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.