Uniquely geodesic spaces

Question

The purpose of this list issue is to better understand the class of uniquely geodesic spaces.
I'm looking for two different things :

• Overclass : for example geodesic space or contractible space.
• Subclass : for example CAT(0) space.

Of course, I'm looking for overclass as small as possible, and subclass as large as possible. If you know just an example $E$ not contained in the union $U$ of all the subclasses cited, then $\{E\} \cup U$ is of course an acceptable subclass and analogously for the counterexamples contained in the intersection of all the overclasses cited.

Also, if you know an interesting thing not classifiable as overclass, subclass or example, please write, these are also welcome (for example rephrasing of the definition if it gives a new lighting in your opinion).

Remark : If you estimate it's too broad, you can first restrict to "complete", "locally compact" and "intrinsic metric" spaces (implying "geodesic" by the Hopf-Rinow theorem), but I'm open to all the interesting things beyond this restriction (in particular beyond "locally compact").

P.S. If possible, please give a title in bold to your answer so that it will be easier to find overclasses, subclasses, examples and out of category posts.

Answer 1

Example

(see this answer of Jean-Marc Schlenker)

However, as pointed out by HenrikRüping, the CAT($0$) asumption is not necessary, you can for instance perturb the hyperbolic plane by putting a small lump of positive curvature near a point and the resulting metric will not be CAT(0) but will still be uniquely geodesic.

A deformation of the euclidean plane with soft bumps, is uniquely geodesic, but no more CAT($0$).

What's the natural subclass containing this example ?

Answer 2

Example

(see this answer of Lee Mosher)

A good example is the Teichmuller space of a closed oriented surface S.
It is uniquely geodesic by Teichmuller's theorem, but it is not CAT($0$).

Answer 3

Example

(see this answer of Roberto Frigerio)

Here is another counterexample. Let us endow the Euclidean space $V=\mathbb{R}^n$ with the distance induced by the usual $\ell^p$-norm. Then, if $1<p< \infty$, then $V$ is uniquely geodesic. However, $V$ is CAT($0$) if and only if $p=2$. (The proofs of these statements are very easy; as far as I remember, they may be found in the book by Bridson and Haefliger on metric spaces of non-positive curvature).