Extended Legendrian Dualities Theorem in Singularity Theory

Abstract

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.

1. Introduction

Singularity theory is a young branch of analysis that currently occupies a central place in mathematics. It is a wide-ranging generalization of the theory of the minima and maxima of functions and a descendant of differential calculus. Whitney firstly noticed this field and Mather set its foundations by the actions of group theory, which is a powerful tool to study symmetry [1,2]. Arnold and Zakalyukin developed the theory of singularity from the viewpoints of symplectic geometry and contact geometry [3]. Since most of the important properties of submanifolds are characterized and distinguished by their singularities, it is important to deal with singularities. However, there are many difficulties in doing this, because the usual mathematical tools fail at the singularity. With the rapid development of singularity theory, it has acted as a microscope to observe the geometric and topological properties of submanifolds near the singularities. Many mathematicians have contributed to this field, including Thom, Porteous, Bruce, Giblin, Izumiya, Romer Fuster, Tari, Pei and Chen, etc. [2,3]. They have considered geometric and topological properties caustics and wavefronts, including evolute, parallel curve, pedal curve, symmetry sets, Gauss map, focal surface, parallel surface, umbilic, foliations, etc. There are two typical applications of singularity theory in symmetry. The first one is that Gutsu gave the notion of the simple symmetric function of germs at a critical point, which are reducible to normal forms by the action of the group of symmetric diffeomorphisms in [3]. The normal forms of the simple symmetric function germs are classified in [3], where more details on many questions of the theory of symmetric critical points can be found. Another interesting application of singularity to symmetry is the singularities of symmetry sets. A symmetric set of a curve (respectively, surface) arises as the locus of centers of circles (respectively, spheres), which have contact with the curve (respectively, surface) in two places. A local version of a symmetric set can be found in [4], where the authors give many examples and trace symmetric sets by using a computer. The main results indicate that there are many simple singularities in symmetric sets, which can be detected by the powerful tool of the theory of singularity. For example, the symmetric sets of quartic curves consist of many cusps. It is incredible that these quartic curves are not closed, but the parts of their symmetric sets would not be affected by closing them up. Hereafter, we focus on the Legendrian duality theorem, which is one of the most important results in singularity theory. It has been an important tool to study the geometric properties of degenerate submanifolds. 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 2007, Izumiya showed four Legendrian dualities between pseudo-spheres in Minkowski space [5]. Then, he and his coauthors extended them to the cases of semi-Euclidean spaces with general index [6], one-parameter families of pseudo-spheres in Lorentz–Minkowski space [7], and the spherical Legendrian duality [8]. It is well known that Legendrian dualities provide a new way of constructing frames from the viewpoints of contact geometry and Legendrian singularity theory, which have been widely used for studying the geometric properties of curves, surfaces and other submanifolds with singularities in Euclidean and pseudo-Euclidean spaces. These dualities have become a core tool for studying the geometric and topological properties of submanifolds with singularities. Some typical applications are the research of submanifolds with singularities in Euclidean space [8,9], Minkowski space [10,11,12], Anti-de Sitter space [13,14,15,16,17] and pseudo-spheres [18,19], respectively. Recently, we conducted some works on applications of Legendrian duality theory [20,21,22,23,24,25].

In this paper, we extend the theorem of Legendrian dualities to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space and add some new information. We construct all contact diffeomorphisms among the contact manifolds and display them in a

Table 1. A table on Legendrian dualities.

	${\mathbf{\Delta }}_1$	${\mathbf{\Delta }}_2^{\pm }$	${\mathbf{\Delta }}_3^{\pm }$	${\mathbf{\Delta }}_4^{\pm }$	${\mathbf{\Delta }}_{12}^{\pm }\left(\mathbf{\theta }\right)$	${\mathbf{\Delta }}_{13}^{\pm }\left(\mathbf{\theta }\right)$	${\mathbf{\Delta }}_{14}^{\pm }\left(\mathbf{\theta }\right)$	${\mathbf{\Delta }}_{23}^{\pm }\left(\mathbf{\theta }\right)$	${\mathbf{\Delta }}_{24}^{\pm }\left(\mathbf{\theta }\right)$	${\mathbf{\Delta }}_{34}^{\pm }\left(\mathbf{\theta }\right)$
${\Delta }_1$	$id$	${\mathcal{L}}_{12}^{\pm }$	${\mathcal{L}}_{13}^{\pm }$	${\mathcal{L}}_{14}^{\pm }$	${\mathcal{L}}_{1\left(12\right)}^{\pm }$	${\mathcal{L}}_{1\left(13\right)}^{\pm }$	${\mathcal{L}}_{1\left(14\right)}^{\pm }$	${\mathcal{L}}_{1\left(23\right)}^{\pm }$	${\mathcal{L}}_{1\left(24\right)}^{\pm }$	${\mathcal{L}}_{1\left(34\right)}^{\pm }$
${\Delta }_2^{\pm }$	${\mathcal{L}}_{21}^{\pm }$	$id$	${\mathcal{L}}_{23}^{\pm }$	${\mathcal{L}}_{24}^{\pm }$	${\mathcal{L}}_{2\left(12\right)}^{\pm }$	${\mathcal{L}}_{2\left(13\right)}^{\pm }$	${\mathcal{L}}_{2\left(14\right)}^{\pm }$	${\mathcal{L}}_{2\left(23\right)}^{\pm }$	${\mathcal{L}}_{2\left(24\right)}^{\pm }$	${\mathcal{L}}_{2\left(34\right)}^{\pm }$
${\Delta }_3^{\pm }$	${\mathcal{L}}_{31}^{\pm }$	${\mathcal{L}}_{32}^{\pm }$	$id$	${\mathcal{L}}_{34}^{\pm }$	${\mathcal{L}}_{3\left(12\right)}^{\pm }$	${\mathcal{L}}_{3\left(13\right)}^{\pm }$	${\mathcal{L}}_{3\left(14\right)}^{\pm }$	${\mathcal{L}}_{3\left(23\right)}^{\pm }$	${\mathcal{L}}_{3\left(24\right)}^{\pm }$	${\mathcal{L}}_{3\left(34\right)}^{\pm }$
${\Delta }_4^{\pm }$	${\mathcal{L}}_{41}^{\pm }$	${\mathcal{L}}_{42}^{\pm }$	${\mathcal{L}}_{43}^{\pm }$	$id$	${\mathcal{L}}_{4\left(12\right)}^{\pm }$	${\mathcal{L}}_{4\left(13\right)}^{\pm }$	${\mathcal{L}}_{4\left(14\right)}^{\pm }$	${\mathcal{L}}_{4\left(23\right)}^{\pm }$	${\mathcal{L}}_{4\left(24\right)}^{\pm }$	${\mathcal{L}}_{4\left(34\right)}^{\pm }$
${\Delta }_{12}^{\pm }\left(\theta \right)$	${\mathcal{L}}_{\left(12\right)1}^{\pm }$	${\mathcal{L}}_{\left(12\right)2}^{\pm }$	${\mathcal{L}}_{\left(12\right)3}^{\pm }$	${\mathcal{L}}_{\left(12\right)4}^{\pm }$	$id$	${\mathcal{L}}_{\left(12\right)\left(13\right)}^{\pm }$	${\mathcal{L}}_{\left(12\right)\left(14\right)}^{\pm }$	${\mathcal{L}}_{\left(12\right)\left(23\right)}^{\pm }$	${\mathcal{L}}_{\left(12\right)\left(24\right)}^{\pm }$	${\mathcal{L}}_{\left(12\right)\left(34\right)}^{\pm }$
${\Delta }_{13}^{\pm }\left(\theta \right)$	${\mathcal{L}}_{\left(13\right)1}^{\pm }$	${\mathcal{L}}_{\left(13\right)2}^{\pm }$	${\mathcal{L}}_{\left(13\right)3}^{\pm }$	${\mathcal{L}}_{\left(13\right)4}^{\pm }$	${\mathcal{L}}_{\left(13\right)\left(12\right)}^{\pm }$	$id$	${\mathcal{L}}_{\left(13\right)\left(14\right)}^{\pm }$	${\mathcal{L}}_{\left(13\right)\left(23\right)}^{\pm }$	${\mathcal{L}}_{\left(13\right)\left(24\right)}^{\pm }$	${\mathcal{L}}_{\left(13\right)\left(34\right)}^{\pm }$
${\Delta }_{14}^{\pm }\left(\theta \right)$	${\mathcal{L}}_{\left(14\right)1}^{\pm }$	${\mathcal{L}}_{\left(14\right)2}^{\pm }$	${\mathcal{L}}_{\left(14\right)3}^{\pm }$	${\mathcal{L}}_{\left(14\right)4}^{\pm }$	${\mathcal{L}}_{\left(14\right)\left(12\right)}^{\pm }$	${\mathcal{L}}_{\left(14\right)\left(13\right)}^{\pm }$	$id$	${\mathcal{L}}_{\left(14\right)\left(23\right)}^{\pm }$	${\mathcal{L}}_{\left(14\right)\left(24\right)}^{\pm }$	${\mathcal{L}}_{\left(14\right)\left(34\right)}^{\pm }$
${\Delta }_{23}^{\pm }\left(\theta \right)$	${\mathcal{L}}_{\left(23\right)1}^{\pm }$	${\mathcal{L}}_{\left(23\right)2}^{\pm }$	${\mathcal{L}}_{\left(23\right)3}^{\pm }$	${\mathcal{L}}_{\left(23\right)4}^{\pm }$	${\mathcal{L}}_{\left(23\right)\left(12\right)}^{\pm }$	${\mathcal{L}}_{\left(23\right)\left(13\right)}^{\pm }$	${\mathcal{L}}_{\left(23\right)\left(14\right)}^{\pm }$	$id$	${\mathcal{L}}_{\left(23\right)\left(24\right)}^{\pm }$	${\mathcal{L}}_{\left(23\right)\left(34\right)}^{\pm }$
${\Delta }_{24}^{\pm }\left(\theta \right)$	${\mathcal{L}}_{\left(24\right)1}^{\pm }$	${\mathcal{L}}_{\left(24\right)2}^{\pm }$	${\mathcal{L}}_{\left(24\right)3}^{\pm }$	${\mathcal{L}}_{\left(24\right)4}^{\pm }$	${\mathcal{L}}_{\left(24\right)\left(12\right)}^{\pm }$	${\mathcal{L}}_{\left(24\right)\left(13\right)}^{\pm }$	${\mathcal{L}}_{\left(24\right)\left(14\right)}^{\pm }$	${\mathcal{L}}_{\left(24\right)\left(23\right)}^{\pm }$	$id$	${\mathcal{L}}_{\left(24\right)\left(34\right)}^{\pm }$
${\Delta }_{34}^{\pm }\left(\theta \right)$	${\mathcal{L}}_{\left(34\right)1}^{\pm }$	${\mathcal{L}}_{\left(34\right)2}^{\pm }$	${\mathcal{L}}_{\left(34\right)3}^{\pm }$	${\mathcal{L}}_{\left(34\right)4}^{\pm }$	${\mathcal{L}}_{\left(34\right)\left(12\right)}^{\pm }$	${\mathcal{L}}_{\left(34\right)\left(13\right)}^{\pm }$	${\mathcal{L}}_{\left(34\right)\left(14\right)}^{\pm }$	${\mathcal{L}}_{\left(34\right)\left(23\right)}^{\pm }$	${\mathcal{L}}_{\left(34\right)\left(24\right)}^{\pm }$	$id.$

