In this paper, we study the following “slice rigidity property”: given two Kobayashi complete hyperbolic manifolds M, N and a collection of complex geodesics \({\mathscr {F}}\) of M, when is it true that every holomorphic map \(F:M\rightarrow N\) which maps isometrically every complex geodesic of \({\mathscr {F}}\) 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 \({\mathscr {F}}\) contains a given point of \({\overline{M}}\) and \({\mathscr {F}}\) spans all of M. More general results are provided in dimension 2 and for the unit ball.

在本文中，我们研究以下“切片刚度属性”：给定两个Kobayashi完整的双曲流形M，  N和M的复杂测地线\（{\ mathscr {F}} \\）的集合，何时每个同胚映射\（F：M \ rightarrow N \）等轴测图将\（{{\ mathscr {F}} \）的每个复杂测地线映射到N的复杂测地线是双全纯的吗？除其他外，我们证明如果M，  N是严格有界的（线性）凸域，则\（{\ mathscr {F}} \）的每个元素都包含\（{\ overline { M}} \）和\（{\ mathscr {F}} \）跨越所有的中号。在尺寸2和单位球中提供了更一般的结果。