Envelopes of families of Legendre mappings in the unit tangent bundle over the Euclidean space

Abstract

For families of hypersurfaces with singular points, a classical definition of an envelope is vague. In order to define an envelope for a family of hypersurfaces with singular points, we consider r -parameter families of frontals and of Legendre mappings in the unit tangent bundle over the Euclidean space. We define an envelope for the r -parameter family of Legendre mappings. Then the envelope is also a frontal. We investigate properties of the envelopes. As an application, we give a condition that the projection of a singular solution of a first order partial differential equation is an envelope.