Bundles and pencils of nets on a surface

Abstract

A net of curves on a surface consists of two one-parameter families of curves such that through each point of the portion of the surface under consideration there passes just one curve of each family, the two tangents at each point being distinct. It is a characteristic property of conjugate nets that at every surface point the two tangents of every conjugate net separate the tangents of the asymptotic net harmonically. In other words, at every point of a surface the cross ratio of the two asymptotic tangents and any two conjugate tangents is -1. Recent developments in projective differential geometry seem to indicate that these ideas should be generalized. In the first place, the number -1 may be replaced by any number r, and we may then consider the class of all nets on a surface, every one of which has the property that at every surface point its tangents form with the asymptotic tangents the constant cross ratio r. In the second place, the asymptotic net may be replaced by an arbitrary fundamental net. We are thus led to introduce the following definition. The class of all nets on a surface, every one of which has the property that at every surface point its two tangents form with the two tangents of a fundamental net the same cross ratio, will be called a bundle of nets. A bundle of nets on a surface is therefore determined by a fundamental net and a number r. For example, all conjugate nets on a surface constitute a bundle, for which the fundamental net is the asymptotic net, and for which r =-1. The one-parameter family of conjugate nets on a surface, each of which has the property that, at every surface point, its tangents form with the tangents of a fundamental conjugate net a constant cross ratio, has been calledt by Wilczynski a pencil of conjugate nets. This definition can be extended from the bundle of conjugate nets to any bundle of nets. A pencil of nets in a bundle is a one-parameter family of nets in the bundle, each of which has the property that, at every surface point, its two tangents form with the two tangents of a fundamental net of the bundle a constant cross ratio. This cross ratio is different for different nets of the pencil, but for a given