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Considering a topological manifold $\mathcal S$, let's say specificly of ${\rm dim} = 2$ (i.e. "a surface") we can also identify curves (the set of curves $\{ \mathcal K_j \}$) "in" this surface, i.e. $\mathcal K_j \subset \mathcal S$.

For any point $p \in \mathcal S$ we can distinguish curves "through" this point (i.e. set $\{ \mathcal K_j : p \in K_j \} \equiv \{ \mathcal K^p_j \}$ from all other curves.
(Note that this set $\{ \mathcal K^p_j \}$ of curves through point $p$ generally contains pairs of curves $\mathcal K^p_a, \mathcal K^p_b \in \{ \mathcal K^p_j \}$ for which there exists a neighbourhood $\mathcal U_p \subset \mathcal S$ of point $p$ such that point $p$ is the only point of this neigborhood which these two curves have in common: $$((\mathcal K^p_a \cap \mathcal U_p) \cap (\mathcal K^p_b \cap \mathcal U_p)) = p.$$ If so, then, or course, there are also many other neighborhoods with the same property in respect to these two specific curves, $\mathcal K^p_a$ and $\mathcal K^p_b$.)

Now, it may be useful to consider some (non-zero but perhaps not necessarily proper) subset $\Gamma^p \subseteq \{ \mathcal K^p_j \}$ of curves through point $p$, and to partition set $\Gamma^p$ in disjoint equivalence classes, regarding equivalence by (pairwise mutual) tangentiality of curves of each one of such classes of curves "tangent to each other in point $p$". However, the method described in Wikipedia, as well as some variant method described and to be considered below, is not directly applicable to just any surface $\mathcal S$ (given as topological manifold of ${\rm dim} = 2$), but these methods require "a $C^k$ differentable manifold (with smoothness $k \ge 1$)".

As far as I understand, each surface $\mathcal S$ which is characterized as a topological manifold by its topology thereby has a unique maximal $C^0$-atlas $\mathsf A^{(0)}$ of "smoothness" $k = 0$; a.k.a. the maximal continuous atlas of $\mathcal S$. (If this happens to be wrong, then please let me know; and I'd like you to consider in the following just one partcular maximal $C^0$-atlas $\mathsf A^{(0)}$ of the given surface $\mathcal S$.)

So, in order to apply the method mentioned above, as well as its variant, for establishing disjoint equivalence classes of curves "tangent to each other in point $p$" on a suitable set of curves $\Gamma^p$, we should select one specific (not necessarily maximal) $C^1$-atlas $\mathsf A^{(1)}$ of smoothness $k = 1$, as subset of the given maximal $C^0$-atlas $\mathsf A^{(0)}$.

The method under consideration then requires to pick a particular coordinate chart $\varphi_w \in (\mathcal U^p_w, \varphi_w) \in \mathsf A^{(1)}$, where of course $p \in \mathcal U^p_w$, and shall then proceed as follows:

  • To each curve $\mathcal K^p_j \in \{ \mathcal K^p_j \}$ assign a corresponding parametrization $\gamma^{\mathsf A}_j : (-1, 1) \leftrightarrow \mathcal K^p_j$ such that $\gamma^{\mathsf A}_j[ \, 0 \, ] := p$ and, if at all possible, such that the coordinate representation of curve $\mathcal K^p_j$ in coordinate chart $\varphi_w$ is differentiable with respect to variable $t \in (-1, 1)$ at least at value $t = 0$, i.e at least at (the image of) point $p$; and such that the value of the derivative at value $t = 0$ is different from $(0, 0) \in \mathbb R^2$ :

$$ {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma^{\mathsf A}_j) \, \right]_{(t = 0)} \text{ exists and is } \ne (0,0).$$

  • Those curves $\mathcal K^p_j$ for which such a suitable parametrization $\gamma_j^{\mathsf A}$ can be found become members of the set $\Gamma^p_{\mathsf A}$ of (images of parametrized) curves through point $p$.

