1

1

So let $n,m$ be two strictly positive numbers. Let $A\subseteq\mathbf{R}^n$ be a compact $C^{\infty}$-submanifold (possibly with boundary and of real dimension not necessarily equal to $n$). Let $f:A\rightarrow \mathbf{R}^m$ be a smooth function. Then is it always possible to extend $f$ in a smooth way to all of $\mathbf{R}^n$?

Is there a good textbooks where such extension results are discussed extensively?

flag
8 
 The answer is always. Most intro manifold theory texts have this result, but Hirsch's "Differential Topology" text goes quite far in this direction. Using the tubular neighbourhood theorem and a bump function there's rather explicit such functions you can write down. – Ryan Budney Jan 21 2011 at 4:31
1 
 Although it's not a theorem in Milnor's "Topology from a Differentiable Viewpoint," once you get half-way through that book (which is short) you'll see more rudimentary partition of unity style arguments for such constructions. – Ryan Budney Jan 21 2011 at 4:35
@Ryan: Can I request posting that as an answer. – Greg Kuperberg Jan 23 2011 at 19:05
Yes, the key point that I realised is this: if the manifold is orientable then one may find a tubular neighboorhood of $A$ which looks like $A\times D^{n-k}$ where where $k=dim(A)$ and from there it is easy to extend $f$ un such a tubular neighboorhood. So roughly speaking, it works because we have a nice Riemannian metric in the ambient space. – Hugo Chapdelaine Jan 23 2011 at 21:57

Your Answer

Get an OpenID
or

Browse other questions tagged or ask your own question.