Timeline for What are the sufficient and necessary conditions for surjective submersions to be locally trivial
Current License: CC BY-SA 4.0
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| Oct 10, 2023 at 23:52 | comment | added | Jochen Trumpf | The proof in Michor's book (just like the other known proofs in the literature) has a critical gap as pointed out by Matias del Hoyo in Complete connections on fiber bundles, Indagationes Mathematicae, Volume 27, Issue 4, 2016, Pages 985-990. He provides an alternative proof for this result (the result is true). The issue with the existing proofs is that convex combinations of complete Ehresmann connections or fibred Riemannian metrics need not be complete. Del Hoyo gives counterexamples for these assertions. His alternative construction is well worth a detailed read! | |
| Nov 11, 2020 at 12:04 | vote | accept | alexpglez98 | ||
| Nov 11, 2020 at 10:10 | comment | added | Sebastian | Yes, I meant surjective submersion. I would also guess that one can recover a complete connections by patching locally complete connections, but I haven't looked at the details. I will have a look into Michor's book. The one proof of Ehresmann theorem I am aware of uses connections, and the simple observation that any connection on a proper surjective submersion is complete. | |
| Nov 11, 2020 at 10:04 | history | edited | Sebastian | CC BY-SA 4.0 |
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| Nov 11, 2020 at 9:51 | comment | added | alexpglez98 | Thanks! By the word fibration do you mean surjective submersion? In topology it means something different. I find this observation very beautiful. In Michor's book, there's a proof that all the fiber bundles (locally trivial surjective submersions) have complete connections. I think we can recover a complete connection from these complete locally connections. I was thinking if there is on another criteria similar than $\pi$ to be proper, but weaker. | |
| Nov 11, 2020 at 9:51 | vote | accept | alexpglez98 | ||
| Nov 11, 2020 at 12:04 | |||||
| Nov 11, 2020 at 7:46 | history | answered | Sebastian | CC BY-SA 4.0 |