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# Asian Journal of Mathematics

## Volume 20 (2016)

### Number 2

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

Pages: 353 – 382

DOI: https://dx.doi.org/10.4310/AJM.2016.v20.n2.a6

#### Authors

#### Abstract

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 $G_2$.

#### Keywords

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

#### 2010 Mathematics Subject Classification

Primary 58K40. Secondary 53A20, 57R45.

Published 18 March 2016