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Jun 5, 2023 at 18:15 comment added Aidan I think this is certainly along the lines of what I'm looking for. I'll have to read it in more detail, but I very much appreciate the suggestion
Jun 4, 2023 at 17:50 comment added Thomas Kurbach Do you want an honest connection on $E$ or a relative connection with respect to $\pi$? There is a somewhat related paper by Biswas & Singh entitled "On the relative Connections" (2019). It is on arXiv. As far as I understand it they only show the existence of a relative connection under the assumption of connections along the fibers and this connection may not agree with the connections along the fibers. But it might still be a place to start. I do not know that paper in depth.
Jun 2, 2023 at 11:12 history asked Aidan CC BY-SA 4.0