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.
To Hans Fischer and Raoul Bott
Keywords
partial differential equation, generalized immersion,
contact form, second fundamental form, connection,
foliation, developable