on Legendrian dualities. In particular, we calculate the general expressions of contact diffeomorphisms ${\mathcal{L}}_{2\left(\alpha \beta \right)}^{\pm }:{\Delta }_2^{\pm }⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)$ and their converse mappings ${\mathcal{L}}_{\left(\alpha \beta \right)2}^{\pm }:{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)⟶{\Delta }_2^{\pm }.$ We take ${\mathcal{L}}_{2\left(\alpha \beta \right)}^{\pm }:{\Delta }_2^{\pm }⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)$ as an example to illustrate the symbols in this paper. In fact, ${\mathcal{L}}_{2\left(\alpha \beta \right)}^{\pm }:{\Delta }_2^{\pm }⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)$ denotes that ${\mathcal{L}}_{2\left(\alpha \beta \right)}^{+}:{\Delta }_2^{+}⟶{\Delta }_{\alpha \beta }^{+}\left(\theta \right)$ or ${\mathcal{L}}_{2\left(\alpha \beta \right)}^{-}:{\Delta }_2^{-}⟶{\Delta }_{\alpha \beta }^{-}\left(\theta \right),$ ${\Delta }_2^{\pm }$ denotes ${\Delta }_2^{+}$ or ${\Delta }_2^{-}$, where

$${\mathbb{H}}_{r-1}^n(-1)\times {\Lambda }^n\supset {\Delta }_2^{+}=\{(\mathbf{n},\mathbf{b})\mid {\langle \mathbf{n},\mathbf{b}\rangle }_r=+1\}$$

and

$${\mathbb{H}}_{r-1}^n(-1)\times {\Lambda }^n\supset {\Delta }_2^{-}=\{(\mathbf{n},\mathbf{b})\mid {\langle \mathbf{n},\mathbf{b}\rangle }_r=-1\}.$$

We remark that a null vector $\mathbf{b}$ can be constructed by a timelike vector $\mathbf{n}$ and a spacelike vector $\mathbf{e},$ where ${\langle \mathbf{n},\mathbf{n}\rangle }_r=-1$, ${\langle \mathbf{e},\mathbf{e}\rangle }_r=1$ and ${\langle \mathbf{n},\mathbf{e}\rangle }_r=0.$ If we take $\mathbf{b}=\mathbf{e}-\mathbf{n}$, then ${\langle \mathbf{n},\mathbf{b}\rangle }_r=+1.$ However, if we take $\mathbf{b}=\mathbf{e}+\mathbf{n}$, then ${\langle \mathbf{n},\mathbf{b}\rangle }_r=-1.$ Therefore, we give the definitions of ${\Delta }_2^{+}$ and ${\Delta }_2^{-}$, respectively, and denote them by ${\Delta }_2^{\pm }.$ Other cases follow in the same way. More detailed definitions can be found in Section 2.

We also calculate ${\mathcal{L}}_{3\left(\alpha \beta \right)}^{\pm }:{\Delta }_3^{\pm }⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)$ and their converse mappings ${\mathcal{L}}_{\left(\alpha \beta \right)3}^{\pm }:{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)⟶{\Delta }_3^{\pm },$ ${\mathcal{L}}_{4\left(\alpha \beta \right)}^{\pm }:{\Delta }_4^{\pm }⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)$ and their converse mappings ${\mathcal{L}}_{\left(\alpha \beta \right)4}^{\pm }:{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)⟶{\Delta }_4^{\pm }.$ Finally, we construct expressions of ${\mathcal{L}}_{\left(\alpha \beta \right)\left(\gamma \delta \right)}^{\pm }:{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)⟶{\Delta }_{\gamma \delta }^{\pm }\left(\theta \right)$ and their converse mappings

$${\mathcal{L}}_{\left(\gamma \delta \right)\left(\alpha \beta \right)}^{\pm }:{\Delta }_{\gamma \delta }^{\pm }\left(\theta \right)⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)(\alpha ,\beta ,\gamma ,\delta =1,2,3,4,\alpha <\beta ,\gamma <\delta ).$$

Thus, the table of Legendrian dualities contains all information about Legendrian dualities. The mapping in this table has some operational rules. Firstly, the mapping of the main diagonal symmetry is a diffeomorphism that is inverse to each other. For example, ${\mathcal{L}}_{12}^{\pm }\circ {\mathcal{L}}_{21}^{\pm }=id$ and ${\mathcal{L}}_{21}^{\pm }\circ {\mathcal{L}}_{12}^{\pm }=id.$ Secondly, starting from any of the ten contact manifolds, we can always calculate the contact diffeomorphism between the main diagonal symmetry mapping and any other manifold by using the composite operation of the mapping in the table. Contact diffeomorphisms are used to establish the contact diffeomorphism relations between these manifolds. We give the general expressions of these contact diffeomorphisms.

In Section 2, we give some basic concepts. In Section 3, we extend the theorem of Legendrian dualities to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space. To do this, we construct some new contact diffeomorphisms among the contact manifolds. In Section 4, we give two applications of the extended Legendrian dualities theorem. In Section 5, we summarize this paper.

