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

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

    Why can we just take limits inside? It is not obvious to me that the map $\displaystyle t \mapsto Y_{\phi_t (p)}(h_t)$ 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 $\displaystyle M$ be a smooth manifold and let $\displaystyle X,Y$ be vector fields. Let $\displaystyle \phi_t$ be the flow of $\displaystyle X$. So in my notation I have $\displaystyle Y_{p} \in T_p M$ and for a smooth function $\displaystyle g \in C^{\infty}(M)$ then $\displaystyle Y(g) \in C^{\infty}(M)$. Let $\displaystyle h_t := \displaystyle\frac{g \circ \phi_t - g}{t}$, so that $\displaystyle \displaystyle\lim_{t\rightarrow 0} h_t = X(g)$ pointwise.

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

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