orbits are open in Manifold ? group action on manifold.

Question

I need to show: for a differentiable manifold $M$, and $Aut(M)$ acts on $M$, orbit of a point $a\in M$ is open in $M$, please help.

Answer 1

This is a local problem:
The key point is to prove that given an open ball in $B\subset \mathbb R^n$ and two points $a,b\in B$, there exists a diffeomorphism $f:\mathbb R^n\to \mathbb R^n$ which is is the identity outside of $B$ and such that $f(a)=b$.
This can be done by constructing a suitable compactly supported (and thus complete) vector field on $\mathbb R^n$ and using the flow it generates.
The details are in Conlon's Differentiable Manifolds, Lemma 4.1.14, page 134. [And this is your lucky day: Google generously allows you to access that very page and the two following ones.]

As a corollary one immediately obtains that the automorphism group $Aut(M)$ of a connected manifold $M$ acts transitively on $M$: the orbit of any point $p \in M$ under $Aut(M)$ is the whole manifold $M$.

Edit
In Shastri's Element of Differential Topology (Corollary 6.3.3, page 166) you will find the following amazing generalization :
Given a connected manifold $M$ of dimension $\gt 1$ and an integer $k\geq 1$, one can send any $k$-element subset $\{a_1,...,a_k\}\subset M$ to any $k$-element subset$\{b_1,...,b_k\}\subset M$ by some diffeomorphism $f$ of $M$, i.e. $f\in Diff(M)$ satisfies $f(a_i)=b_i$.