Can any vector be extended locally to a geodesic vector field

Question

Given a (pseudo-)Riemannian manifold $(M,g)$ and some vector $X_p\in T_pM$ at $p\in M$, can one always extend $X_p$, locally, to a geodesic vector field $X$, in the sense that any integral curve of $X$ is a geodesic?

If so, why? Does this extend to more general contexts then Riemannian geometry? For instance given only some arbitrary, possibly non-linear, connection.

I can imagine a construction of $X$ might go along the following lines. First construct the geodesic $\gamma$ starting at $p$ with velocity $X_p$. Then specify a vector field $Y$ along $\gamma$ which is orthogonal to $X_p$ and extend $X_p$ along $Y$ by parallel translation. Now we have a geodesic surface, so specify a vector field $Z$ along this surface, orthogonal to the surface, and extend the the tangent vectors along this vector field, again by parallel translation. And so on. I have no idea if this can be made rigorous, but intuitively it seems to me it should be possible.

Answer 1

Yes; such a vector field exists. Here's an outline of one way to construct one (locally, of course).

• Choose a codimension 1 hypersurface $S$ containing $p$ such that $X_p$ is not tangent to $S$.
• Extend $X_p$ to a vector field $X_S$ on $S$ which is nowhere tangent to $S$.
• Let $F:S\times \mathbb{R}\to M$ be defined by letting $F(s,t)=\gamma_s(t)$, where $\gamma_s$ is the geodesic with $\gamma_s(0)=s$, $\dot{\gamma}_s(0)=X_S(s)$. This map is a diffeomorphism on a neighborhood of $(p,0)$.
• Finally, define the extension $X$ as the velocity of these geodesics, equivalently the pushforward of $\frac{d}{dt}$ by $F$.