1
$\begingroup$

I was just reading the Ehresmann connection wikipedia page and noticed that it defines an Ehresmann connection to be complete if a curve in the base can be horizontally lifted over its entire domain. I was under the impression that this was always true!

It is always true for the frame bundle of a vector bundle. In this case, Gronwall's Inequality tells you that the paralell transport of a vector cannot blow up in finite time.

Question 1: Is it true that any principal bundle connection is complete? I haven't been able to prove this.

Question 2: Are there any interesting examples of Ehresmann connections which are not complete?

$\endgroup$
1
  • 1
    $\begingroup$ A comment just for completeness: I think Ehresmann defined his connections to always be complete. And as soon as the distribution is complete, the bundle must be a fibre bundle. I guess the fact that 1) is true is just a generalisation of the fact that every homogeneous space is complete. $\endgroup$ Feb 16, 2015 at 21:02

1 Answer 1

2
$\begingroup$

Question 1: yes. See 19.6 of here.

Question 2: Not really. Most of them are of the kind of projecting an open disk to the open interval with the horizontal connection.

Remark: 17.9 of the same source proves that every fiber bundle admits a complete connection. See also 17.11. The first published version of this is 9.9 of this book

$\endgroup$
2
  • $\begingroup$ I just wanted to say that I have been reading your topics in differential geometry notes and am learning a lot! Thanks for writing them $\endgroup$ Feb 20, 2015 at 13:34
  • $\begingroup$ @Peter Michor do you know where I can read more about the relationship between the holonomy along a path and the homotopy class of this path? By the way, your books are really good, congratulations!! $\endgroup$
    – PtF
    Jul 19, 2016 at 21:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.