For $M^n$ a riemannian manifold and $S$ a hypersurface, if we consider $$S_t=\{\exp^\perp(v):v\in T(S)^\perp,\;|v|=t\}$$ and $$f_t:S\rightarrow S_t:p\mapsto \exp^\perp(t\eta)$$ with $\eta$ the unit normal vector, then what is the relation of $f_t$ and the focal points of a geodesic $\gamma\perp S$?
We say that $f_t$ is a diffeomorphism for small $t$. How can we determine the value of $t$ where it stops being a diffeomorphism?
Any references on this material?
My uneducated view (disposable if a real answer is posted) is that I think we can continue this process right until a focal point is reached, of the apparently discrete set of such points (references therein might be helpful too). Consider the unit sphere, we start from the equator to parallel disks at each latitude up until the north pole when $ t = 1 $.