Asian Journal of Mathematics

Volume 20 (2016)

Number 2

Singularities of tangent surfaces in Cartan’s split $G_2$-geometry

Pages: 353 – 382



Goo Ishikawa (Department of Mathematics, Hokkaido University, Sapporo, Japan)

Yoshinori Machida (Numazu College of Technology, Shizuoka, Japan)

Masatomo Takahashi (Muroran Institute of Technology, Muroran, Japan)


In the split $G_2$-geometry, we study the correspondence found by E. Cartan between the Cartan distribution and the contact distribution with Monge structure on spaces of five variables. Then the generic classification is given on singularities of tangent surfaces to Cartan curves and to Monge curves via the viewpoint of duality. The present paper completes the generic classification of singularitites for simple Lie algebras of rank $2$, namely, for $A_2$, $C_2 = B_2$ and $G_2$.


split octonion, Cartan distribution, null Grassmannian, Engel curve, Monge curve, tangent surface

2010 Mathematics Subject Classification

Primary 58K40. Secondary 53A20, 57R45.

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