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 $\alpha$ is a geodesic ? 

(If $d(c(t),n_i)=d(c(t),N),\ n_i\in N$, then we choose any $n_i$ for $\alpha(t)$ So we can make $\alpha$ to be continuous by choosing suitable point)

Thank you in anticipation