Given a nowhere vanishing vector field $X\in\Gamma TM$ on a smooth manifold $M$, is it always possible to find a connction $\nabla$ on $M$ so that the parallel transport induced by $\nabla$ preserves $X$?
By this, I mean that given some smooth curve $\gamma:[0,1]\longrightarrow M$, and denoting the parallel transport from $\gamma(0)$ to $\gamma(t)$ along $\gamma$ by $P^\gamma_{0,t}$, one has
$$P^\gamma_{0,t}(X) = X(\gamma(t))$$ for all $t\in[0,1]$.
My initial guess would be that this should be possible at least locally using first local frames and then a partition of unity for the whole manifold, but on the other hand it seems to be too strong of a statement.
Best Answer
Yes, this is true. Choose a Riemannian metric $g$ and a connection $\nabla$ on $M$. Then $X$ is parallel for the connection $\bar\nabla$ which for $Y,Z\in\Gamma TM$ is defined by $$\bar\nabla_ZY:=\nabla_ZY-g(X,Y)/g(X,X)\,\nabla_ZX$$