# Hyperbolic geometry - isometry

#### Arnold

##### New member
Hello.

I'm currently reading a paper by Jason Jeffers titled "Lost Theorems of Geometry" published in The American Mathematical Monthly vol. 107, no.9 (Nov. 2000) pages 800-812.

The authors proves three theorems there and I have problems understanding the proof of this one:

Suppose that $$\displaystyle f: \mathbb{H}^2 \rightarrow \mathbb{H}^2$$ is a bijection which preserves geodesics. Then $$\displaystyle f$$ is an isometry of $$\displaystyle \mathbb{H}^2$$.

$$\displaystyle \mathbb{H}^n$$ - hyperbolic $$\displaystyle n$$-space

The author says (quote) :

"The essential idea is to show first that the map is continuous, and then that it is identical to an isometry on a dense subset of points. Actually, all of this takes place on the boundary of $$\displaystyle \mathbb{H}^2$$., so our first task is to show that it makes sense to refer to $$\displaystyle f$$ on the boundary".

Then we prove that it makes sense to think of $$\displaystyle f$$ as a map on $$\displaystyle \mathbb{H}^2$$ and its boundary by proving a lemma which says that if two geodesics in $$\displaystyle \mathbb{H}^2$$ have a common endpoint on the boundary, then their images also have a common endpoint on the boundary. We only assume that $$\displaystyle f$$ is a bijection which preserves geodesics.

Up to this point the only thing I don't fully understand is why is it so important to know that we can think of $$\displaystyle f$$ as a map on the hyperbolic space plus its boundary. What does the author mean by "Actually, all of this takes place on the boundary"?

The crucial lemma, though, is:

Suppose that $$\displaystyle x_1, x_2, y_1, y_2 \in \partial \mathbb{H}^2$$ and suppose that $$\displaystyle y_1, y_2$$ tohether separate $$\displaystyle x_1$$ from $$\displaystyle x_2$$ in $$\displaystyle \partial \mathbb{H}^2$$. Then $$\displaystyle f(y_1), f(y_2)$$ together separate $$\displaystyle f(x_1)$$ from $$\displaystyle f(x_2)$$ in $$\displaystyle \partial \mathbb{H}^2$$.

It is supposed to imply $$\displaystyle f$$'s continuity.
I think that this lemma may mean that however small "interval" $$\displaystyle (x_1, x_2)$$ we'll take and $$\displaystyle y_1$$ inside it, $$\displaystyle f(y_1)$$ will end up inside a small "interval" $$\displaystyle (f(x_1), f(x_2))$$.

Could you help me with my questions?

Thank you.