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 F of M, when is it true that every holomorphic map F : M → N which maps isometrically every complex geodesic of F onto a complex geodesic of N is a biholomorphism? Amongotherthings,weprovethatthisisthecaseif M, N aresmoothboundedstrictly (linearly) convex domains, every element of F contains a given point of M and F spans all of M. More general results are provided in dimension 2 and for the unit ball.