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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.

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  • $\begingroup$ 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". $\endgroup$ – tparker Aug 25 '18 at 17:41
  • $\begingroup$ 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. $\endgroup$ – Stanislav Hronek Aug 25 '18 at 18:02
  • $\begingroup$ 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. $\endgroup$ – tparker Aug 25 '18 at 18:24
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    $\begingroup$ Related: physics.stackexchange.com/q/342821/2451 and links therein. $\endgroup$ – Qmechanic♦ Aug 25 '18 at 18:52
  • $\begingroup$ @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. $\endgroup$ – doetoe Aug 26 '18 at 1:21
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For any non-degenerate coordinate system, we can define a "coordinate connection" under which the individual components of a parallel-transported vector (with respect to that coordinate system) are left unchanged under parallel transport. For example, in polar coordinates on the plane, the polar and azimuthal components would be left unchanged under parallel transport, so that a parallel-transported vector either moves "straight out", or in circles around the origin (or a combination of both). This connection is not compatible with the standard Euclidean metric.

I don't know of any physics applications that use non-metric-compatible connections. Usually if there's a metric then it defines a natural geometry on your system, and it doesn't make sense to think of "parallel" transport that isn't compatible with that geometry.

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