2. Basic Notions on Legendrian Dualities in Semi-Euclidean Space

In this section, we introduce some basic notions of Legendrian dualities in semi-Euclidean $(n+1)$-space with index r. Let ${\mathbb{R}}^{n+1}=\{({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_{n+1})\mid {\alpha }_{\tau }\in \mathbb{R},\tau =1,2,\dots ,n+1\}$ be an $(n+1)$-dimensional vector space. For any vectors $\alpha =({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_{n+1})$ and $\beta =({\beta }_1,{\beta }_2,\cdots ,{\beta }_{n+1})$ in ${\mathbb{R}}^{n+1},$ the pseudo scalar product of $\alpha$ and $\beta$ is defined by

$${\langle \alpha ,\beta \rangle }_r=-\sum _{\tau =1}^r{\alpha }_{\tau }{\beta }_{\tau }+\sum _{\tau =r+1}^{n+1}{\alpha }_{\tau }{\beta }_{\tau }.$$

We call $({\mathbb{R}}^{n+1},{\langle ,\rangle }_r)$ a semi-Euclidean $(n+1)$-space with index r and denote it by ${\mathbb{R}}_r^{n+1}$. A non-zero vector $\alpha \in {\mathbb{R}}_r^{n+1}$ is called spacelike, null or timelike if ${\langle \alpha ,\alpha \rangle }_r>0$, ${\langle \alpha ,\alpha \rangle }_r=0$ or ${\langle \alpha ,\alpha \rangle }_r<0$, respectively. The norm of the vector $\mathbf{v}\in {\mathbb{R}}_r^{n+1}$ is defined by $\Vert \mathbf{v}\Vert =\sqrt{{|\langle \mathbf{v},\mathbf{v}\rangle }_r|}$. We now define the n-hyperbolic space with index $r-1$ by ${\mathbb{H}}_{r-1}^n(-c^2)=\{\mathbf{v}\in {\mathbb{R}}_r^{n+1}\mid {\langle \mathbf{v},\mathbf{v}\rangle }_r=-c^2\},$ a unit pseudo n-sphere with index r by ${\mathbb{S}}_r^n\left(c^2\right)=\{\mathbf{v}\in {\mathbb{R}}_r^{n+1}\mid {\langle \mathbf{v},\mathbf{v}\rangle }_r=c^2\};$ and the open nullcone by ${\Lambda }^n=\{\mathbf{v}\in {\mathbb{R}}_r^{n+1}\backslash \left\{0\right\}\mid {\langle \mathbf{v},\mathbf{v}\rangle }_r=0\}$ for any real number $c.$ We also need some basic notions of contact geometry. A $(2n+1)$-dimensional manifold E with a contact structure K is called a contact manifold and is denoted by $(E,K).$ Let $\mathcal{L}$ be an n-dimensional submanifold of E; if the tangent space of $\mathcal{L}$ at any point p is a subspace of $K_p,$ we call $\mathcal{L}$ a Legendrian submaifold. If the fibers of the fiber bundle $\pi :E\to N$ are Legendrian submanifolds of E, we call $\pi$ a Legendrian fibration.

For our purpose, we should consider the following extended Legendrian dualities in general semi-Euclidean space.

$$\begin{array}{cc}\hfill \left(1\right)\left(\alpha \right)& {\mathbb{H}}_{r-1}^n(-1)\times {\mathbb{S}}_r^n\left(1\right)\supset {\Delta }_1=\{(\mathbf{n},\mathbf{b})\mid {\langle \mathbf{n},\mathbf{b}\rangle }_r=0\},\hfill \\ \hfill \left(\beta \right)& {\pi }_{11}:{\Delta }_1⟶{\mathbb{H}}_{r-1}^n(-1),{\pi }_{12}:{\Delta }_1⟶{\mathbb{S}}_r^n\left(1\right),\hfill \\ \hfill \left(\gamma \right)& {\eta }_{11}={\langle d\mathbf{n},\mathbf{b}\rangle }_r{\mid }_{{\Delta }_1},{\eta }_{12}={\langle \mathbf{n},d\mathbf{b}\rangle }_r{\mid }_{{\Delta }_1}.\hfill \\ \hfill \left(2\right)\left(\alpha \right)& {\mathbb{H}}_{r-1}^n(-1)\times {\Lambda }^n\supset {\Delta }_2^{\pm }=\{(\mathbf{n},\mathbf{b})\mid {\langle \mathbf{n},\mathbf{b}\rangle }_r=\pm 1\},\hfill \\ \hfill \left(\beta \right)& {\pi }_{21}^{\pm }:{\Delta }_2^{\pm }⟶{\mathbb{H}}_{r-1}^n(-1),{\pi }_{22}^{\pm }:{\Delta }_2^{\pm }⟶{\Lambda }^n,\hfill \\ \hfill \left(\gamma \right)& {\eta }_{21}^{\pm }={\langle d\mathbf{n},\mathbf{b}\rangle }_r{\mid }_{{\Delta }_2^{\pm }},{\eta }_{22}^{\pm }={\langle \mathbf{n},d\mathbf{b}\rangle }_r{\mid }_{{\Delta }_2^{\pm }}.\hfill \\ \hfill \left(3\right)\left(\alpha \right)& {\Lambda }^n\times {\mathbb{S}}_r^n\left(1\right)\supset {\Delta }_3^{\pm }=\{(\mathbf{n},\mathbf{b})\mid {\langle \mathbf{n},\mathbf{b}\rangle }_r=\pm 1\},\hfill \\ \hfill \left(\beta \right)& {\pi }_{31}^{\pm }:{\Delta }_3^{\pm }⟶{\Lambda }^n,{\pi }_{32}^{\pm }:{\Delta }_3^{\pm }⟶{\mathbb{S}}_r^n\left(1\right),\hfill \\ \hfill \left(\gamma \right)& {\eta }_{31}^{\pm }={\langle d\mathbf{n},\mathbf{b}\rangle }_r{\mid }_{{\Delta }_3^{\pm }},{\eta }_{32}^{\pm }={\langle \mathbf{n},d\mathbf{b}\rangle }_r{\mid }_{{\Delta }_3^{\pm }}.\hfill \\ \hfill \left(4\right)\left(\alpha \right)& {\Lambda }^n\times {\Lambda }^n\supset {\Delta }_4^{\pm }=\{(\mathbf{n},\mathbf{b})\mid {\langle \mathbf{n},\mathbf{b}\rangle }_r=\pm 2\},\hfill \\ \hfill \left(\beta \right)& {\pi }_{41}^{\pm }:{\Delta }_4^{\pm }⟶{\Lambda }^n,{\pi }_{42}^{\pm }:{\Delta }_3^{\pm }⟶{\Lambda }^n,\hfill \\ \hfill \left(\gamma \right)& {\eta }_{41}^{\pm }={\langle d\mathbf{n},\mathbf{b}\rangle }_r{\mid }_{{\Delta }_4^{\pm }},{\eta }_{42}^{\pm }={\langle \mathbf{n},d\mathbf{b}\rangle }_r{\mid }_{{\Delta }_4^{\pm }}.\hfill \\ \hfill \left(5\right)\left(\alpha \right)& {\mathbb{H}}_{r-1}^n(-1)\times {\mathbb{S}}_r^n\left(co{s}^2\theta \right)\supset {\Delta }_{12}^{\pm }\left(\theta \right)=\{(\mathbf{n},\mathbf{b})\mid {\langle \mathbf{n},\mathbf{b}\rangle }_r=\pm sin\theta \},\hfill \\ \hfill \left(\beta \right)& \pi {\left[\theta \right]}_{\left(12\right)1}^{\pm }:{\Delta }_{12}^{\pm }\left(\theta \right)⟶{\mathbb{H}}_{r-1}^n(-1),\pi {\left[\theta \right]}_{\left(12\right)2}^{\pm }:{\Delta }_{12}^{\pm }\left(\theta \right)⟶{\mathbb{S}}_r^n\left(co{s}^2\theta \right),\hfill \\ \hfill \left(\gamma \right)& \eta {\left[\theta \right]}_{\left(12\right)1}^{\pm }={\langle d\mathbf{n},\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{12}^{\pm }\left(\theta \right)},\eta {\left[\theta \right]}_{\left(12\right)2}^{\pm }={\langle \mathbf{n},d\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{12}^{\pm }\left(\theta \right)}.\hfill \\ \hfill \left(6\right)\left(\alpha \right)& {\mathbb{H}}_{r-1}^n(-co{s}^2\theta )\times {\mathbb{S}}_r^n\left(1\right)\supset {\Delta }_{13}^{\pm }\left(\theta \right)=\{(\mathbf{n},\mathbf{b})\mid {\langle \mathbf{n},\mathbf{b}\rangle }_r=\pm sin\theta \},\hfill \\ \hfill \left(\beta \right)& \pi {\left[\theta \right]}_{\left(13\right)1}^{\pm }:{\Delta }_{13}^{\pm }\left(\theta \right)⟶{\mathbb{H}}_{r-1}^n(-co{s}^2\theta ),\pi {\left[\theta \right]}_{\left(13\right)2}^{\pm }:{\Delta }_{13}^{\pm }\left(\theta \right)⟶{\mathbb{S}}_r^n\left(1\right),\hfill \\ \hfill \left(\gamma \right)& \eta {\left[\theta \right]}_{\left(13\right)1}^{\pm }={\langle d\mathbf{n},\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{13}^{\pm }\left(\theta \right)},\eta {\left[\theta \right]}_{\left(13\right)2}^{\pm }={\langle \mathbf{n},d\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{13}^{\pm }\left(\theta \right)}.\hfill \\ \hfill \left(7\right)\left(\alpha \right)& {\mathbb{H}}_{r-1}^n(-co{s}^2\theta )\times {\mathbb{S}}_r^n\left(co{s}^2\theta \right)\supset {\Delta }_{14}^{\pm }\left(\theta \right)=\{(\mathbf{n},\mathbf{b})\mid {\langle \mathbf{n},\mathbf{b}\rangle }_r=\pm 2sin\theta \},\hfill \\ \hfill \left(\beta \right)& \pi {\left[\theta \right]}_{\left(14\right)1}^{\pm }:{\Delta }_{14}^{\pm }\left(\theta \right)⟶{\mathbb{H}}_{r-1}^n(-co{s}^2\theta ),\pi {\left[\theta \right]}_{\left(14\right)2}^{\pm }:{\Delta }_{14}^{\pm }\left(\theta \right)⟶{\mathbb{S}}_r^n\left(co{s}^2\theta \right),\hfill \\ \hfill \left(\gamma \right)& \eta {\left[\theta \right]}_{\left(14\right)1}^{\pm }={\langle d\mathbf{n},\mathbf{w}\rangle }_r{\mid }_{{\Delta }_{14}^{\pm }\left(\theta \right)},\eta {\left[\theta \right]}_{\left(14\right)2}^{\pm }={\langle \mathbf{n},d\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{14}^{\pm }\left(\theta \right)}.\hfill \\ \hfill \left(8\right)\left(\alpha \right)& {\mathbb{H}}_{r-1}^n(-co{s}^2\theta )\times {\mathbb{S}}_r^n\left(si{n}^2\theta \right)\supset {\Delta }_{23}^{\pm }\left(\theta \right)=\{(\mathbf{n},\mathbf{b})\mid {\langle \mathbf{n},\mathbf{b}\rangle }_r=\pm (sin\theta +cos\theta )\},\hfill \\ \hfill \left(\beta \right)& \pi {\left[\theta \right]}_{\left(23\right)1}^{\pm }:{\Delta }_{23}^{\pm }\left(\theta \right)⟶{\mathbb{H}}_{r-1}^n(-co{s}^2\theta ),\pi {\left[\theta \right]}_{\left(23\right)2}^{\pm }:{\Delta }_{23}^{\pm }\left(\theta \right)⟶{\mathbb{S}}_r^n\left(si{n}^2\theta \right),\hfill \\ \hfill \left(\gamma \right)& \eta {\left[\theta \right]}_{\left(23\right)1}^{\pm }={\langle d\mathbf{n},\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{23}^{\pm }\left(\theta \right)},\eta {\left[\theta \right]}_{\left(23\right)2}^{\pm }={\langle \mathbf{n},d\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{23}^{\pm }\left(\theta \right)}.\hfill \\ \hfill \left(9\right)\left(\alpha \right)& {\mathbb{H}}_{r-1}^n(-co{s}^2\theta )\times {\Lambda }^n\supset {\Delta }_{24}^{\pm }\left(\theta \right)=\{(\mathbf{n},\mathbf{b})\mid {\langle \mathbf{n},\mathbf{b}\rangle }_r=\pm (sin\theta +1)\},\hfill \\ \hfill \left(\beta \right)& \pi {\left[\theta \right]}_{\left(24\right)1}^{\pm }:{\Delta }_{24}^{\pm }\left(\theta \right)⟶{\mathbb{H}}_{r-1}^n(-co{s}^2\theta ),\pi {\left[\theta \right]}_{\left(24\right)2}^{\pm }:{\Delta }_{24}^{\pm }\left(\theta \right)⟶{\Lambda }^n,\hfill \\ \hfill \left(\gamma \right)& \eta {\left[\theta \right]}_{\left(24\right)1}^{\pm }={\langle d\mathbf{n},\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{24}^{\pm }\left(\theta \right)},\eta {\left[\theta \right]}_{\left(24\right)2}^{\pm }={\langle \mathbf{n},d\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{24}^{\pm }\left(\theta \right)}.\hfill \\ \hfill \left(10\right)\left(\alpha \right)& {\Lambda }^n\times {\mathbb{S}}_r^n\left(co{s}^2\theta \right)\supset {\Delta }_{34}^{\pm }\left(\theta \right)=\{(\mathbf{n},\mathbf{b})\mid {\langle \mathbf{n},\mathbf{b}\rangle }_r=\pm (sin\theta +1)\},\hfill \\ \hfill \left(\beta \right)& \pi {\left[\theta \right]}_{\left(34\right)1}^{\pm }:{\Delta }_{34}^{\pm }\left(\theta \right)⟶{\Lambda }^n,\pi {\left[\theta \right]}_{\left(34\right)2}^{\pm }:{\Delta }_{34}^{\pm }\left(\theta \right)⟶{\mathbb{S}}_r^n\left(co{s}^2\theta \right),\hfill \\ \hfill \left(\gamma \right)& \eta {\left[\theta \right]}_{\left(34\right)1}^{\pm }={\langle d\mathbf{n},\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{34}^{\pm }\left(\theta \right)},\eta {\left[\theta \right]}_{\left(34\right)2}^{\pm }={\langle \mathbf{n},d\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{34}^{\pm }\left(\theta \right)}.\hfill \end{array}$$

