Congruence of geodesic spheres in H3 and S3 - S. Reyes, Edwin O. (2018)
In [2], was obtained a characterization of the surfaces in R3 which are envelopes of a sphere congruence in R3, in which the other envelope is in R2. In this paper, we characterize the surfaces of H3 and S3 which are envelopes of a congruence of geodesic spheres in H3 and S3, respectively, in which the other envelope is contained in H2 H3and S2 S3. We show that this characterization allows locally to obtain a parameterization of the surfaces contained in H3 and S3, this characterization extends the result obtained in [2]. Moreover, we provide sufficient conditions for these surfaces to be locally associated by a transformation of Ribaucour. Also, we present families of surfaces parameterized by lines of curvature in H3 and S3, which depend on a function of two variables which is solution of a differential equation. Finally, we characterize the surfaces of the spherical type in H3 and S3, as the surfaces where its radius function is the solution of the Helmholtz equation.