Codimension two umbilic points on surfaces immersed in $R^3$

Abstract

Abstract. In this paper is studied the behavior of lines of curvature near umbilic points that appear generically on surfaces depending on two parameters.

Singularities of asymptotic lines on surfaces in R^4

“…The rank-2 singular points of PQD ω for DEAL (3) at the origin is one of the following types: Remark 3.2. Type D 12 corresponds to type D 1 2 with codimension 1 for umbilic points of surfaces in R 3 studied in [11], also type D 1 corresponds to type D 2 1 with codimension 2 studied in [16].…”

“…If the 3-jet of F at the origin vanishes, a condition (i) for type D 23 corresponds to the condition for type D 1 23 with codimension 1 studied in [11]. The classification in the case not satisfying the condition (i) or (ii) (this case corresponds to a case with codimension 2 studied in [16]) shall be studied in a forthcoming paper.…”

“…Especially such differential equations also appear as differential equations of principal curvature lines around umbilic points of surfaces in R 3 ([4], [10], [1]), and the stable configurations of its integral curves are well-known as Darbouxian which are determined by the 1-jet of this differential equation. Moreover Gutierrez,Sotomayor and Garcia ([11], [16]) established umbilic bifurcations. In [3] Bruce and Tari pointed out the 1-jet of DEAL is precisely the 1-jet of (1) giving the principal curvature lines of a surface in R 3 at the umbilic point.…”