Principal Curvature Lines

Abstract

Theory of surface foliations grew from the classical problems of differential geometry. By the end of the 19th century due to the works of Cayley, Darboux and Picard an important class of foliations given by the lines of principal curvature was singled out. To this date theory of such foliations is the hardest one both locally (umbilic points, Caratheodory conjecture) and globally (periodic and non-trivially recurrent leaves). An excellent introduction to the area is the monograph: C. Gutierrez & J. Sotomayor, Lines of Curvature and Umbilical Points on Surfaces, Instituto de Matematica Pura e Aplicada, Rio de Janeiro, 1994, ISBN 85-244-0057-9.