  • Any two curves $\mathcal K^p_a, \mathcal K^p_m \in \Gamma^p_{\mathsf A}$ are then said to be equivalent (and thus members of the same equivalence class) if and only if for the corresponding parametrized curves (a.k.a. paths) $\gamma^{\mathsf A}_a, \gamma^{\mathsf A}_m$ holds:

$$ \exists \, r \in \mathbb R, r \ne 0 : \\ {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma_a^{\mathsf A}) \, \right]_{(t = 0)} = r \, {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma_m^{\mathsf A}) \, \right]_{(t = 0)}.$$

Importantly, it is claimed in connection with the original method, as described in the Wikipedia section linked above, that the equivalence classification (partition) of set $\Gamma^p_{\mathsf A}$ which is thereby achieved "does not depend on the choice of coordinate chart" $\varphi_w$. (This appears to be widely appreciated and proven ... especially clearly here or here.)

Surely, the chart-independence considered and proven refers to chart-choices restricted to charts $\{ \varphi_w : \varphi_w \in (\mathcal U^p_w, \varphi_w) \}$ of neighbourhoods $\{ \mathcal U^p_w \}$ which all must contain point $p$ (i.e. restricted to charts which "map point $p$"), and restricted to charts from one-and-the-same (possibly maximal) $C^1$-atlas $\mathsf A^{(1)}$.

However: along with $C^1$-atlas $\mathsf A^{(1)}$, the given maximal $C^0$-atlas $\mathsf A^{(0)}$ can (and, being maximal, surely does) also contain another $C^1$-atlas, $\mathsf B^{(1)}$, which is non-equivalent to atlas $\mathsf A^{(1)}$, and which in particular contains a chart $\psi_y \in (\mathcal V^p_y, \psi_y) \in \mathsf B^{(1)}$ which is incompatible, at point $p$, to any chart $\varphi_w$ of atlas $\mathsf A^{(1)}$ which is mapping point $p$; i.e. such that for the transition function

$$(\psi_y \circ \varphi_w^{(-1)}) : \{ (\text{image of } \varphi_w[ \, \mathcal U^p_w \cap \mathcal V^p_y \, ]) \subset \mathbb R^2 \} \longleftrightarrow \{ (\text{image of } \psi_y[ \, \mathcal V^p_y \cap \mathcal U^p_w \, ]) \subset \mathbb R^2 \}$$

  • either this transition function $(\psi_y \circ \varphi_w^{(-1)})$ itself fails to be differentiable "at (the image of) point $p$", i.e. at $\varphi_w[ \, p \, ] \in \mathbb R^2$,

  • or the corresponding inverse transition function, $(\psi_y \circ \varphi_w^{(-1)})^{(-1)} := (\varphi_w \circ \psi_y^{(-1)})$ fails to be differentiable "at (the image of) point $p$", i.e. at $\psi_y[ \, p \, ] \in \mathbb R^2$,

  • or both.

(The necessary construction or identification of such an atlas $\mathsf B^{(1)}$ and chart $\psi_y$ can be as easy as sketched here.)

The described method variant for establishing equivalence classes of curves "tangent to each other in point $p$" can now similarly be carried out by

  • picking out chart $\psi_y$,

  • assigning the required parametrizations, if at all possible, to each curve $\mathcal K^p_j$ with respect to chart $\psi_y$,

  • collecting the (images of) suitably parametrized curves as set $\Gamma^p_{\mathsf B}$,

  • partitioning set $\Gamma^p_{\mathsf B}$ into disjoint equivalence classes; explicitly:

    Any two curves $\mathcal K^p_b, \mathcal K^p_n \in \Gamma^p_{\mathsf B}$ are said to be equivalent (and thus members of the same equivalence class) if and only if for the corresponding paths $\gamma^{\mathsf B}_b, \gamma^{\mathsf B}_n $ holds:

$$ \exists \, s \in \mathbb R, s \ne 0 : \\ {\frac{d}{dt}} \left[ \, (\psi_y \circ \gamma_b^{\mathsf B}) \, \right]_{(t = 0)} = s \, {\frac{d}{dt}} \left[ \, (\psi_y \circ \gamma_n^{\mathsf B}) \, \right]_{(t = 0)}.$$

My questions:

(1) Given any surface $\mathcal S$ and picking any chart $\varphi_w$ (which maps point $p$) from a suitable $C^1$-atlas $\mathsf A^{(1)}$, can always an inequivalent $C^1$-atlas $\mathsf B^{(1)}$ be found or constructed, and a suitable chart $\psi_y$ be picked from it, such that there exists at least one curve $\mathcal K^p_j$ at all which can be suitably parametrized wrt. chart $\varphi_w$ as well as wrt. chart $\psi_y$ (i.e. such that the image of one corresponding path $\gamma_j^{\mathsf A}$ becomes a member of set $\Gamma^p_{\mathsf A}$, and the image of the other corresponding path $\gamma_j^{\mathsf B}$ becomes a member of set $\Gamma^p_{\mathsf B}$) ?

