Study of hypergeometric functions

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1) Original Schwarz map has the Riemann sphere as its target. I proposed the hyperbolic and de Sitter Schwarz maps, whose targets are hyperbolic 3-space and de Sitter 3-space, respectively. The unit normals of the hyperbolic Schwarz image surface are de Sitter Schwarz maps, and vice versa. Geodesic extension of the normals hit the ideal boundary which turn out to be the original and the derived Schwarz maps. These new maps have singularities outside the singularities of the differential equation. These singularities offer geometric invariants.2) Hyperplane arrangements in the real projective spaces. n+3 hyperplanes in general position in n-space is combinatorially unique. I studied the action of the cyclic group of order n+3 on the chambers cut out by these hyperplanes.