Isometries preserving distance but not angles

Question

In Euclidean geometry, every distance-preserving map (isometry) also preserves angles between two vectors.

Is there any example of a non-Euclidean geometry in which an isometry need not preserve the angle between two vectors?

Answer 1

No. In any Riemmanian geometry, the angle $\theta \in [0, \pi]$ between two nonzero vectors ${\bf a}, {\bf b}$ satisfies $$\cos \theta = \frac{{\bf a} \cdot {\bf b}}{||{\bf a}|| \, ||{\bf b}||};$$ any isometry preserves the r.h.s. and hence the angle.

Conversely, there are maps that preserve angles by are not isometries, for example, any scaling of $\mathbb{R}^n$.

Answer 2

In Riemannian geometry, once you have a surjective distance preserving continuous map (note, no assumptions on differentiability) then you are automatically a smooth isometry (i.e. preserve inner products of tangent vectors). This is one of Myers-Steenrood theorems. Conversely, every isometry clearly preserves distances.