On the index of principal foliations of surfaces in R^3 with corank 1 singularities

J. C. F. Costa, L. F. Martins, and J. J. Nuño-Ballesteros

Journal of Singularities
volume 22 (2020), 1-16

Received: 25 February 2019. Received in revised form: 16 February 2020.

DOI: 10.5427/jsing.2020.22a

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Abstract:

It is well known that the index associated to the principal foliations at a cross-cap point is 1/2. In this work we study the index for other corank 1 singularities from (R^2,0) to (R^3,0) which either are simple or are non-simple but included in strata of A_e-codimension ≤ 3. We show that the index, under certain conditions, is always 0 or 1, bearing out that the Loewner conjecture could be true for all corank 1 singularities.

2010 Mathematical Subject Classification:

Primary 58K05; Secondary 34A09, 53A05

Key words and phrases:

Singular surfaces, Lowener conjecture, index, principal lines

Author(s) information:

J. C. F. Costa
Departamento de Matemática
UNESP - Universidade Estadual Paulista
R. Cristóvão Colombo, 2265
CEP 15054-000, São José do Rio Preto-SP, Brazil
email: joao.costa@unesp.br

L. F. Martins
Departamento de Matemática
UNESP - Universidade Estadual Paulista
R. Cristóvão Colombo, 2265
CEP 15054-000, São José do Rio Preto-SP, Brazil
email: luciana.martins@unesp.br

J. J. Nuño-Ballesteros
Departament de Geometria i Topologia
Universitat de València
Campus de Burjassot 46100, Spain
email: Juan.Nuno@uv.es