Riemannian hypermaps to complex space forms

Abstract

In this paper, we propose the notion of a \textit{hypermap}, defined as a Riemannian map whose range distribution is tangent to a real hypersurface of a complex space form. This construction leads to a new direction in Riemannian map theory. We establish the fundamental Gauss and Codazzi equations for a hypermap and use them to investigate the extrinsic geometry of its range leaves. In particular, when the ambient complex space form is non-Euclidean, we prove that these leaves cannot be totally umbilical and that their shape operator fails to be parallel. We further incorporate the Hopf condition and define \textit{Hopf hypermap}, thereby obtaining the analog of Hopf hypersurfaces. We also examine the umbilical case and show that an umbilical hypermap cannot exist to a non-Euclidean complex space form. This motivates introducing the \textit{$η$-umbilical hypermap}, which broadens the class of examples for studying the interaction between the range geometry and the ambient complex structure. Finally, we analyze the principal curvatures associated with these hypermaps, including results on their multiplicities and constancy.