Let $M$ be a Riemannian manifold $M$ (or any manifold with enough structure to talk about geodesics).
Definition. Let a geodesic surface $X\subseteq M$ be submanifold, so that any geodesic in $M$ through a point $p\in X$ and with direction $v\in T_p X$ lies completely in $X$.
For me, this feels like some kind of higher dimensional geodesic in $M$. For example, affine subspaces in $\Bbb R^n$, or great-spheres in $\Bbb S^n$ are geodesic surfaces.
Quesion: Is there a standard name for such submanifold, and what can be said about their existence?
For example, if I am given a point $p\in M$ and a direction $v\in T_p M$, there is a (maximal) geodesic $\gamma_{p,v}$ through $p$ with direction $v$. Equivalently, for linearly independent $v_1,...,v_k\in T_p M$ I could ask for the existence of such a (maximal) geodesic surface $X_{v_1,...,v_k}$ of dimension $k$ and containing all $\gamma_{p,v_i}$. Do these surfaces always exist?
Naively, I would probably set $X_{v_1,...,v_k}\subseteq M$ to be the union of all geodesic $\gamma_{p,v}$ with $v\in\mathrm{span}\{v_1,...,v_k\}\subseteq T_p M$. Any geodesic surface must contain the $\gamma_{p,v}$, so this is a good start, but I am far from proving that any geodesics of $M$ that does not pass through $p$ is also contained in this set.