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The hyperbolic Schwarz map for the hypergeometric equation. (English) Zbl 1157.33305

Summary: The Schwarz map of the hypergeometric differential equation has been studied since the beginning of the last century. Its target is the complex projective line, the 2-sphere. This paper introduces the hyperbolic Schwarz map, whose target is hyperbolic 3-space. This map can be considered to be a lifting to 3-space of the Schwarz map. In this paper, we study the singularities of this map, and attempt to visualize its image when the monodromy group is a finite group or a typical Fuchsian group. General cases will be treated in forthcoming papers.

MSC:

33C05 Classical hypergeometric functions, \({}_2F_1\)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)