(2) In case that there are several curves $\mathcal K^p_j$ which can be suitably parametrized wrt. chart $\varphi_w$ as well as wrt. chart $\psi_v$, are the resulting partitions of $\Gamma^p_{\mathsf A}$ and of $\Gamma^p_{\mathsf B}$ guaranteed to be consistent for these curves, and thereby "independent of the choice of charts" even from inequivalent atlases $\mathsf A^{(1)}$ and $\mathsf B^{(1)}$ ?

Or more formally: is guaranteed that

$$ {\mathbf {if}} \, \, \exists \, r \in \mathbb R, r \ne 0 : \\ {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma_a^{\mathsf A}) \, \right]_{(t = 0)} = r \, {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma_m^{\mathsf A}) \, \right]_{(t = 0)} \\ {\mathbf {then}} \, \, \exists \, s \in \mathbb R, s \ne 0 : \\ {\frac{d}{dt}} \left[ \, (\psi_y \circ \gamma_a^{\mathsf B}) \, \right]_{(t = 0)} = s \, {\frac{d}{dt}} \left[ \, (\psi_y \circ \gamma_m^{\mathsf B}) \, \right]_{(t = 0)},$$

for applicable curves $\mathcal K^p_a$ and $\mathcal K^p_m$, and

$$ {\mathbf {if}} \, \, \forall \, r \in \mathbb R, r \ne 0 : \\ {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma_a^{\mathsf A}) \, \right]_{(t = 0)} \ne r \, {\frac{d}{dt}} \left[ \, (\varphi_w \circ \gamma_b^{\mathsf A}) \, \right]_{(t = 0)} \\ {\mathbf {then}} \, \, \forall \, s \in \mathbb R, s \ne 0 : \\ {\frac{d}{dt}} \left[ \, (\psi_y \circ \gamma_a^{\mathsf B}) \, \right]_{(t = 0)} \ne s \, {\frac{d}{dt}} \left[ \, (\psi_y \circ \gamma_b^{\mathsf B}) \, \right]_{(t = 0)}$$

for applicable curves $\mathcal K^p_a$ and $\mathcal K^p_b$ ?

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    $\begingroup$ I suggest, you think of curves not as subsets of the surface but as maps to that surface. This is the approach used in differential topology. Otherwise, you run into trouble the various ways. Also, forget about the $C^0$-atlases, they are mostly useless. $\endgroup$ Jan 14 at 1:38
  • $\begingroup$ @Moishe Kohan: "I suggest, you think of curves not as subsets of the surface but as maps to that surface. This is the approach used in differential topology." -- Well, I do appreciate your usage, but I rather conform to the notion emphasized in Wikipedia: "A curve is the image of {... a path}.". Surely, the images of the maps you mentioned do exist as such? (Also, being a physicist, always considering the application to sets of events, there "we do have" worldlines outright, as sets of observable events in which an observable material point took part. $\endgroup$
    – user12262
    Jan 14 at 4:52
  • $\begingroup$ @Moishe Kohan: "$C^0$-atlases are mostly useless." -- Funny!, I'm used to getting by with $C^{-1}$, a.k.a. "coordinate dust". $\endgroup$
    – user12262
    Jan 14 at 4:52
  • $\begingroup$ @Moishe Kohan: "Otherwise, you run into trouble the various ways." -- I confirm that I had run into big trouble with my notation: Each path (a.k.a. parametrized curve) should be called $\gamma$, as usual and as shown in the linked Wikipedia sections, of course. Hopefully the relevant equations in my OP question now look more readable, more familiar. $\endgroup$
    – user12262
    Jan 14 at 7:45
  • $\begingroup$ I do not understand the question. You are now mixing curves and their images. Suppose for instance that you have a continuous map $\gamma$ which is a "Peano curve" in the plane. What would $\gamma^A$ would mean? This is no $C^1$-map of an interval whose image is the same as that of $\gamma$ (since the latter image is the "solid" square). Also, your equivalence relation is very unusual. Where id you find it and why do you use it? From my viewpoint, it is useless and should be abandoned. Another example to think about the the Koch snowflake in the plane. $\endgroup$ Jan 16 at 22:00

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