Let $g_{ab}$ a Riemaniann ( Lorentzian ) metric in a $n-$dimensional manifold $N$ and let $M$ be a submanifold of $N$. In general, the Levi-Civitta connection induced by the induced metric in $M$ isn't given just by restriction. This happen only when $M$ is a totally geodesic submanifold. My question is:
How many totally geodesic submanifolds of codimension $k$ are enough to recover the Christoffel symbols at a point in $N$?
It's very rare to have totally geodesic submanifolds of dimension $>1$. Indeed, if you have one through each $p$ for every $2$-plane in $T_pN$, then $N$ must have constant curvature.