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

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?

1 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 $$\mathbb R^n$$.
• Thank you very much! Commented Feb 28, 2019 at 19:14
• You’re welcome! Commented Feb 28, 2019 at 19:15