a tangent vector which does not fall on any geodesic

Question

Given a point on a manifold, is it possible that there is a tangent vector at that point which does not correspond to any local velocity of some geodesic? That is in that direction no geodesic exists whatsoever? If the answer is positive what is a condition which assures the above does not happen at a point? That all tangent vectors correspond to some geodesic?

Answer 1

Given any point $p$ and $v \in T_pM$, there is a unique geodesic $\gamma$ starting at $p$ with velocity $v$, meaning $\gamma(0) = p$ and $\gamma'(0) = v$. This is a consequence of the theory of ODE's. Since every tangent vector is tangent to a point, we can use the above to conclude that the answer to your initial question is no.