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.

Re: Differential Geometry - Question about Limit

Quote:

Originally Posted by

**slevvio** 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*?

Re: Differential Geometry - Question about Limit

I can't really see why it's continuous in t :(

Re: Differential Geometry - Question about Limit

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

Thanks very much