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The following appears in ANALYTICAL MECHANICS Classical, Lagrangian and Hamiltonian Mechanics, Stability Theory & Special Relativity, by Valter Moretti

Definition 1.22 A differentiable manifold of dimension $n$ and class $\mathcal{C}^{k}$, for given $n\in\left\{ 1,2,3,\dots\right\} $ and $k\in\left\{ 1,2,3,\dots\right\} \cup\left\{ \infty\right\} $, is a locally Euclidean, Hausdorff and second countable topological space $M$, whose elements are called points, equipped with a differentiable structure of class $\mathcal{C}^{k}$ and dimension $n$. The differentiable structure is a collection of $n$-dimensional local charts $\mathcal{A}=\left\{ \left(U_{i},\phi_{i}\right)\right\} _{i\in I}$ satisfying the following:

(1) $\cup_{i\in I}U_{i}=M;$

(2) the local charts in $\mathcal{A}$ must be pairwise $\mathcal{C}^{k}$ -compatible;

(3) $\mathcal{A}$ is maximal with respect to (ii): if $\left(U,\phi\right)$ is an $n$-dimensional local chart on $M$ compatible with every chart of $\mathcal{A}$, then $\left(U,\phi\right)\in\mathcal{A}.$

It is evident that the statement of item (3) of this definition is not what was intended. I have contacted the author, but I suspect he appropriately is busy with paying students. How might this definition be amended so that it is coherent and correct? There is nothing in the context labeled "(ii):".

I'm thinking it should be something like:

(3) (i) $\mathcal{A}$ is maximal with respect to $M;$

(ii) if $\left(U,\phi\right)$ is an $n$-dimensional local chart on $M$ compatible with every chart of $\mathcal{A}$, then $\left(U,\phi\right)\in\mathcal{A}.$

Even if my suggested (3)(i) is correct, I am still unsure about (3)(ii).

RobPratt
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Steven Thomas Hatton
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    My first thought would be that $(ii)=(2)$. Not sure I've seen that notion of maximality before, though. – lulu Jan 07 '25 at 20:12
  • What is your notion of maximality in this context? Assuming that you believe there is one. – Steven Thomas Hatton Jan 07 '25 at 20:14
  • That every local chart, compatible with the designated class of open charts, is a member of that class. Again, I'm not sure I've seen that before. – lulu Jan 07 '25 at 20:16
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    I had the same thought as @lulu: (ii) = (2). – md2perpe Jan 07 '25 at 20:22
  • So, something like: $\mathcal{A}$" is maximal with respect to (2): In other words, if $\left(U,\phi\right)$ is an $n$-dimensional local chart on $M$ compatible with every chart of $\mathcal{A}$, then $\left(U,\phi\right)\in\mathcal{A}.$?

    This notion of maximality regarding charts and atlases (my term) is unfamiliar to me.

     – Steven Thomas Hatton Jan 07 '25 at 20:33
  • Simply change (ii) to (2). This is the standard concept of a maximal atlas. For similar posts here on MSE see this and this and this. – Lee Mosher Jan 07 '25 at 20:33
  • @LeeMosher I will look at the other posts. I'm still unsure how to interpret the colon. – Steven Thomas Hatton Jan 07 '25 at 20:40
  • The colon indicates the definition of "maximal" (which is otherwise undefined). The bit after the colon explains that maximal means, loosely, that any chart compatible with the given class is a member of that class. – lulu Jan 07 '25 at 20:44
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    Regarding the colon ":" simply put some words before it "which means:" Mathematical writing can sometimes be very terse. – Lee Mosher Jan 07 '25 at 20:56
  • That's pretty much what I thought. I think its time to read Frank Warner's book, which I have found quite lucid in the bits I've read. – Steven Thomas Hatton Jan 07 '25 at 21:04
  • You see, typos happen. I'm rather sure you have one, too, in your copy of the definition, when you wrote $n\in {1,2,3,\ldots} \cup { \infty }$. I think it should read $k\in {1,2,3,\ldots} \cup { \infty }$. As has been pointed out in many of the comments, you should simply replace $(ii)$ by $(2)$. – Thomas Jan 07 '25 at 21:13
  • @Thomas Fixed. I sometimes post a screen-scrape of the text for comparison. Some books have very few typos, e.g., Misner, Thorne and Wheeler. Some are practically unreadable due to typos, e.g., the English translation of Hermann Weyl's Space-Time-Matter. Typos in math are easy to make, hard to spot, and often very consequential. – Steven Thomas Hatton Jan 07 '25 at 21:27

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Based on comments and Frank Warner's Foundations of Differentiable Manifolds and Lie Groups this should read:

Definition 1.22 A differentiable manifold of dimension $n$ and class $\mathcal{C}^{k}$, for given $n\in\left\{ 1,2,3,\dots\right\} $ and $k\in\left\{ 1,2,3,\dots\right\} \cup\left\{ \infty\right\} $, is a locally Euclidean, Hausdorff and second countable topological space $M$, whose elements are called points, equipped with a differentiable structure of class $\mathcal{C}^{k}$ and dimension $n$. The differentiable structure is a collection of $n$-dimensional local charts $\mathcal{A}=\left\{ \left(U_{i},\phi_{i}\right)\right\} _{i\in I}$ satisfying the following:

(1) $\cup_{i\in I}U_{i}=M;$

(2) the local charts in $\mathcal{A}$ must be pairwise $\mathcal{C}^{k}$ -compatible;

(3) $\mathcal{A}$ is maximal with respect to (2):[that is,] if $\left(U,\phi\right)$ is an $n$-dimensional local chart on $M$ compatible with every chart of $\mathcal{A}$, then $\left(U,\phi\right)\in\mathcal{A}.$

Also, my use of the term atlas was incorrect. The collection of all charts compatible with a differentiable atlas is a differentiable structure. https://www3.cs.stonybrook.edu/~gu/tutorial/ComputationalConformalGeometry.pdf

Steven Thomas Hatton
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