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What are the definitions of

  1. cubical chart for a smooth manifold
  2. cube in $\mathbb{R}^n$

I am reading John Lee's Introduction to Smooth Manifolds 2nd edition, and the author seems to use these mathematical objects often but I'm unable to get a precise definition for them.

I have an interpretation of those definitions as

  1. cube in $\mathbb{R}^n$ is a product of connected open sets in $\mathbb{R} \times \mathbb{R} \ldots \times \mathbb{R}$
  2. cubical chart is $(U,\varphi)$ where $U$ is open in the manifold and $\varphi(U)$ is a cube in $\mathbb{R}^n$

Are these interpretations correct?

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1 Answer 1

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The index is your friend! Cube is defined on page 649, coordinate cube on page 4, and smooth coordinate cube on page 15. (Note that, as I wrote in the preface, most readers should read, or at least skim, the appendices before the rest of the book.) "Cubical" is the adjective form of "cube," so a cubical chart is just a chart whose domain is a coordinate cube, or equivalently whose image is an open cube in $\mathbb R^n$.

So your interpretations are close, but not exactly right. More precisely,

  1. An open cube in $\mathbb R^n$ is a product of bounded open intervals that all have the same length.
  2. A cubical chart is a coordinate chart $(U,\varphi)$, where $U$ is open in the manifold and $\varphi(U)$ is an open cube in $\mathbb R^n$.
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  • $\begingroup$ Thank you very much! $\endgroup$ Commented Feb 28, 2019 at 19:14
  • $\begingroup$ You’re welcome! $\endgroup$
    – Jack Lee
    Commented Feb 28, 2019 at 19:15

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