where

$${\pi }_{11}(\mathbf{n},\mathbf{b})=\mathbf{n},{\pi }_{12}(\mathbf{n},\mathbf{b})=\mathbf{b},{\pi }_{i1}^{\pm }(\mathbf{n},\mathbf{b})=\mathbf{n},{\pi }_{\alpha 2}^{\pm }(\mathbf{n},\mathbf{b})=\mathbf{b}(\alpha =2,3,4),$$

$${\langle d\mathbf{n},\mathbf{b}\rangle }_r=-\sum _{i=1}^r{b}_{i}d{n}_i+\sum _{i=r+1}^{n+1}{b}_{i}d{n}_i,{\langle \mathbf{n},d\mathbf{b}\rangle }_r=-\sum _{i=1}^r{n}_{i}d{b}_i+\sum _{i=3}^{n+1}{n}_{i}d{b}_i$$

are 1-forms on ${\mathbb{R}}_r^{n+1}\times {\mathbb{R}}_r^{n+1}.$ It is easy to know that ${\eta }_{11}^{-1}\left(0\right)$ and ${\eta }_{12}^{-1}\left(0\right)$ define the same tangent hyperplane on ${\Delta }_1,$ denoted by $K_1.$ For the same reason, ${\left({\eta }_{\alpha 1}^{\pm }\right)}^{-1}\left(0\right)$ and ${\left({\eta }_{\alpha 2}^{\pm }\right)}^{-1}\left(0\right)$ define the same tangent hyperplane on ${\Delta }_{\alpha }^{\pm },$ denoted by $K_{\alpha }^{\pm }(\alpha =2,3,4).$ Note that $\pi {\left[\theta \right]}_{\left(\alpha \beta \right)1}^{\pm }(\mathbf{n},\mathbf{b})=\mathbf{n},\pi {\left[\theta \right]}_{\left(\alpha \beta \right)2}^{\pm }(\mathbf{n},\mathbf{b})=\mathbf{b}.$ ${\left(\eta {\left[\theta \right]}_{\left(\alpha \beta \right)1}^{\pm }\right)}^{-1}\left(0\right)$ and ${\left(\eta {\left[\theta \right]}_{\left(\alpha \beta \right)2}^{\pm }\right)}^{-1}\left(0\right)$ also define the same tangent hyperplane $K{\left[\theta \right]}_{\left(\alpha \beta \right)}^{\pm }(\alpha ,\beta =1,2,3,4;\alpha <\beta ).$

3. Extended Legendrian Dualities Theorem

In this section, we extend the theorem of Legendrian dualities to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space. We can obtain the following extended theorem.

Theorem 1. 

