Vol. 225, No. 2, 2006

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Generalized immersions and the rank of the second fundamental form

Robert J. Fisher and H. Turner Laquer

Vol. 225 (2006), No. 2, 243–272
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.

 To Hans Fischer and Raoul Bott
Keywords
partial differential equation, generalized immersion, contact form, second fundamental form, connection, foliation, developable
Mathematical Subject Classification 2000
Primary: 53C15, 53C42
Secondary: 35F20, 53C12, 58A30