Differentiability of functions defined on manifolds

[yifli]

Quoted from a book I'm reading:

if f is any function defined on a manifold M with values in Banach space, then f is differentiable if and only if it is differentiable as a map of manifolds.

what does it mean by 'differentiable as a map of manifolds'?

 

[micromass]

Hi yifli!


(I assume that you have defined what a differentiable map is between Banach spaces).

Differentiability as a map of manifolds means:

Let $\Phi:M\rightarrow X$ be your map from M to a Banach space. And let $x\in M$, then x has an open neighbourhood which is homeomorphic to an open set of $\mathbb{R}^n$. Thus there exists a homeomorphism $a:U\rightarrow V$ with U an open set in M that contaisn x and V open in $\mathbb{R}^n$.

Now, $\Phi$ is differentiable in x if and only if $\Phi\circ a^{-1}$ is differentiable.