Abstract

Partial differential equations and differential geometry come together in the idea of a generalized immersion. This concept, defined by means of Grassmann bundles and contact forms, allows for “immersions” with “singularities.” Sophus Lie’s generalized solutions to partial differential equations are an important special case.

The classical second fundamental form has a natural generalization in the context of generalized immersions. The rank of the form is then meaningful. A constant rank assumption on the generalized second fundamental form leads to a natural foliation of the generalized immersion, at least when the ambient space is a space of constant curvature. Questions about the total geodesy and regularity of the foliation are also addressed.

Keywords

partial differential equation, generalized immersion, contact form, second fundamental form, connection, foliation, developable

Mathematical Subject Classification

Primary: 53C15, 53C42

Secondary: 35F20, 53C12, 58A30, 58G30

Authors