On the index of principal foliations of surfaces in R3with corank 1 singularities

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 (R2, 0) to (R3, 0) which either are simple or are non-simple but included in strata of Ae-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.