John Lee : Cubical charts and cube in $\mathbb{R}^n$

differential-geometrymanifoldssmooth-manifolds

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?

Best Answer

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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