Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $M$ be a smooth manifold, $x,y\in M$. Must there exist a diffeomorphism $f : M \rightarrow M$ with $f(x) = y$?

I tried proving this via vector fields, i.e. trying to find a vector field whose flow through $x$ passes through $y$, without much success. Besides, this only has a chance of working on complete manifolds. Anyone know the answer to this?

share|improve this question
    
Do you assume that $M$ is connected? –  mixedmath Oct 2 '12 at 0:37
    
Yes, lets assume $M$ is connected. Otherwise this is rather trivially false. –  user15464 Oct 2 '12 at 0:41
add comment

2 Answers 2

up vote 6 down vote accepted

No; take $M$ to be the disjoint union of two smooth manifolds which are not diffeomorphic.

However, the statement is true if $M$ is connected. You do not need completeness. It suffices to show that the set of all points that can be reached from $x$ via some diffeomorphism is both open and closed.

share|improve this answer
    
Yeah, I was thinking an open disk disjoint union and an open interval. –  ncmathsadist Oct 2 '12 at 0:51
    
Maybe a better way to phrase the hint is as follows: define $x \sim y$ if $x$ can be reached from $y$ via some diffeomorphism. Show that this is an equivalence relation and that the equivalence classes are open. –  Qiaochu Yuan Oct 2 '12 at 1:47
add comment

You can find a demonstration of this fact (if M is connected) in the book of Milnor - Topology from the differentiable viewpoint. It is the lemma of homogeneity. In fact you have more :

Homogeneity Lemma: Let $y$ and $z$ be arbitray interior points of the smooth, connected manifold M. Then there exists a diffeomorphism $f:M\rightarrow M$ that is smoothly isotopic to the identity and carries $y$ into $z$.

share|improve this answer
    
Way cool, Kaye. +1 for you! –  ncmathsadist Oct 2 '12 at 0:52
    
@ncmathsadist or Tomas could you please make me understand the lemma? I am not able to grasp after some stage for example from the step where he introduced some differential equation. –  Bunuelian Trick Mar 9 '13 at 9:47
    
@CityOfGod can you please explain exactly what is your doubt? –  Tomás Mar 9 '13 at 13:16
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.