The behavior of the principal distributions on a real-analytic surface

Abstract

The purposes of this paper are: (a) to study the behavior of the principal distributions around an isolated umbilical point on a real-analytic surface of a certain type, which contains all the real-analytic, special Weingarten surfaces; (b) to present one condition such that if a real-analytic surface satisfies the condition, then the index of an isolated umbilical point on the surface is less than or equal to one.