The flat geometry of the I_1 singularity: (x,y) -> (x,xy,y^2,y^3)

P. Benedini Riul and R. Oset Sinha

Journal of Singularities
volume 21 (2020), 1-14

Received: 27 April 2018. Received in revised form: 7 August 2018.

DOI: 10.5427/jsing.2020.21a

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Abstract:

We study the flat geometry of the least degenerate singularity of a singular surface in R^4, the I_1 singularity parametrised by (x,y)->(x,xy,y^2,y^3). This singularity appears generically when projecting a regular surface in R^5 orthogonally to R^4 along a tangent direction. We obtain a generic normal form for I_1 invariant under diffeomorphisms in the source and isometries in the target. We then consider the contact with hyperplanes by classifying submersions which preserve the image of I_1. The main tool is the study of the singularities of the height function.

2000 Mathematical Subject Classification:

Primary 57R45; Secondary 53A05, 58K05

Key words and phrases:

singular surface in 4-space, flat geometry, height function

Author(s) information:

P. Benedini Riul	R. Oset Sinha
Instituto de Ciências Matemáticas	Departament de Matemàtiques
e de Computação - USP	Universitat de València
Av. Trabalhador são-carlense	Campus de Burjasso
400-Centro, CEP: 13566-590, São Carlos-SP	46100 Burjassot
Brazil	Spain
email: pedro.benedini.riul@gmail.com	email: raul.oset@uv.es