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Math Help - Differential Geometry - Question about Limit

  1. #1
    Senior Member slevvio's Avatar
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    Differential Geometry - Question about Limit

    Hello everyone, I had a question I was hoping I could get some help with. Let M be a smooth manifold and let X,Y be vector fields. Let \phi_t be the flow of X. So in my notation I have Y_{p} \in T_p M and for a smooth function g \in C^{\infty}(M) then Y(g) \in C^{\infty}(M). Let h_t := \displaystyle\frac{g \circ \phi_t - g}{t}, so that \displaystyle\lim_{t\rightarrow 0} h_t = X(g) pointwise.

    In a proof in my course there is a statement
    \displaystyle\lim_{t\rightarrow 0} Y_{\phi_t(p)}\left(h_t\right) =^{?} Y_{\phi_0(t)}\left( \lim_{t\rightarrow 0}h_t\right)= Y_p( X(g)).

    Why can we just take limits inside? It is not obvious to me that the map t \mapsto Y_{\phi_t (p)}(h_t) is smooth.

    Thanks for any help.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Differential Geometry - Question about Limit

    Quote Originally Posted by slevvio View Post
    Hello everyone, I had a question I was hoping I could get some help with. Let M be a smooth manifold and let X,Y be vector fields. Let \phi_t be the flow of X. So in my notation I have Y_{p} \in T_p M and for a smooth function g \in C^{\infty}(M) then Y(g) \in C^{\infty}(M). Let h_t := \displaystyle\frac{g \circ \phi_t - g}{t}, so that \displaystyle\lim_{t\rightarrow 0} h_t = X(g) pointwise.

    In a proof in my course there is a statement
    \displaystyle\lim_{t\rightarrow 0} Y_{\phi_t(p)}\left(h_t\right) =^{?} Y_{\phi_0(t)}\left( \lim_{t\rightarrow 0}h_t\right)= Y_p( X(g)).

    Why can we just take limits inside? It is not obvious to me that the map t \mapsto Y_{\phi_t (p)}(h_t) is smooth.

    Thanks for any help.
    No, but do you disagree that it's not continuous?
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  3. #3
    Senior Member slevvio's Avatar
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    Re: Differential Geometry - Question about Limit

    I can't really see why it's continuous in t
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  4. #4
    Senior Member slevvio's Avatar
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    Re: Differential Geometry - Question about Limit

    have you any ideas as to why this might be continuous in t?

    Thanks very much
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