$({\Delta }_1,K_1)$, $({\Delta }_2^{\pm },K_2^{\pm })$,$({\Delta }_3^{\pm },K_3^{\pm })$, $({\Delta }_4^{\pm },K_4^{\pm })$, $({\Delta }_{12}^{\pm }\left[\theta \right],K{\left[\theta \right]}_{\left(12\right)}^{\pm })$, $({\Delta }_{13}^{\pm }\left[\theta \right],K{\left[\theta \right]}_{\left(13\right)}^{\pm })$, $({\Delta }_{14}^{\pm }\left[\theta \right],K{\left[\theta \right]}_{\left(14\right)}^{\pm })$,$({\Delta }_{23}^{\pm }\left[\theta \right],K{\left[\theta \right]}_{\left(23\right)}^{\pm })$,$({\Delta }_{24}^{\pm }\left[\theta \right],K{\left[\theta \right]}_{\left(24\right)}^{\pm })$ and $({\Delta }_{34}^{\pm }\left[\theta \right],K{\left[\theta \right]}_{\left(34\right)}^{\pm })$ are contact manifolds such that ${\pi }_{1\beta }$,${\pi }_{2\beta }^{\pm }$, ${\pi }_{3\beta }^{\pm }$, ${\pi }_{4\beta }^{\pm }$, $\pi {\left[\theta \right]}_{\left(12\right)\beta }^{\pm }$, $\pi {\left[\theta \right]}_{\left(13\right)\beta }^{\pm }$, $\pi {\left[\theta \right]}_{\left(14\right)\beta }^{\pm }$, $\pi {\left[\theta \right]}_{\left(23\right)\beta }^{\pm }$, $\pi {\left[\theta \right]}_{\left(24\right)\beta }^{\pm }$, and $\pi {\left[\theta \right]}_{\left(34\right)\beta }^{\pm }$ $(\beta =1,2)$ are Legendrian fibrations. Moreover, the above manifolds are contact diffeomorphic to each other.

Proof. 

First, we consider $({\Delta }_1,K_1)$, $({\Delta }_{\alpha }^{\pm },K_{\alpha }^{\pm })(\alpha =2,3,4).$ By definition, we can show that ${\Delta }_1$ and ${\Delta }_{\alpha 1}^{\pm },\alpha =2,3,4,$ are smooth submanifolds in ${\mathbb{R}}_r^{n+1}\times {\mathbb{R}}_r^{n+1}$, and all of ${\pi }_{1\beta }$ and ${\pi }_{\alpha \beta }^{\pm },\beta =1,2,$ are smooth Legendrian fibrations. It also follows from the definition of ${\theta }_{\alpha \beta }$ that each fiber of ${\pi }_{\alpha \beta }$ is an integral submanifold of $K_{\alpha }^{\pm },\alpha =1,2,3,4.$ In [6], it was shown that $({\Delta }_1,{\eta }_{11}^{-1}\left(0\right))$ is a contact manifold. We need to prove that $({\Delta }_{\alpha 1}^{\pm },K_{\alpha }^{\pm }),\alpha =2,3,4,$ are contact manifolds. To do this, we can prove that they are contact diffeomorphic to $({\Delta }_1,{\eta }_{11}^{-1}\left(0\right)).$ We construct smooth mappings ${\mathcal{L}}_{1\alpha }^{\pm }:{\Delta }_1⟶{\Delta }_{\alpha 1}^{\pm }$ and their inverse mappings ${\mathcal{L}}_{\alpha 1}^{\pm }:{\Delta }_{\alpha 1}^{\pm }⟶{\Delta }_1$ as follows:

$$\begin{array}{cc}\hfill {\mathcal{L}}_{12}^{\pm }(\mathbf{n},\mathbf{b})& =(\mathbf{n},\pm \mathbf{n}+\mathbf{b}),{\mathcal{L}}_{21}^{\pm }(\mathbf{n},\mathbf{b})=(\mathbf{n},\mp \mathbf{n}+\mathbf{b}),\hfill \\ \hfill {\mathcal{L}}_{13}^{\pm }(\mathbf{n},\mathbf{b})& =(\mathbf{n}\pm \mathbf{b},\mathbf{b}),{\mathcal{L}}_{31}^{\pm }(\mathbf{n},\mathbf{b})=(\mathbf{n}\mp \mathbf{b},\mathbf{b}),\hfill \\ \hfill {\mathcal{L}}_{14}^{\pm }(\mathbf{n},\mathbf{b})& =(\mathbf{n}\pm \mathbf{b},\mp \mathbf{n}+\mathbf{b}),{\mathcal{L}}_{41}^{\pm }(\mathbf{n},\mathbf{b})=(1/2)(\mathbf{n}\mp \mathbf{b},\pm \mathbf{n}+\mathbf{b}).\hfill \end{array}$$

We can also construct smooth mappings ${\mathcal{L}}_{\alpha \beta }^{\pm }:{\Delta }_{\alpha }^{\pm }⟶{\Delta }_{\beta }^{\pm }$ and their inverse mappings ${\mathcal{L}}_{\beta \alpha }^{\pm }:{\Delta }_{\beta }^{\pm }⟶{\Delta }_{\alpha }^{\pm },\alpha ,\beta =2,3,4,$ as follows:

$$\begin{array}{cc}\hfill {\mathcal{L}}_{23}^{\pm }(\mathbf{n},\mathbf{b})& =(\pm \mathbf{b},\mp \mathbf{n}+\mathbf{b}),{\mathcal{L}}_{32}^{\pm }(\mathbf{n},\mathbf{b})=(\mathbf{n}\mp \mathbf{b},\pm \mathbf{n}),\hfill \\ \hfill {\mathcal{L}}_{24}^{\pm }(\mathbf{n},\mathbf{b})& =(\pm \mathbf{b},\mp 2\mathbf{n}+\mathbf{b}),{\mathcal{L}}_{42}^{\pm }(\mathbf{n},\mathbf{b})=[(1/2)(\mathbf{n}\mp \mathbf{b}),\pm \mathbf{n}],\hfill \\ \hfill {\mathcal{L}}_{34}^{\pm }(\mathbf{n},\mathbf{b})& =(\mathbf{n},\mp \mathbf{n}+2\mathbf{b}),{\mathcal{L}}_{43}^{\pm }(\mathbf{n},\mathbf{b})=[\mathbf{n},(1/2)(\pm \mathbf{n}+\mathbf{b})].\hfill \end{array}$$

Therefore, they are diffeomorphisms. We should check that they are contact diffeomorphisms. Taking ${\mathcal{L}}_{13}^{\pm }$, for example, we have

$$\begin{array}{cc}\hfill {\mathcal{L}}_{13}^{\pm *}{\eta }_{31}^{\pm }(\mathbf{n},\mathbf{b})& ={\eta }_{31}^{\pm }(\mathbf{n}\pm \mathbf{b},\mathbf{b})\hfill \\ \hfill & =\langle d(\mathbf{n}\pm \mathbf{b}),\mathbf{b}\rangle {\mid }_{{\Delta }_1}\hfill \\ \hfill & =\langle d\mathbf{n},\mathbf{b}\rangle {\mid }_{{\Delta }_1}\pm \langle d\mathbf{b},\mathbf{w}\rangle {\mid }_{{\Delta }_1}\hfill \\ \hfill & =\langle d\mathbf{n},\mathbf{b}\rangle {\mid }_{{\Delta }_1}\hfill \\ \hfill & ={\eta }_{11}.\hfill \end{array}$$

This indicates that each of $({\Delta }_{31}^{\pm },K_3^{\pm })$ is a contact manifold and each of ${\mathcal{L}}_{13}^{\pm }$ is a contact diffeomorphism. By similar calculations, $({\Delta }_{\alpha 1}^{\pm },K_{\alpha }^{\pm }),\alpha =2,4,$ are contact manifolds.

