I am working on my master thesis about harmonic and geodesic mappings and I am looking for some examples with physical meaning. I want to find some geodesic mapping, so a map between manifolds endowed with affine connections $$ \phi:(M,\nabla)\rightarrow (N,\bar{\nabla}), $$ which preserves the geodesics, the problem is that for my thesis I need some examples for non-metric connections. Actually the connection $\bar{\nabla}$ on the target space can be metrizable. If anyone knows any examples of such mappings or just parts of physics that might include such mappings. I know a lot of examples for metrizable connections but I haven“t found any with non-metrizable.
Thank you.