# Physical example of geodesic mapping without metric

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.

• I don't understand - are you looking for a non-metric-compatible connection, or a non-metrizable connection? You say "the connection ... can be metrizable", but then two sentences later you say "I haven't found with non-metrizable". – tparker Aug 25 '18 at 17:41
• There are two connections, one on the source space and one on the target space the one on the target space can me metrizable. By metrizable I mean that it comes from a metric. I am not sure about your first part of the question what exactly is the difference ? Thank you. – Stanislav Hronek Aug 25 '18 at 18:02
• A connection is said to be "metric-compatible" (or just a "metric connection") if the covariant derivative of the metric vanishes: $\nabla_\mu g^{\mu \nu} \equiv 0$. The word "metrizable" is unrelated: it usually refers to a topological space whose topology is generated by the open balls of some metric. – tparker Aug 25 '18 at 18:24
• Related: physics.stackexchange.com/q/342821/2451 and links therein. – Qmechanic Aug 25 '18 at 18:52
• @tparker I think a connection on the tangent bundle is quite generally called metrizable if there exists a metric for which it is the Levi-Civita connection. – doetoe Aug 26 '18 at 1:21