# Fixed point theorems involving isometries

A fixed point theorem is a theorem which establishes some conditions that guarantee a certain function has a fixed point.

I would like to know if there exists a fixed point theorem involving metric spaces and isometries. Certainly one of these theorems should at least give some restrictive hypothesis about the metric space and/or the function. It's easy to see that there exist isometries which don't have fixed points, like any translation in $\mathbb{R}^{n}$.

So do we know what is needed to let an isometry have a fixed point? What's the most general setting in which we can state such a theorem?

Obviously if a fixed point theorem involving isometries exists in a more specific setting like normed space or Banach space it would be interesting anyway.

Thank you all if you can help me and sorry if I did some language mistake, I'm not an English native speaker.

• It is not just translations that are isometries without fixed points; there are plenty of them. A rotation on a closed ball in $\mathbb R^{2}$ is an isometry on a compact metric space without fixed. points. Mar 29, 2018 at 11:46
• @KaviRamaMurthy Correct me if I'm wrong, but to me it seems that a rotation on a closed ball in $\mathbb{R}^2$ has its centre of rotation as a fixed point. A rotation on an annulus in $\mathbb{R}^2$ doesn't have fixed points instead. Don't know if I interpreted your comment wrong. Mar 29, 2018 at 15:21
• I think there is a theorem saying that if $f : M→M$ is an isometry with $M$ a closed manifold, then the Lefschetz number of $f$ is the Euler characteristic of the submanifold $\{x ∈ M\,|\,f(x)=f\}$. I don't know enough to provide an answer and I don't find any source. But the existence of a fixed point given that the Lefschetz number is not $0$ does not require $f$ to be an isometry. Mar 29, 2018 at 18:29
• @ Stefano Rando Thanks for the correction. I should have taken an annulus and not a ball. Mar 30, 2018 at 22:52

Specifically, if $$K$$ is a nonempty, closed, bounded, convex subset of a Banach space and $$K$$ has normal structure, then every isometry $$T : K \to K$$ has a unique common fixed point. See "Fixed point theorems for mappings which do not increase distances" by Art Kirk, one of the foundational results in modern fixed point theory.