Timeline for Existence of a twice differentiable curve on a manifold
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 21, 2013 at 20:37 | comment | added | Ryan Budney | You should really edit your question rather than continually making revisions in the comments. | |
| Mar 21, 2013 at 20:24 | comment | added | Shake Baby | Correction: $H:\mathbb{R}^n\to\mathbb{R}^m$ with no relation between $m$ and $n$. | |
| Mar 21, 2013 at 19:55 | answer | added | Peter Michor | timeline score: 1 | |
| Mar 21, 2013 at 18:16 | comment | added | Shake Baby | $H$ is a $C^2$ function. Does this mean that $S$ is a $C^2$-differentiable manifold? What does it mean that the question is trivial? Are all tangent curves C^2? How do I see that? I'm not very familiar to differential geometry, as you may notice. | |
| Mar 21, 2013 at 17:44 | comment | added | Lee Mosher | It would certainly allay confusion if Shake would specify the level of differentiability, as suggested by Misha's comment. | |
| Mar 21, 2013 at 17:36 | comment | added | Lee Mosher | @Ryan: Yes, I was just pointing out a special case, given that the OP did not specify the level of differentiability of $S$. In some contexts, to say that $S \subset \mathbb{R}^n$ is differentiable means that it is $C^\infty$ differentiable, and I was unsure whether the OP knew that. | |
| Mar 21, 2013 at 17:18 | comment | added | Ryan Budney | @Lee: $S$ in in $\mathbb R^n$ so we can talk about differentiability of the curve of any order regardless of what $S$ is. | |
| Mar 21, 2013 at 15:32 | comment | added | Misha | @Shake: What do you mean by a smooth function (how many derivatives)? If you mean infinitely differentiable, then the answer is trivial. If you mean a function that is merely $C^1$ then it becomes interesting. | |
| Mar 21, 2013 at 15:10 | comment | added | Shake Baby | The manifold $S$ is given by the set of zeros of a smooth function $H:\mathbb{R}^n\to\mathbb{R}^n$ with locally constant rank jacobian. | |
| Mar 21, 2013 at 14:55 | comment | added | Lee Mosher | It depends on what amount of differentiability you assume for the manifold $S$. If $S$ is a $C^k$-differentiable manifold then there is such a $C^k$-curve $\eta$, as you can easily see in local coordinates at $x$. | |
| Mar 21, 2013 at 14:53 | history | asked | Shake Baby | CC BY-SA 3.0 |