Second, we consider $({\Delta }_1,K_1)$ and $({\Delta }_{\alpha \beta }^{\pm }\left[\theta \right],K{\left[\theta \right]}_{\left(\alpha \beta \right)}^{\pm })(\alpha ,\beta =1,2,3,4;\alpha <\beta ).$ We can construct ${\mathcal{L}}_{1\left(\alpha \beta \right)}^{\pm }:{\Delta }_1^{\pm }⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)$ and their converse mappings ${\mathcal{L}}_{\left(\alpha \beta \right)1}^{\pm }:{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)⟶{\Delta }_1$ with $d{\mathcal{L}}_{\left(12\right)1}^{\pm }\left(K{\left[\theta \right]}_{\left(12\right)}^{\pm }\right)=K_1.$ For any $(\mathbf{n},\mathbf{b})\in {\Delta }_1,$ we have

$${\langle \mp sin\theta \mathbf{n}+\mathbf{b},\mp sin\theta \mathbf{n}+\mathbf{b}\rangle }_r=-si{n}^2\theta +1=co{s}^2\theta$$

and

$${\langle \mathbf{n},\mp sin\theta \mathbf{n}+\mathbf{b}\rangle }_r=\pm sin\theta .$$

Therefore, we find that ${\mathcal{L}}_{1\left(12\right)}^{\pm }\left({\Delta }_1\right)\subset {\Delta }_{12}^{\pm }\left(\theta \right).$ For any $(\mathbf{n},\mathbf{b})\in {\Delta }_{12}^{\pm }\left(\theta \right),$ we also have

$$\begin{array}{c}\hfill {\langle \pm sin\theta \mathbf{n}+\mathbf{b},\pm sin\theta \mathbf{n}+\mathbf{b}\rangle }_r=-si{n}^2\theta +2si{n}^2\theta +co{s}^2\theta =1\end{array}$$

and ${\langle \mathbf{n},\pm sin\theta \mathbf{n}+\mathbf{b}\rangle }_r=\mp sin\theta \pm sin\theta =0.$ Therefore, we have ${\mathcal{L}}_{\left(12\right)1}^{\pm }\left({\Delta }_{12}^{\pm }\left(\theta \right)\right)\subset {\Delta }_1.$ Thus, we find that ${\mathcal{L}}_{1\left(12\right)}^{\pm }\circ {\mathcal{L}}_{\left(12\right)1}^{\pm }{\mid }_{{\Delta }_{12}^{\pm }\left(\theta \right)}=i{d}_{{\Delta }_{12}^{\pm }\left(\theta \right)}$ and ${\mathcal{L}}_{\left(12\right)1}^{\pm }\circ {\mathcal{L}}_{1\left(12\right)}^{\pm }{\mid }_{{\Delta }_1}=i{d}_{{\Delta }_1}.$ We also have

$${\left({\mathcal{L}}_{\left(12\right)1}^{\pm }\right)}^{*}{\eta }_{11}={\langle d\mathbf{n},\pm sin\theta \mathbf{n}+\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{12}^{\pm }\left(\theta \right)}={\langle d\mathbf{n},\mathbf{b}\rangle }_r{\mid }_{{\Delta }_{12}^{\pm }\left(\theta \right)}=\eta {\left[\theta \right]}_{\left(12\right)1}^{\pm }.$$

Therefore, $K{\left[\theta \right]}_{\left(12\right)}^{\pm }$ is a contact structure on ${\Delta }_{12}^{\pm }\left(\theta \right)$ such that ${\mathcal{L}}_{1\left(12\right)}^{\pm }$ is a contact diffeomorphism. Then, for other cases, we consider smooth mappings ${\mathcal{L}}_{1\left(\alpha \beta \right)}^{\pm }:{\Delta }_1^{\pm }⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)$ and their converse mappings ${\mathcal{L}}_{\left(\alpha \beta \right)1}^{\pm }:{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)⟶{\Delta }_1.$ Moreover, we have to consider some mappings ${\mathcal{L}}_{\left(\alpha \beta \right)\left(\gamma \delta \right)}^{\pm }:{\Delta }_{\alpha \beta }^{\pm }⟶{\Delta }_{\gamma \delta }^{\pm }$ and their converse mappings

$${\mathcal{L}}_{\left(\gamma \delta \right)\left(\alpha \beta \right)}^{\pm }:{\Delta }_{\gamma \delta }^{\pm }⟶{\Delta }_{\alpha \beta }^{\pm }(\alpha ,\beta ,\gamma ,\delta =1,2,3,4,\alpha <\beta ,\gamma <\delta ).$$

We only prove that ${\mathcal{L}}_{\left(12\right)\left(14\right)}^{\pm }$ is a contact diffeomorphism as an example. For any $(\mathbf{n},\mathbf{b})\in {\Delta }_{12}^{\pm }\left(\theta \right),$ we have

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\left(12\right)\left(14\right)}^{\pm *}\eta {\left[\theta \right]}_{\left(14\right)1}^{\pm }(\mathbf{n},\mathbf{b})& =& \eta {\left[\theta \right]}_{\left(14\right)1}^{\pm }[(1+si{n}^2\theta )\mathbf{n}\pm sin\theta \mathbf{b},\mathbf{b}]\hfill \\ & =& {\langle d[(1+si{n}^2\theta )\mathbf{n}\pm sin\theta \mathbf{b}],\mathbf{b}\rangle }_r{\mid }_{\Delta {\left[\theta \right]}_{12}^{\pm }}\hfill \\ & =& (1+si{n}^2\theta ){\langle d\mathbf{n},\mathbf{b}\rangle }_r{\mid }_{\Delta {\left[\theta \right]}_{12}^{\pm }}\pm sin\theta {\langle d\mathbf{b},\mathbf{b}\rangle }_r{\mid }_{\Delta {\left[\theta \right]}_{12}^{\pm }}\hfill \\ & =& (1+si{n}^2\theta ){\langle d\mathbf{n},\mathbf{b}\rangle }_r{\mid }_{\Delta {\left[\theta \right]}_{12}^{\pm }}\hfill \\ & =& (1+si{n}^2\theta )\eta {\left[\theta \right]}_{\left(12\right)1}.\hfill \end{array}$$

This means that $(\Delta {\left[\theta \right]}_{14}^{\pm },K{\left[\theta \right]}_{14}^{\pm })$ is a contact manifold such that ${\mathcal{L}}_{\left(12\right)\left(14\right)}^{\pm }$ is a contact diffeomorphism. Finally, for other cases, we have a similar calculation, so that $(\Delta {\left[\theta \right]}_{\alpha \beta }^{\pm },K{\left[\theta \right]}_{\alpha \beta }^{\pm })$$(\alpha =1,2,3;\beta =2,3,4;\alpha <\beta )$ are contact manifolds. We remark that one has to construct the expressions of the contact diffeomorphisms ${\mathcal{L}}_{1\left(\alpha \beta \right)}^{\pm }:{\Delta }_1⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)$ and their converse mappings ${\mathcal{L}}_{\left(\alpha \beta \right)1}^{\pm }:{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)⟶{\Delta }_1$ as follows:

$$\begin{array}{l}{\mathcal{L}}_{1\left(12\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left(\mathbf{n},\mp sin\theta \mathbf{n}+\mathbf{b}\right),\\ {\mathcal{L}}_{\left(12\right)1}^{\pm }(\mathbf{n},\mathbf{b})=\left(\mathbf{n},\pm sin\theta \mathbf{n}+\mathbf{b}\right),\\ {\mathcal{L}}_{1\left(13\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left(\mathbf{n}\pm sin\theta \mathbf{b},\mathbf{b}\right),\\ {\mathcal{L}}_{\left(13\right)1}^{\pm }(\mathbf{n},\mathbf{b})=\left(\mathbf{n}\mp sin\theta \mathbf{b},\mathbf{b}\right),\\ {\mathcal{L}}_{1\left(14\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left(\mathbf{n}\pm sin\theta \mathbf{b},\mp sin\theta \mathbf{n}+\mathbf{b}\right),\\ {\mathcal{L}}_{\left(14\right)1}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{{sin}^2\theta +1}\left(\mathbf{n}\mp sin\theta \mathbf{b},\pm sin\theta \mathbf{n}+\mathbf{b}\right),\\ {\mathcal{L}}_{1\left(23\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left(\mathbf{n}\pm sin\theta \mathbf{b},\mp cos\theta \mathbf{n}+\mathbf{b}\right),\\ {\mathcal{L}}_{\left(23\right)1}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta cos\theta }\left(\mathbf{n}\mp sin\theta \mathbf{b},\pm cos\theta \mathbf{n}+\mathbf{b}\right),\\ {\mathcal{L}}_{1\left(24\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left(\mathbf{n}\pm sin\theta \mathbf{b},\mp \mathbf{n}+\mathbf{b}\right),\\ {\mathcal{L}}_{\left(24\right)1}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left(\mathbf{n}\mp sin\theta \mathbf{b},\pm \mathbf{n}+\mathbf{b}\right),\\ {\mathcal{L}}_{1\left(34\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left(\mathbf{n}\pm \mathbf{b},\mp sin\theta \mathbf{n}+\mathbf{b}\right),\\ {\mathcal{L}}_{\left(34\right)1}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left(\mathbf{n}\mp \mathbf{b},\pm sin\theta \mathbf{n}+\mathbf{b}\right).\end{array}$$

