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?