What are locally parallelizable manifolds? I came across this concept on this wiki page regarding killing vector field. The last sentence in section "Cartan Involution" says that "Equivalently, the curvature tensor is covariantly constant on locally symmetric spaces, and so these are locally parallelizable; this is the Cartan–Ambrose–Hicks theorem."
I'm really confused that apparently every manifold is "locally parallelizable", since you simply take a local chart and $\frac{\partial}{\partial x^i}$ automatically form a local parallelization. Therefore, I think it actually says that the manifold admits $n$ linearly independent local killing vector fields, not some general vector fields.
However, the wiki page doesn't cite anything here, so I don't know where that result come from, and of course I don't know how to prove it either. Besides, I don't understand its connection with Cartan–Ambrose–Hicks theorem.
Could someone explain that to me?
 A: Recall that each locally symmetric space is locally homogeneous, and, since the question is local, I will limit myself to the discussion of homogeneous Riemannian manifolds. Namely, given a homogeneouse manifold $M$ and a point $x\in M$, I will construct a neighbourhood $U$ of $x$ and a collection of Killing fields on $M$ whose restriction to $U$ defines a parallelization of $U$.
The condition that a Riemannian manifold $M$ is homogeneous is equivalent to the assumption that this manifold is of the form $G/H$, where $G$ is a Lie group, $H< G$ is a compact Lie subgroup and $G$ is equipped with a left-$G$-invariant Riemannian metric. Since $M$ is homogeneous, this identification can be made so that $x\in M$ corresponds to the projection of the identity element $e\in G$.  (The diffeomorphism $G/H\to M$ comes from the orbit map $g\mapsto gx, g\in G$.)
Now, given this description, let ${\mathfrak h}\subset {\mathfrak g}$ denote the Lie algebras of $H$ and $G$ respectively. Take any linear subspace ${\mathfrak p}\subset {\mathfrak g}$ such that we have a direct sum decomposition
$$
{\mathfrak g}= {\mathfrak h}\oplus {\mathfrak p}. 
$$
In particular, the dimension $d$ of ${\mathfrak p}$ is the same as the dimension of $M$. Taking a basis  $v_1,...,v_d$ in ${\mathfrak p}$, we see that these vectors project to Killing fields $X_1,...,X_d$ on $M$ which are linearly independent at $x$. By continuity, the same linear independence will hold  at all points in a sufficiently small neighborhood $U$ of $x$ thereby, providing a "Killing" parallelization of $U$.
According to the comments, this observation appears to answer your actual question. Needless to say, this is not what the Cartan–Ambrose–Hicks theorem says. Looking at the record of a person who made the edit in the Wikipedia page claiming that "this is the Cartan–Ambrose–Hicks theorem," this person is likely to be a physicist and the quote should not be taken seriously. This is also a general remark regarding Wikipedia articles: Anybody can edit these and, while, in general, the quality of math Wikipedia articles is quite good, occasionally what's written is wrong or misleading or unclear. Reading Wikipedia is not a substitute to reading math books.
