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.
Naoya ANDO. "The behavior of the principal distributions on a real-analytic surface." J. Math. Soc. Japan 56 (1) 201 - 214, January, 2004. https://doi.org/10.2969/jmsj/1191418703