In addition, we need to construct the expressions of ${\mathcal{L}}_{2\left(\alpha \beta \right)}^{\pm }:{\Delta }_2^{\pm }⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)$ and their converse mappings ${\mathcal{L}}_{\left(\alpha \beta \right)2}^{\pm }:{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)⟶{\Delta }_2^{\pm }.$ They are contact diffeomorphisms and are denoted, respectively, by

$$\begin{array}{l}{\mathcal{L}}_{2\left(12\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left(\mathbf{n},\mp (1+sin\theta )\mathbf{n}+\mathbf{b}\right),\\ {\mathcal{L}}_{\left(12\right)2}^{\pm }(\mathbf{n},\mathbf{b})=\left(\mathbf{n},\pm (1+sin\theta )\mathbf{n}+\mathbf{b}\right),\\ {\mathcal{L}}_{2\left(13\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left((1-sin\theta )\mathbf{n}+\mathbf{w},\mp \mathbf{n}+\mathbf{b}\right),\\ {\mathcal{L}}_{\left(13\right)2}^{\pm }(\mathbf{n},\mathbf{b})=\left(\mathbf{n}\mp sin\theta \mathbf{b},\pm \mathbf{n}+(1-sin\theta )\mathbf{b}\right),\\ {\mathcal{L}}_{2\left(14\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1-sin\theta )\mathbf{n}\pm sin\theta \mathbf{b},\mp (1+sin\theta )\mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{\left(14\right)2}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{{sin}^2\theta +1}\left[\mathbf{n}\mp sin\theta \mathbf{b},\pm (1+sin\theta )\mathbf{n}+(1-sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{2\left(23\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1-sin\theta )\mathbf{n}\pm sin\theta \mathbf{b},\mp (1+cos\theta )\mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{\left(23\right)2}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta cos\theta }\left[\mathbf{n}\mp sin\theta \mathbf{b},\pm (1+cos\theta )\mathbf{n}\pm (1\mp sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{2\left(24\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1-sin\theta )\mathbf{n}\pm sin\theta \mathbf{b},\mp 2\mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{\left(24\right)2}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left[\mathbf{n}\mp sin\theta \mathbf{b},\pm 2\mathbf{n}+(1-sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{2\left(34\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\pm \mathbf{b},\mp (1+sin\theta )\mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{\left(34\right)2}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left[\mathbf{n}\mp \mathbf{b},\pm (1+sin\theta )\mathbf{n}\right].\end{array}$$

Furthermore, we also need to construct the expressions of ${\mathcal{L}}_{3\left(\alpha \beta \right)}^{\pm }:{\Delta }_3^{\pm }⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)$ and their converse mappings ${\mathcal{L}}_{\left(\alpha \beta \right)3}^{\pm }:{\Delta }_{ij}^{\pm }\left(\theta \right)⟶{\Delta }_3^{\pm }.$ They are contact diffeomorphisms and are denoted, respectively, by

$$\begin{array}{l}{\mathcal{L}}_{3\left(12\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n}\mp \mathbf{b},\mp sin\theta \mathbf{n}+(1+sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(12\right)3}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1+sin\theta )\mathbf{n}\pm \mathbf{b},\pm sin\theta \mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{3\left(13\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n}\mp (1-sin\theta )\mathbf{b},\mathbf{b}\right],\\ {\mathcal{L}}_{\left(13\right)3}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n}\pm (1-sin\theta )\mathbf{b},\mathbf{b}\right],\\ {\mathcal{L}}_{3\left(14\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n}\mp (1-sin\theta )\mathbf{b},\mp sin\theta \mathbf{n}+(1+sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(14\right)3}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{{sin}^2\theta +1}\left[(1+sin\theta )\mathbf{n}\pm (1-sin\theta )\mathbf{b},\pm sin\theta \mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{3\left(23\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n}\mp (1-sin\theta )\mathbf{b},(1\mp cos\theta )\mathbf{n}+cos\theta \mathbf{b}\right],\\ {\mathcal{L}}_{\left(23\right)3}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta cos\theta }\left[(1+cos\theta )\mathbf{n}\pm (1-sin\theta )\mathbf{b},\pm cos\theta \mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{3\left(24\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n}\mp (1-sin\theta )\mathbf{b},\mp \mathbf{n}+2\mathbf{b}\right],\\ {\mathcal{L}}_{\left(24\right)3}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left[2\mathbf{n}\pm (1-sin\theta )\mathbf{b},\pm \mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{3\left(34\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n},\mp sin\theta \mathbf{n}+(1+sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(34\right)3}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left[(1+sin\theta )\mathbf{n},\pm sin\theta \mathbf{n}+\mathbf{b}\right].\end{array}$$

Moreover, we have to construct the expressions of ${\mathcal{L}}_{4\left(\alpha \beta \right)}^{\pm }:{\Delta }_4^{\pm }⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)$ and their converse mappings ${\mathcal{L}}_{\left(\alpha \beta \right)4}^{\pm }:{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)⟶{\Delta }_4^{\pm }.$ They are contact diffeomorphisms and are denoted, respectively, by

$$\begin{array}{l}{\mathcal{L}}_{4\left(12\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1/2)(\mathbf{n}\mp \mathbf{b}),\pm (1/2)(1-sin\theta )\mathbf{n}+(1/2)(1+sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(12\right)4}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1+sin\theta )\mathbf{n}\pm \mathbf{b},\mp (1-sin\theta )\mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{4\left(13\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1/2)(1+sin\theta )\mathbf{n}-(1/2)(1\mp sin\theta )\mathbf{b},(1/2)(\mp \mathbf{n}+\mathbf{b})\right],\\ {\mathcal{L}}_{\left(13\right)4}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n}\pm (1+sin\theta )\mathbf{b},\mp \mathbf{n}+(1+sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{4\left(14\right)}^{\pm }(\mathbf{n},\mathbf{b})=(1/2)\left[(1+sin\theta )\mathbf{n}\mp (1-sin\theta )\mathbf{b},\pm (1\mp sin\theta )\mathbf{n}+(1+sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(14\right)4}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{{sin}^2\theta +1}\left[(1+sin\theta )\mathbf{n}\pm (1-sin\theta )\mathbf{b},\mp (1-sin\theta )\mathbf{n}+(1+sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{4\left(23\right)}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{2}\left[(1+sin\theta )\mathbf{n}\mp (1-sin\theta )\mathbf{b},\pm (1-cos\theta )\mathbf{n}+(1+cos\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(23\right)4}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta cos\theta }\left[(1+cos\theta )\mathbf{n}\pm (1-sin\theta )\mathbf{b},\mp (1-cos\theta )\mathbf{n}+(1+sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{4\left(24\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1/2)(1+sin\theta )\mathbf{n}\mp (1/2)(1-sin\theta )\mathbf{b},\mathbf{b}\right],\\ {\mathcal{L}}_{\left(24\right)4}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left[2\mathbf{n}\pm (1-sin\theta )\mathbf{b},(1+sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{4\left(34\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n},\pm (1/2)(1-sin\theta )\mathbf{n}+(1/2)(1+sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(34\right)4}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left[(1+sin\theta )\mathbf{n},\mp (1-sin\theta )\mathbf{n}+2\mathbf{b}\right].\end{array}$$

Last but not least, we construct the expressions of ${\mathcal{L}}_{\left(\alpha \beta \right)\left(\gamma \delta \right)}^{\pm }:{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)⟶{\Delta }_{\gamma \delta }^{\pm }\left(\theta \right)$ and their converse mappings

$${\mathcal{L}}_{\left(\gamma \delta \right)\left(\alpha \beta \right)}^{\pm }:{\Delta }_{\gamma \delta }^{\pm }\left(\theta \right)⟶{\Delta }_{\alpha \beta }^{\pm }\left(\theta \right)(\alpha ,\beta ,\gamma ,\delta =1,2,3,4,\alpha <\beta ,\gamma <\delta ).$$

