Differentiable function germs on differentiable manifolds

2009-12-06T00:26:10Z

Hello everyone, I was wondering if anyone knew how to prove that the map from $C^{\infty}(M)$ to $\xi (p)$, that is, from the infinitely differentiable functions on a manifold M to the space of (once)-differentiable function germs, where the map is associating to each f in $C^{\infty}(M)$ its class in $\xi (p)$ is onto. By the way, since you ask, the reason I'm interested in this is because its a question that WAS on my final for differential topology, I've tried to work it out since then but no luck so far, this is not homework it's just curiosity now, hope its ok ill have to check the post regultaions, sorry, if not just tell me and i'll delete the question...

Answer by Andrew Critch for Differentiable function germs on differentiable manifolds

2009-12-06T00:38:23Z

I think your question is mis-stated, because this map is not onto:

There is no smooth function on the reals whose germ at $0$ is the germ of $x|x|$, a once-differentiable function germ.

Assuming this is not what you wanted, please restate the question, and include an explanation of why you are interested in it ;)

Differentiable function germs on differentiable manifolds - MathOverflow

Differentiable function germs on differentiable manifolds

Answer by Andrew Critch for Differentiable function germs on differentiable manifolds