I have a connection $\nabla$ in the tangent bundle of a manifold $M$, and a vector field $v$ on $M$ satisfying $\nabla_w(v)=w$, for all vector fields $w$ on $M$. Let $P\in M$ be a point where $v$ vanishes.
My hope is that I can find an isomorphism of $M$ with a vector space, identifying $v$ with the Euler field (at least locally near $P$).
