The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat" spacelike surfaces

Abstract

Using the Legnedrian duarities between surfaces in pseudo-spheres in Lorentz-Minkowski 4-space, we study various kind of flat surfaces in pseudo-spheres. We consider a surface in the pseudo-sphere and its dual surface. Flatness of a surface is defined by the degeneracy of the dual surface similar to the case for the Gauss map of a flat surface in the Euclidean space. We study singularities of these flat surfaces and dualities of singularities.

Orientability of linear Weingarten surfaces, spacelike CMC‐1 surfaces and maximal surfaces

“…It should be remarked that Aledo and Espinar 1 recently classified complete linear Weingarten immersions of Bryant type with non‐negative Gaussian curvature in S 3 1 . In this paper, almost all formulas are of linear Weingarten surfaces in H 3 , but one can find the several corresponding formulas of linear Weingarten surfaces in S 3 1 in 1 (see also 16).…”

“…Fundamental properties of flat fronts are given in 13, 22 and 23. The duality between flat surfaces in H 3 and those in S 3 1 is pointed out in 16.…”

“…Fundamental properties of flat fronts are given in 13, 22 and 23. The duality between flat surfaces in H 3 and those in S 3 1 is pointed out in 16. (Hyperbolic type) A front f ∈ W ε ( M 2 ) is called a linear Weingarten front of hyperbolic type if ε > 0 (i.e., a /( a + 2 b ) > 0). The parallel fronts of f are also in the same class \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\bigcup _{\varepsilon >0} W_{\varepsilon }(M^2)$\end{document}.…”

Visualization of Minkowski Patch

“…The dimension of A is the dimension of the vector space of translations V. Then the vector space of translations is the space of vectors in the plane or in space [1]. Proof: Let E denote an affine space which is different from real vector space V = ,!…”

Dualities of Differential Geometric Invariants on Cuspidal Edges on Flat Fronts in the Hyperbolic Space and the de Sitter Space

“…It is introduced in [11], that a formulation considering this duality as a double Legendrian fibration in contact geometry. See [13,14] for studies of inear Weingarten surfaces from this viewpoint. On the other hand, a front is a surface in a 3-space with well-defined unit normal vector even on the set of singular points.…”