Definition talk:Regular Curve

A note on terminology

In cases like this, which hinge upon specific details of a particular case, it is unwise and potentially inflammatory to stamp an exposition as "wrong". --prime mover (talk) 12:57, 20 October 2022 (UTC)

With curve endpoints included we may need to consider a smooth manifold with a boundary. In the literature sometimes this information is mentioned in the parenthesis, i.e. "smooth manifold (with or without a boundary)", but sometimes it has to be understood from the context. I will tell more when I come back from work.--Julius (talk) 13:34, 20 October 2022 (UTC)

Definition of $\gamma'$

Is the definition of velocity $\map {\gamma '} t$ for curves on a manifold still missing in $\mathsf{Pr} \infty \mathsf{fWiki}$?

This Definition:Velocity of Smooth Curve is only for curves in $\R^n$. --Usagiop (talk) 15:42, 10 July 2023 (UTC)

For personal reasons I copied mostly from Lee's book on Riemannian manifolds. In the same book, the topics for smooth manifold are covered not to full depth. I copied only the Euclidean definition. If instead we open Lee's book on smooth manifolds, then for velocity of a curve we have the following:

Let $\gamma : I \to M$.

Let $t_0 \in I$

Then the velocity of $\gamma$ is the map $\gamma' : I \to T_\gamma M$ such that:

$\ds \map {\gamma'} {t_0} = \map {\rd \gamma} {\valueat{\dfrac d {dt} }{t \mathop = t_0} }$

Furthermore, suppose $\tuple {U, \phi}$ is a smooth chart with coordinate functions $\tuple {x^i}$.

Also suppose $\map \gamma {t_0} \in U$.

Then we can write the coordinate representation of $\gamma$ as $\map \gamma t = \tuple {\map {\gamma^1} t, \map {\gamma^2} t, \ldots \map {\gamma^n} t}$

And the coordinate formula for velocity reads $\map {\gamma'} {t_0} = \map {\dfrac {d \gamma^i}{d t} } {t_0} \valueat {\dfrac {\partial}{\partial x^i} } {\map \gamma {t_0} }$ with Einstein summation implied.

At the moment I cannot make a proper page out from it. I would need to nit pick a few more things before I have a nice article. You can start building something. Maybe later in the evening I will put together something with a few red links.--Julius (talk) 13:27, 11 July 2023 (UTC)

Before that, Definition:Differential of Mapping/Manifolds is also missing. --Usagiop (talk) 18:16, 12 July 2023 (UTC)

Same reason. Is the following what you have in mind?

Let $M$ be a manifold, $p \in M$, and $T_p M$ the tangent space of $M$ at $p$. Let $f : U \to \R^n$ be a smooth real-valued function on open $U \supseteq M$. Then the differential of $f$, denoted by $d f$, is defined as a covector field such that $\forall p \in U : \forall v \in T_p M : \map {df_p} v = v f$.

No, I am looking for:

$(1)$ definition of $df_p : T_p M \to T_{\map f p} N$ for $f : M \to N$, and

I think I also have this in my book. It's just a small change of what I wrote above.--Julius (talk) 08:25, 13 July 2023 (UTC)

I wrote up a sketch. There are a few minors things I want to add. Is there anything major missing?--Julius (talk) 23:54, 14 July 2023 (UTC)

$(2)$ definition of $T_p M$ as roughly like $T_p M = \set {\map {\gamma'} 0 : \gamma : I \to M, \map \gamma 0 = p }$

I am familiar with this interpretation, but my book does not have it. At least not in such explicit way.--Julius (talk) 08:29, 13 July 2023 (UTC)

It is hard to imagine that these are not yet in $\mathsf{Pr} \infty \mathsf{fWiki}$. --Usagiop (talk) 20:35, 12 July 2023 (UTC)

I guess this site was not lucky enough with attracting people with knowledge in differential geometry and manifolds. There were some preliminary works years ago, but not that much. Then I came. I was studying General Relativity, and while writing my thesis I started writing up the parts related to Riemannian geometry. There was not enough time to cover both topological and smooth manifolds, so I decided to skip them. I still have my books, but plowing through every single thing is quite exhausting. Is there a certain goal you want to achieve with this topic? I may cherry pick the most important parts if you really need them.--Julius (talk) 08:25, 13 July 2023 (UTC)

OK, then I will put such basis definitions when I have time. I have no goal. I just added a wanted proof for In Connected Smooth Manifold Any Two Points can be Joined by Admissible Curve, where I could not fully explain why my curve $\ell_r$ is regular, because the definition of ${\ell_r}'$ could not be found. --Usagiop (talk) 15:38, 13 July 2023 (UTC)

If you need definitions for a particular exposition, and they're not there, then you are encouraged to add those definitions yourself. --prime mover (talk) 22:54, 13 July 2023 (UTC)

See The differential of a smooth map --Usagiop (talk) 20:38, 12 July 2023 (UTC)