Nonalgebraic Anosov diffeos. Classify 3-dimensional Anosov flows; Is the variable curvature surface geodesic flow conjugate to constant curvature?
See the introduction of https://arxiv.org/pdf/1505.06259.pdf for a quick account on results about classification of Anosov flows in dimension 3.
The geodesic flow for a variable, non-constant negative curvature surface is never $C^2$ conjugate to a constant curvature flow. See the papers by [ghys1987flotsanosov] and [hurderkatok1991anosovflows].
For non-algebraic Anosov diffeomorphisms. See Farrell, F. T.; Jones, L. E.
Anosov diffeomorphisms constructed from π1Diff(Sn). Topology 17 (1978), no. 3, 273–282.