Can we use Gram-Schmidt to estimate principal directions of curvature at a point p on a manifold?
Suppose for any $tV \in T_pM$, we use the exponential map exp(V_t) to get a curve $\gamma: \Re \to M$, where \gamma(0) = p. We then calculate the curvature of this curve at p, and pick a direction $V$ that maximizes the curvature. We can use Gram-Schmidt to pick orthogonal directions based on the curvature. Does this very special (normal) coordinate system based on curvature have a name?