I was just reading the [Ehresmann connection wikipedia page](http://en.wikipedia.org/wiki/Ehresmann_connection) 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](http://en.wikipedia.org/wiki/Gronwall%27s_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.
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> 
>  **Question 2:**  Are there any interesting examples of Ehresmann connections which are not complete?
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> 



A question about horizontal lifts for an Ehresmann connection
Question 1: yes. See 19.6 of [here](http://www.mat.univie.ac.at/~michor/dgbook.pdf).


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](http://www.mat.univie.ac.at/~michor/kmsbookh.pdf)


