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.