If $M$ has nonnegative sectional curvature and if $N$ is a totally geodesic submanifold, then assume that $c$ is a geodesic. Define $$d(\alpha (t), c(t))= d(N, c(t)) ,\ \alpha (t) \in N$$

Then **image** of $\alpha$ is an image of some geodesic ? 

Here we give more condition for well-definedness of $\alpha$ : $c(0)\ \in N$ and $t\in [0,\epsilon]$ 


Thank you in anticipation