The lightcone Gauss map of a spacelike surface in Minkowski 4-space. (English) Zbl 1084.53048

The authors study the lightcone Gauss map of space-like surfaces in Minkowski 4-space. They first define a Lorentzian invariant called light-like Gaussian curvature of the space-like surface. Then they introduce the notion of lightcone height function and using it they show that the lightcone Gauss map has a singular point if and only if the Gaussian curvature vanishes at such point. Moreover, they show that the lightcone Gauss map is a constant map if and only if the surface is contained in a light-like hyperplane. They also define the notion of lightcone pedal surface of a space-like surface and show that its singularities are in correspondence with that of the lightcone Gauss map of the surface. Applying Montaldi’s methods to the extended lightcone height function, they study the generic contacts between space-like surfaces and light-like hyperplanes.
Finally, they give a generic classification of lightcone pedal surfaces and lightcone Gauss maps.


53C40 Global submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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