We study Gauss maps of order k, associated to a projective variety X embedded in projective space via a line bundle L. We show that if X is a smooth, complete complex variety and L is a k-jet spanned line bundle on X , with k > 1, then the Gauss map of order k has finite fibers, unless X = Pn is embedded by the Veronese embedding of order k. In the case where X is a toric variety, we give a com...

For a smooth strictly convex closed hypersurface Σ in R, the Gauss map n : Σ → S is a diffeomorphism. A fundamental question in classical differential geometry concerns how much one can recover through the inverse Gauss map when some information is prescribed on S ([27]). This question has attracted much attention for more than a hundred years. The most notable example is probably the Minkowski...

For an oriented isometric immersion f : M → S the spherical Gauss map is the Legendrian immersion of its unit normal bundle UM⊥ into the unit sphere subbundle of TS, and the geodesic Gauss map γ projects this into the manifold of oriented geodesics in S (the Grassmannian of oriented 2-planes in R), giving a Lagrangian immersion of UM⊥ into a Kähler-Einstein manifold. We give expressions for the...

In this work we extend the Weierstrass representation for maximal spacelike surfaces in the 3-dimensional Lorentz–Minkowski space to spacelike surfaces whose mean curvature is proportional to its Gaussian curvature (linear Weingarten surfaces of maximal type). We use this representation in order to study the Gaussian curvature and the Gauss map of such surfaces when the immersion is complete, p...

The main idea of this chapter is to try to measure to which extent a surface S is different from a plane, in other words, how “curved” is a surface. The idea of doing this is by assigning to each point P on S a unit normal vector N(P ) (that is, a vector perpendicular to the tangent plane at P ). We are measuring to which extent is the map from S to R given by P 7→ N(P ) (called the Gauss map) ...