Show that if $X$ is a connected manifold then for any $x,y\in X$, there exists a diffeomorphism $h$ sending $x\xrightarrow{h}y$. This is what I have so far:
Given a connected manifold $X$, for any $x,y\in\mathbb{R}$ there is an embedding $\gamma_x:[0,1]=I\rightarrow X$ with $\gamma_x(0) = x, \gamma_x(1) = y$. Let $K = \gamma(I)$, and note that this is compact. Then we can define a vector field $v$ on $X$ to be $v(r) = \gamma'_x(t_r)$ if $\gamma_x(t_r) = r$ i.e. $r\in K$ and $0$ otherwise. Then the results follows from the following theorem:
If $v$ is a vector field on $X$ such that $v$ is zero outside of a compact $K\subset int X$. Then $\exists !$ flow $H$ on $X$ whose velocity field is $v$.
From this we get a diffeomorphism $h_t$ such that $h_t(x) = y$ so $X$ is homogeneous.
A couple issues I have with this so far: 1) is $v$ a smooth vector field as defined or do I need to use partitions of unity to smooth it out? 2) Since $K$ is required to be in the interior of $X$, does this not work if either $x$ or $y$ are in $\partial X$?