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In this paper, we find some new information on Legendrian dualities and extend them to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space. In particular, we construct all contact diffeomorphic mappings between the contact manifolds and display them in a table that contains all information about Legendrian dualities.

The universal covering group of Euclidean motion group E(2) with the general left-invariant metric is denoted by $$(\widetilde{E(2)},g_L(\lambda _1,\lambda _2)),$$ ( E ( 2 ) ~ , g L ( λ 1 , λ 2 ) ) , where $$\lambda _1\ge \lambda _2>0.$$ λ 1 ≥ λ 2 > 0 . It is one of three-dimensional unimodular Lie groups which are classified by Milnor. In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean $$C^2$$ C 2 -smooth surface in $$(\widetilde{E(2)},g_L(\lambda _1,\lambda _2))$$ ( E ( 2 ) ~ , g L ( λ 1 , λ 2 ) ) away from characteristic points and signed geodesic curvature for Euclidean $$C^2$$ C 2 -smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the universal covering group of Euclidean motion group E(2) with the general left-invariant metric.

The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by E 1 , 1 , g λ 1 , λ 2 , where λ 1 ≥ λ 2 > 0 . It provides a natural 2 -parametric deformation family of the Riemannian homogeneous manifold Sol 3 = E 1 , 1 , g 1 , 1 which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in E 1 , 1 , g L λ 1 , λ 2 away from characteristic points and signed geodesic curvature for the Euclidean C 2 -smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with a general left-invariant metric.

The aim of this paper is to classify the singularities of caustics, which have implications for a wide range of physical applications, of translation surfaces. In addition, we give a particular study on ridge point, sub-parabolic ridge point, and constant curvature line on translation surface and we find that there is no elliptic umblic on translation surface.

We consider the sub-Lorentzian geometry of curves and surfaces in the Lie group E 1 , 1 . Firstly, as an application of Riemannian approximants scheme, we give the definition of Lorentzian approximants scheme for E 1 , 1 which is a sequence of Lorentzian manifolds denoted by E λ 1 , λ 2 L . By using the Koszul formula, we calculate the expressions of Levi-Civita connection and curvature tensor in the Lorentzian approximants of E λ 1 , λ 2 L in terms of the basis E 1 , E 2 , E 3 . These expressions will be used to define the notions of the intrinsic curvature for curves, the intrinsic geodesic curvature of curves on surfaces, and the intrinsic Gaussian curvature of surfaces away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove two generalized Gauss-Bonnet theorems in E λ 1 , λ 2 L .

In this paper, we consider the Lorentzian approximations of rigid motions of the Minkowski plane E L 2 1,1 . By using the method of Lorentzian approximations, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surface, and the intrinsic Gaussian curvature of Lorentzian surface in E 1,1 with the second Lorentzian metric away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss–Bonnet theorem for the Lorentzian surface in E 1,1 with the second left-invariant Lorentzian metric g 2 .

The evolutes of regular curves in the Euclidean plane are given by the caustics of regular curves. In this paper, we define the generalized evolutes of planar curves which are spatial curves, and the projection of generalized evolutes along a fixed direction are the evolutes. We also prove that the generalized evolutes are the locus of centers of slant circles of the curvature of planar curves. Moreover, we define the generalized parallels of planar curves and show that the singular points of generalized parallels sweep out the generalized evolute. In general, we cannot define the generalized evolutes at the singular points of planar curves, but we can define the generalized evolutes of fronts by using moving frames along fronts and curvatures of the Legendre immersion. Then we study the behaviors of generalized evolutes at the singular points of fronts. Finally, we give some examples to show the generalized evolutes.

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