Timeline for Taylor's theorem for embedded manifolds
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 19, 2021 at 1:22 | comment | added | user35593 | Proposition 1.2.8 of research-collection.ethz.ch/bitstream/handle/20.500.11850/… seems related. | |
| Feb 17, 2021 at 11:03 | comment | added | Uzu Lim | I think there is a misunderstanding. My question is on whether the error term of the 1st order Taylor expansion of $\exp_x$ can be realized as $(t^2/2) Q_x(w)$ for some appropriate $w \in T_x M$ with $\|w\| \le 1$. I don't think that this follows directly from Taylor's theorem? The best shot seems to be applying Taylor's theorem coordinate-wise to $t \mapsto \exp_x(tv)$ but that won't imply the desired result directly. Also, $J_x: T_x M \rightarrow \mathbb{R}^n$ is a required notation if we regard the tangent space as not a priori embedded in $\mathbb{R}^n$. | |
| Feb 17, 2021 at 3:15 | comment | added | Ryan Budney | Your question looks essentially like the definition of the Taylor expansion of 1st order. So, provided I'm not misunderstanding your question, the answer is yes. Moreover, J_x(v)=v for any reasonable definition of the tangent space. | |
| Feb 17, 2021 at 1:39 | comment | added | Uzu Lim | I think I have a resolution to the question asked, by applying Taylor's theorem coordinate-wise to the exponential map $t \mapsto \exp_x(tv)$ and bounding the error term crudely. This seems to yield that the error term is bounded by $Bt^2$ where $B = \frac n2 \sup_{w \in TM} \| \nabla_w w \| $ and $n$ is the ambient dimension. I'll clarify after I get some sleep. | |
| Feb 17, 2021 at 1:28 | history | edited | Uzu Lim | CC BY-SA 4.0 |
typo deleted
|
| Feb 17, 2021 at 1:27 | comment | added | Uzu Lim | ah, someone added a typo while editing. I'm going to delete that $>$. | |
| Feb 17, 2021 at 1:15 | comment | added | Ryan Budney | Your notation does not make sense to me. What does the inequality mean between a point of $M$ and the function on the right, which appears to be vector-valued? | |
| Feb 16, 2021 at 21:27 | history | edited | Francesco Polizzi | CC BY-SA 4.0 |
added 9 characters in body
|
| Feb 16, 2021 at 19:13 | history | asked | Uzu Lim | CC BY-SA 4.0 |