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i don't get the example of the 2-sphere and euclidean plane in the discussion section: a local diffeomorphism is not required to be onto, so i see no contradiction in the fact that the image of the 2-sphere under the local diffeo is compact. please can someone explain? thanks. — Preceding unsigned comment added by 147.122.45.24 ( talk) 19:51, 5 July 2011 (UTC) reply

oh now i see. compactness of source space implies surjectivity (replied to myself here in case anybody had the same doubt). 147.122.45.24 ( talk) 20:00, 5 July 2011 (UTC) reply

Given a vector field on a manifold ${\displaystyle X}$, we have its time-${\displaystyle t}$ local flow ${\displaystyle \Phi ^{t}}$. This will be a "locally-defined" diffeomorphism, namely a diffeomorphism between open subsets of ${\displaystyle X}$. If ${\displaystyle \Phi ^{t}}$ is defined on all of ${\displaystyle X}$, the it is a diffeomorphism.

Thus, in general, I am not sure how a local diffeomorphism (that is not a diffeomorphism) might arise from a flow. SkiingArcher ( talk) 23:58, 11 June 2024 (UTC) reply