I have a general question in Riemannian geometry:
Let M be a compact manifold and $\partial M \neq \emptyset$. Then shoot a geodesic from any boundary point perpendicularly into the interior of M. How can one prove it will end at boundary? If so, it induces a transformation of $\partial M$, does anyone know any result about this transformation? For example, one can ask rigidity property, i.e. if the transformation is an isometry, can one say anything about M?



Transformations induced by geodesics of boundary
It doesn't seem true to me that the geodesic will always return to the boundary. It may end up accumulating around a closed geodesic in the interior. 


For example, consider the hyperboloid of one sheet 
$$x^2+y^2-z^2=1$$
in $\mathbb{R}^3$. This is a surface of revolution and so one can talk of the angular momentum of a path with respect to the axis. For a geodesic this must be constant (Clairault's theorem). If a geodesic has momentum exactly one (in some well-chosen units!) it will accumulate on the "waist" of the surface, where $z=0$. 


Of course this is a non-compact example, but if a geodesic accumulates on the waist, it stays inside some compact region. So it is easy to imagine a compact surface with boundary which contains the relevant compact region and for which this accumulating geodesic meets the boundary at a right angle. This gives an example where the geodesic doesn't return to the boundary. (I think it's even possible to find an example of a metric on the disc of this sort.)


(In case you're curious, a geodesic on the hyperboloid starting with $z>0$, pointing "down" and with momentum less than 1 will pass through the waist and end up asymptotic to a meridian at z = -infinity. If the momentum is bigger than 1 it will not reach the waist and swing back upwards, ending up asymptotic to a meridian at z = +infinity.)


