Geometry of D_4 conformal triality and singularities of tangent surfaces

Goo Ishikawa, Yoshinori Machida, and Masatomo Takahashi

Journal of Singularities
volume 12 (2015), 27-52

Received 25 March 2014. Received in revised form 29 October 2014.

DOI: 10.5427/jsing.2015.12c

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It is well known that projective duality can be understood in the context of geometry of A_n-type. In this paper, as D_4-geometry, we construct explicitly a flag manifold, its triple-fibration and differential systems which have D_4-symmetry and conformal triality. Then we give the generic classification for singularities of the tangent surfaces to associated integral curves, which exhibits the triality. The classification is performed in terms of the classical theory on root systems combined with the singularity theory of mappings. The relations of D_4-geometry with G_2-geometry and B_3-geometry are mentioned. The motivation of the tangent surface construction in D_4-geometry is provided.

Mathematical Subject Classification:

Primary 58K40; Secondary 57R45, 53A20

Author(s) information:

Goo Ishikawa Yoshinori Machida Masatomo Takahashi
Department of Mathematics Numazu College of Technology Muroran Institute of Technology
Hokkaido University Shizuoka 410-8501, Japan Muroran 050-8585, Japan
Sapporo 060-0810, Japan
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