On congruences of Laguerre cycles and their envelopes

In the cyclographic model of Laguerre geometry, the isotropy projection establishes a correspondence between submanifolds in the Lorentz (n + 1)-space and oriented hypersphere congruences in the Euclidean n-space. In this paper, we investigate the relation between the Lorentzian geometry of the sphere congruences and the Euclidean geometry of the corresponding oriented envelopes. We introduce the notion and discuss the existence of marginally outer trapped submanifolds associated to an (n − 1)-dimensional Legendre submanifold of the unit tangent bundle over the Euclidean n-space. This extends the notion of Laguerre–Gauss map of an oriented surface. Two special classes of marginally outer trapped surfaces are characterized in terms of their oriented envelopes. The Lorentzian geometry of several families of spheres whose envelopes are well-known surfaces in the Euclidean three-space is investigated. (paper)