Slice rigidity property of holomorphic maps Kobayashi-isometrically preserving complex geodesics

In this paper we study the following slice rigidity property: given two Kobayashi complete hyperbolic manifolds $M, N$ and a collection of complex geodesics $mathcal F$ of $M$, when is it true that every holomorphic map $F:Mto N$ which maps isometrically every complex geodesic of $mathcal F$ onto a complex geodesic of $N$ is a biholomorphism? Among other things, we prove that this is the case if $M, N$ are smooth bounded strictly (linearly) convex domains, every element of $mathcal F$ contains a given point of $overline{M}$ and $mathcal F$ spans all of $M$. More general results are provided in dimension $2$ and for the unit ball.