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.
 A: 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
