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A problem about integral curves on a manifold

  1. Nov 8, 2008 #1

    quasar987

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    I must demonstrate in two ways that if c(t) is an integral curve of a smooth vector field X on a smooth manifold M with c'(t_0)=0 for some t_0, then c is a constant curve.

    I found one way: If [itex]\theta[/itex] denotes the flow of X, then because X is invariant under its own flow, we have

    [tex]c'(t)=X_{c(t)} = (\theta_{t-t_0})_*X_{c(t_0)}=(\theta_{t-t_0})_*c'(t_0)=0[/tex]

    for all t.

    Does anyone see another way?
     
  2. jcsd
  3. Nov 10, 2008 #2

    quasar987

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    (Solved. I had initially thought that and argument based directly on the fundamental theorem on EDO does not work, but it does.)
     
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