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

Goo Ishikawa, Yoshinori Machida, Masatomo Takahashi
2016 Asian Journal of Mathematics  
In the split G2-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 geometric singularity theory for simple Lie algebras of rank 2, namely, for A2, C2 = B2 and G2 is established. Introduction In this paper we present a duality of
more » ... t a duality of certain singularities appearing in the correspondence for split G 2 -geometry found by E. Cartan and formulated by R. L. Bryant [6] . The complex simple Lie algebras are classified by Dynkin diagrams through root systems and in the case of rank 2, there are exactly three cases, namely A 2 , C 2 = B 2 , and G 2 . Figure 1: Dynkin diagrams of types A 2 , C 2 and G 2 We associate an explicit pair of fibrations with each type A 2 , C 2 or G 2 : The fibration induces canonical geometric structures on the three spaces Z, Y, X in each case. In particular the completely non-integrable plane field E = Ker(Π Y * ) ⊕ Ker(Π X * ) on Z is associated. Then parametrized integral curves f : I → Z of the plane field E project to curves Π Y • f and Π X • f in Y and X respectively. Moreover each curve Π Y • f (resp. Π X • f ) is embedded in a surface ruled by the "tangent lines" Π Y Π −1 X Π X (f (t)) (resp. Π X Π −1 Y Π Y (f (t))), t ∈ I, which we call the tangent surface. Note that both two curves Π −1 X Π X (f (t 0 )) and f have tangent lines in the The tangent surfaces are naturally appear in the G 2 -geometry and they are regarded as solutions for certain involutive systems of partial differential equations (see [7] ). It is classically known that the tangent surfaces necessarily have singularities. However the singularities appearing in such surfaces had never been studied in detail. In this paper, for the G 2 case, we describe the duality explicitly and provide generic classification results on tangent surfaces, or more exactly, the tangent mappings which parametrize tangent surfaces, under local diffeomorphisms using singularity theory of mapping. Then, as a result, we
doi:10.4310/ajm.2016.v20.n2.a6 fatcat:mw2umno4frh5nodcqjy543xiwi