Let X, Y be real analytic vector fields on a real analytic n-dimensional manifold M. Consider a control system on M with dynamics described by \[(1)\qquad {{dx} / {dt}} = X(x(t)) + u(t)Y(x(t)),\quad x(0) = p,\]where an admissible control is Lebesgue measurable with values $| {u(t)} | \leqq 1$. The relationship between the map \[(2)\qquad (s_1 , \cdots ,s_n )\to (\exp s_1 Y) \circ \cdots \circ (\exp s_n ({\text{ad}}^{n - 1} X,Y)) \circ (\exp tX)(p)\] being one-one on a neighborhood of $0 \in \mathbb{R}^n $ for each $0 < t < \varepsilon $ and other known necessary conditions for local controllability are studied. In dimension $ n =2 $, many necessary conditions are equivalent, and also sufficient. For $n \geqq 3$, the map (2) being locally one-one implies many necessary conditions are satisfied, but these need not be sufficient. Examples which illustrate what occurs geometrically are given.