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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.

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    $\begingroup$ 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. $\endgroup$
    – Kavi Rama Murthy
    Mar 29, 2018 at 11:46
  • $\begingroup$ @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. $\endgroup$
    – Stefano Rando
    Mar 29, 2018 at 15:21
  • $\begingroup$ 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. $\endgroup$
    – Dabouliplop
    Mar 29, 2018 at 18:29
  • $\begingroup$ @ Stefano Rando Thanks for the correction. I should have taken an annulus and not a ball. $\endgroup$
    – Kavi Rama Murthy
    Mar 30, 2018 at 22:52

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There is one big fixed point theorem for isometries that I am aware of, and it is due to Brodskii and Milman ("On the center of a convex set" (1947)) in which they introduce the notion of normal structure for a set in a normed linear space and prove that, under certain conditions, all isometric self-mappings of classes of sets in Banach spaces will have a unique point which is fixed by every such mapping.

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.

One of the biggest fixed-point results that I am aware of that takes place in the metric space (as opposed to Banach space) setting is the theorem of Sine and Soardi ("On linear contraction semigroups in sup norm space" and "Existence of fixed points of nonexpansive mappings in certain Banach lattices," resp.) which states that nonexpansive mappings on bounded hyperconvex metric spaces have fixed points.

The theorems which are true for nonexpansive mappings are also trivially true for isometries.

I've included some other references below:

c.f. "Maps which preserve equalities of distance" by Andrew Vogt, "Fixed points of isometries" by Shoshichi Kobayashi

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