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.

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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 .