Covariant Derivatives along smooth Maps

[mjungmath]

In the current implementation of connections, it is not possible to take the covariant derivative along a map. But this is especially useful when the maps are curves, most prominently geodesics.

Since this construction is nothing but taking the pull-back of a connection along a map in terms of bundles, I'd suggest to utilize the implementation over general vector bundles, which already support the desired action (see #30209). The covariant derivative along a curve would then correspond to the pull-back connection on the corresponding pulled-back vectorfield module. Different approaches are most welcome for discussion.

CC: @egourgoulhon @tscrim

Component: manifolds

Issue created by migration from https://trac.sagemath.org/ticket/30781

[mjungmath]

Description changed:

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+++ 
@@ -1,3 +1,3 @@
 In the current implementation of connections, it is not possible to take the covariant derivative along a map. But this is especially useful when the maps are curves, most prominently geodesics.

-Since this construction is simply taking the pull-back of a connection along a map in terms of bundles, I'd suggest to utilize the implementation for general vector bundles, which already support the desired action (see #30209). The covariant derivative along a curve would then correspond to the pull-back connection on the corresponding pulled-back vectorfield module. Different approaches are most welcome.
+Since this construction is nothing but taking the pull-back of a connection along a map in terms of bundles, I'd suggest to utilize the implementation over general vector bundles, which already support the desired action (see #30209). The covariant derivative along a curve would then correspond to the pull-back connection on the corresponding pulled-back vectorfield module. Different approaches are most welcome for discussion.

[mkoeppe]

comment:3

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.