riemannian geometry - Surjectivity of the normal exponential map

Given an isometric (in the Riemannian way) immersion $f:Nrightarrow M$ between complete, smooth riemannian manifolds, are there conditions on $M$, $N$, $f$, such that the normal exponential map $mathrm{exp}^{nu}:nu(N)rightarrow M$ is surjective?

I'm interested in the case of $f$ being not closed. An example of non surjectivity is given by $f:mathbb{R}rightarrowmathbb{R}^2$, where f is the logarithmic spiral. In this case, the normal exponential map misses the origin.