- #1

- 970

- 3

All I can think of is that if the Jacobian at a point is non-zero, then the map is bijective around that point. For example, if:

[tex]f(x)=x_0+J(x_0)(x-x_0)[/tex]

where J(x

_{0}) is the Jacobian matrix at the point x

_{0}, x is the coordinate on one manifold, f(x) is the mapped coordinate to the other manifold, then this can easily be inverted for a non-zero Jacobian:

[tex]J^{-1}(x_0)[f(x)-x_0]+x_0=x[/tex]

So it seems that having a non-vanishing Jacobian only proves that f is a bijection and that f is C

^{1}, but does not prove that f is infinitely differentiable.