Intereting Posts

Proving $\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} dx=1-\frac{\gamma}{2}-\ln2$
Fixed Points and Graphical Analysis
Can we show that $1+2+3+\dotsb=-\frac{1}{12}$ using only stability or linearity, not both, and without regularizing or specifying a summation method?
No extension to complex numbers?
Every matrix can be written as a sum of unitary matrices?
Prove that a non-abelian group of order $pq$ ($p<q$) has a nonnormal subgroup of index $q$
Is Pythagoras the only relation to hold between $\cos$ and $\sin$?
prove $( \lnot \lnot p \Rightarrow p) \Rightarrow (((p \Rightarrow q ) \Rightarrow p ) \Rightarrow p )$ with intuitionistic natural deduction
Integrating $\int_{0}^{\infty} \frac{p^6 dp }{1 + a p^4 + b p^6 } \int_{0}^{\pi}\frac{\sin^5 \theta \,d\theta}{1 + a |p-k|^4 + b |p-k|^6 }$
Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$
Why is $L_A$ not $\mathbb K$ linear (I can prove that it is)
Show $(A^o)^c=\overline{A^c}$
Uniform convergence of infinite series
Galois Field GF(4)
Zorn's Lemma And Axiom of Choice

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?

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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.

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 $z$.**

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