- #1

Korybut

- 48

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I am reading "Differential Geometry and Mathematical Physics" by Rudolph and Schmidt. And they have and example of manifold (projective space). I believe that there is a typo in the book, but perhaps I miss something deep.

Definitions are the following

$$\mathbb{K}^n_\ast=\{\mathbf{x}\in \mathbb{K}^n : \mathbf{x}\neq 0\},\;\; \mathbb{K}^n_1=\{\mathbf{x}\in \mathbb{K} : \| x \|=1\}$$

They pick up two representatives ##\mathbf{x}\neq \mathbf{y}## both with unit norm ##\| \mathbf{x}\|=\|\mathbf{y}\|=1## in ##\mathbf{K}^{n+1}_\ast##

To proceed with the neighborhoods they define function

$$l(\mathbf{x},\mathbf{y}):=\min_{\lambda \in \mathbb{K}_1} \|\mathbf{x} \lambda-\mathbf{y}\|$$

Then the pre-images of the neighborhoods are defined as follows

$$K_{\mathbf{x}}:=\{ \mathbf{z}\in \mathbb{K}^{n+1}_\ast : \max_{\lambda \in \mathbb{K}_1} \| \frac{\mathbf{z}}{\|\mathbf{z}\|}\lambda-\mathbf{x}\|<\frac{1}{2} l(\mathbf{x},\mathbf{y})\} \in \mathbb{K}^{n+1}_\ast$$

I've considered the case ##\mathbb{K}=\mathbb{R}## and ##\mathbb{R}P^1##. Then there are only two possibilities for ##\lambda=\pm 1##. And I found that set ##K_{\mathbf{x}}## is simply empty. This ##\max## in definition of ##K_{\mathbf{x}}## looks like a typo. Am I right?