms of the higher order derivatives of $\iota$ and $\mu$ as well (higher derivatives are usually described via "jets" in differential geometry). | |
| Jul 17 at 15:44 | vote | accept | Isaac | ||
| Jul 17 at 15:42 | comment | added | Isaac | @or Plus, I think your argument can be generalized for higher derivatives. That is, $g \to \lVert d^n_g \iota \rVert$ for any $n \in \mathbb{N}$. Here, I have used the notation $d^n_g \iota$ for $n$-order Frechet derivative of $\iota$ evaluated at $g \in G$. | |
| Jul 17 at 15:40 | comment | added | Isaac | @or By $\lVert d \iota \rVert$, you mean $g \to \lVert d_g \iota \rVert$ for $g \in G$? | |
| Jul 17 at 15:30 | history | rollback | o r |
Rollback to Revision 1
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| Jul 17 at 15:30 | comment | added | o r | @Isaac the norms $\|d\iota\|$ and $\|d\mu\|$ are continuous $\Bbb R$-valued functions on $G$ and $G\times G$. Compactness of $G$ guarantees that their image is bounded. | |
| Jul 17 at 9:48 | comment | added | Isaac | @mme I am confused about the domain and range of your differential. Are you thinking pointwise on $G$ or the whole tangent bundle? | |
| Jul 17 at 0:18 | comment | added | mme | $d\iota: G \to \Bbb R^\ell$ is a continuous function with compact domain. | |
| Jul 16 at 22:46 | history | edited | Isaac | CC BY-SA 4.0 |
deleted 9 characters in body
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| Jul 16 at 22:44 | comment | added | Isaac | I do not clearly see how we can use compactness of $G$ to show that $\iota: G\to G$ has bounded derivatives. Could you provide more details? | |
| Jul 16 at 22:16 | history | answered | o r | CC BY-SA 4.0 |