Given an isometric (in the Riemannian way) immersion $f:N\rightarrow M$ 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 $f:\mathbb{R}\rightarrow\mathbb{R}^2$, where f is the logarithmic spiral. In this case, the normal exponential map misses the origin.