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January  2010, 26(1): 225-249. doi: 10.3934/dcds.2010.26.225

Quartic differential forms and transversal nets with singularities

Carlos Gutierrez 1, , Víctor Guíñez 2, and  Alvaro Castañeda 2,

1. 

ICMC-USP, São Carlos, Caixa Postal 668, CEP 13560-970, São Carlos, SP
2. 

Universidad de Santiago de Chile, Departamento de Matemática y C.C., Casilla 307, Correo 2, Santiago, Chile, Chile

Received  December 2008 Revised  August 2009 Published  October 2009

We consider a class $ \mathcal{Q}(M) \,$ consisting of smooth quartic differential forms which are defined on an oriented two-manifold $ M $, to each of which we associate a pair of transversal nets with common singularities. These quartic forms are related to geometric objects such as curvature lines, asymptotic lines of surfaces immersed in $\R^4.$ Local problems around the rank-2 singular points of the elements of $ \mathcal{Q}(M) \,$, such as stability, normal forms, finite determinacy, versal unfoldings, are studied in [2]. Here we make a similar study for a rank-1 singular point that is analogous to the saddle-node singularity of vector fields.
Keywords: Quartic Differential Forms, nets..
Mathematics Subject Classification: Primary: 53A07, 58D10; Secondary: 37G1.
Citation: Carlos Gutierrez, Víctor Guíñez, Alvaro Castañeda. Quartic differential forms and transversal nets with singularities. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 225-249. doi: 10.3934/dcds.2010.26.225
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