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Math Help - integral curve

  1. #1
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    Oct 2010
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    integral curve

    Hello,

    Let M be a manifold
    I want to show, that the integral curves of a vector field V are immersed submanifolds of M.

    We assume that the vector field V doesn't vanish anywhere.
    I could show that any integral curve is a immersion.

    If the integral curve c is also injective, that is c is an injective immersion then the image of c is a immersed submanifold.

    But what happens if c isn't injective. For instance if c is periodic?
    How can i show that the image of c is a immersed submanifold, i.e. the inclusion map i: c(a,b)->M is a smooth immersion?

    Please help me!

    Thanks in advance
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  2. #2
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    Oct 2010
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    i have a new idea how to show that a periodic integral curve is a immersed submanifold, but i need a little help.

    My Idea is the following:

    if a,b)->M" alt="ca,b)->M" /> is a periodic integral curve with period p.

    then i need to define a topology and a smooth structure on the image c(a,b), s.t. c(a,b) is a manifold and the inclusion map i:c(a,b)->M is a smooth immersion.

    I think i need to define the topology, s.t. the maps c_{|(0,p)} and c_{|(\epsilon, \epsilon +p)} are diffeomorphisms. with \epsilon >0 sufficient small.

    So we can define the topology as: U\subset c(a,b) is open, if c^{-1}(U \cap c(0,p)), c^{-1}(U \cap c(\epsilon,\epsilon + p)) is open in (a,b).
    Then c_{|(0,p)} and c_{|(\epsilon, \epsilon +p)} and the corresponding inverse maps are all continuous.

    And the smooth structure can be defined as
    if q \in c_{|(0,p)} \subset c(a,b) then we can choose the coordinate system \phi \circ c_{|(0,p)}^{-1} with \phi a coordinate
    and analog if q=c(p).

    Do you mean, that this is enough or right??

    Regards
    Last edited by Sogan; January 10th 2011 at 01:30 PM.
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