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Complete vector field X => X defined on the whole manifold?

  1. Apr 3, 2014 #1
    According to Isham (Differential Geometry for Physics) at page 115 he claims:

    "If X is a complete vector field then V can always be chosen to be the entire manifold M"

    where V is an open subset of a manifold M. He leaves this claim unproved.
    A complete vector field is a vector field which has integral curves defined on the whole of ##\mathbb{R}##. Does the claim somehow follow from the definition?
  2. jcsd
  3. Apr 3, 2014 #2


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    How is [itex]V[/itex] defined? [itex]M[/itex] is always an open subset of itself, but without knowing how [itex]V[/itex] relates to [itex]X[/itex] we really can't help you.
  4. Apr 4, 2014 #3
    Isham defines it as follows:

    "Let X be a vector field defined on an open subset U of a manifold M and let p be a point in U c M . Then a local flow of X at p is a local one-parameter group of local diffeomorphisms defined on some open subset V of U such that p ##\in## V c U and such that the vector field induced by this family is equal to the given field X ."
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