They are contact diffeomorphisms and are denoted, respectively, by

$$\begin{array}{l}{\mathcal{L}}_{\left(12\right)\left(14\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1+{sin}^2\theta )\mathbf{n}\pm sin\theta \mathbf{b},\mathbf{b}\right],\\ {\mathcal{L}}_{\left(14\right)\left(12\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1/1+{sin}^2\theta )(\mathbf{n}\mp sin\theta \mathbf{b}),\mathbf{b}\right],\\ {\mathcal{L}}_{\left(12\right)\left(13\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1+{sin}^2\theta )\mathbf{n}\pm sin\theta \mathbf{b},\pm sin\theta \mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{\left(13\right)\left(12\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n}\mp sin\theta \mathbf{b},\mp sin\theta \mathbf{n}+(1+{sin}^2\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(12\right)\left(23\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1+{sin}^2\theta )\mathbf{n}\pm sin\theta \mathbf{b},\pm (sin\theta -cos\theta )\mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{\left(23\right)\left(12\right)}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta cos\theta }\left[\mathbf{n}\mp sin\theta \mathbf{b},(\mp sin\theta \pm cos\theta )\mathbf{n}+(1+{sin}^2\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(12\right)\left(24\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1+{sin}^2\theta )\mathbf{n}\pm sin\theta \mathbf{b},(\pm sin\theta \mp 1)\mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{\left(24\right)\left(12\right)}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left[\mathbf{n}\mp sin\theta \mathbf{b},(\mp sin\theta \pm 1)\mathbf{n}+(1+{sin}^2\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(12\right)\left(34\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1+sin\theta )\mathbf{n}\pm \mathbf{b},\mathbf{b}\right],\\ {\mathcal{L}}_{\left(34\right)\left(12\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(1/1+sin\theta )(\mathbf{n}\mp \mathbf{b}),\mathbf{b}\right],\\ {\mathcal{L}}_{\left(13\right)\left(14\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n},\mp sin\theta \mathbf{n}+(1+{sin}^2\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(14\right)\left(13\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(\mathbf{n},(1/1+{sin}^2\theta )(\pm sin\theta \mathbf{n}+\mathbf{b})\right],\\ {\mathcal{L}}_{\left(13\right)\left(23\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n},\mp cos\theta \mathbf{n}+(1+sin\theta cos\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(23\right)\left(13\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[(\mathbf{n},(1/1+sin\theta cos\theta \left)\right(\pm cos\theta \mathbf{n}+\mathbf{b})\right],\\ {\mathcal{L}}_{\left(13\right)\left(24\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n},\mp \mathbf{n}+(1+sin\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(24\right)\left(13\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n},(1/1+sin\theta )(\pm \mathbf{n}+\mathbf{b})\right],\\ {\mathcal{L}}_{\left(13\right)\left(34\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left[\mathbf{n}\pm \mathbf{b}(1-sin\theta ),\mp \mathbf{n}sin\theta +(1+{sin}^2\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(34\right)\left(13\right)}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left[(1+{sin}^2\theta )\mathbf{n}\mp (1-sin\theta )\mathbf{b},\pm sin\theta \mathbf{n}+\mathbf{b}\right],\\ {\mathcal{L}}_{\left(14\right)\left(23\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left\{\mathbf{n},(1/1+{sin}^2\theta )[(\mp cos\theta \pm sin\theta )\mathbf{n}+(1+sin\theta cos\theta )\mathbf{b}]\right\},\\ {\mathcal{L}}_{\left(23\right)\left(14\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left\{\mathbf{n},(1/1+sin\theta cos\theta )[(\pm cos\theta \mp sin\theta )\mathbf{n}+(1+{sin}^2\theta )\mathbf{b}]\right\},\\ {\mathcal{L}}_{\left(14\right)\left(24\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left\{\mathbf{n},(1/1+{sin}^2\theta )[(\pm sin\theta \mp 1)\mathbf{n}+(1+sin\theta )\mathbf{b}]\right\},\\ {\mathcal{L}}_{\left(24\right)\left(14\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left\{\mathbf{n},(1/1+sin\theta )[(\mp sin\theta \pm 1)\mathbf{n}+(1+{sin}^2\theta )\mathbf{b}]\right\},\\ {\mathcal{L}}_{\left(34\right)\left(14\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left\{[(1+{sin}^2\theta )/(1+sin\theta )]\mathbf{n},\mathbf{b}\right\},\\ {\mathcal{L}}_{\left(14\right)\left(34\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left\{[(1+sin\theta )/(1+{sin}^2\theta )]\mathbf{n},\mathbf{b}\right\},\\ {\mathcal{L}}_{\left(23\right)\left(24\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left\{\mathbf{n},(1/1+sin\theta cos\theta )\left[\right(\pm cos\theta \mp 1)\mathbf{n}+(1+sin\theta \left)\mathbf{b}\right]\right\},\\ {\mathcal{L}}_{\left(24\right)\left(23\right)}^{\pm }(\mathbf{n},\mathbf{b})=\left\{\mathbf{n},(1/1+sin\theta )\left[\right(\mp cos\theta \pm 1)\mathbf{n}+(1+sin\theta cos\theta \left)\mathbf{b}\right]\right\},\\ {\mathcal{L}}_{\left(23\right)\left(34\right)}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta cos\theta }\left(\mathbf{n},\mathbf{W}\right),\\ {\mathcal{L}}_{\left(34\right)\left(23\right)}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left[(1+{sin}^2\theta )\mathbf{n}+(\mp 1\pm sin\theta )\mathbf{b},(\mp cos\theta \pm sin\theta )\mathbf{n}+(1+cos\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(24\right)\left(34\right)}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left[2\mathbf{n}+\mathbf{b}(\mp sin\theta \pm 1)),(\mp sin\theta \pm 1)\mathbf{n}+(1+{sin}^2\theta )\mathbf{b}\right],\\ {\mathcal{L}}_{\left(34\right)\left(24\right)}^{\pm }(\mathbf{n},\mathbf{b})=\frac{1}{1+sin\theta }\left[(1+{sin}^2\theta )\mathbf{n}+(\mp 1\pm sin\theta )\mathbf{b},(\mp 1\pm sin\theta )\mathbf{n}+2\mathbf{b}\right],\end{array}$$

where $\mathbf{n}=(1+cos\theta )\mathbf{n}+\mathbf{b}(\mp sin\theta \pm 1)$, $\mathbf{W}=(\mp sin\theta \pm cos\theta )\mathbf{n}+(1+{sin}^2\theta )\mathbf{b},$ and $id$ is an identity map. This completes the proof. □

4. Applications

In this section, we give two applications of the extended Legendrian dualities theorem. We focus on an open nullcone defined by

$${\Lambda }^n=\{\mathbf{x}\in {\mathbb{R}}_2^{n+1}\backslash \left\{0\right\}\mid {\langle \mathbf{x},\mathbf{x}\rangle }_2=0\}.$$

It is well known that one of the difficulties in the study of the Lorentzian hypersurface in an open nullcone is that it is impossible to obtain the normal vector of the hypersurface from its tangent space by using a pseudo-wedge operation since the induced metric on the open nullcone is degenerate. As the first application of the extended Legendrian dualities, we will construct one of the most important normal vectors by the extended Legendrian duality theorem in order to solve this question. Furthermore, we construct the nullcone Gaussian image, the anti-de Sitter Gaussian image and the pseudo-sphere Gaussian image of the Lorentzian hypersurface in the open nullcone by the extended Legendrian duality theorem. One of our results (cf. Proposition 1) indicates that there are three kinds of totally umbilic hypersurfaces in the nullcone. A naturally interesting question is whether there are relationships among these totally umbilic hypersurfaces. As the second application of the extended Legendrian dualities, we will establish the relations among these totally umbilic hypersurfaces in the nullcone.

Let ${\mathbb{X}}^t:U⟶{\Lambda }^n$ be a timelike embedding for an open subset $U\subset {\mathbb{R}}_1^{n-1}.$ We denote that $M={\mathbb{X}}^t\left(U\right)$ and identify M with U through the embedding ${\mathbf{X}}^t.$ $M={\mathbb{X}}^t\left(U\right)$ is called a Lorentzian hypersurface. The metric on the open nullcone is degenerate, so that we cannot construct the normal vector of the Lorentzian hypersurface by using a pseudo-wedge operation. To deal with this difficulty, we employ the extended Legendrian duality theorem. We consider the ${\Lambda }^n\times {\Lambda }^n\supset {\Delta }_4^{-}$ duality and define Legendrian embedding

$${\mathcal{L}}_4:U\to {\Delta }_4^{-},{\mathcal{L}}_4\left(\mathbf{u}\right)=(\mathbb{X}\left(\mathbf{u}\right),{\mathbb{X}}^n\left(\mathbf{u}\right)).$$

One can check that $\langle d\mathbb{X}\left(\mathbf{u}\right),{\mathbb{X}}^n\left(\mathbf{u}\right)\rangle =0.$ This indicates that ${\mathbb{X}}^n\left(\mathbf{u}\right)$ lies on the normal space $N_{p}M$ of $M=\mathbb{X}\left(U\right)$ at $p=\mathbb{X}\left(u\right).$ Since $N_{p}M$ is locally isomorphic to the Lorentzian plane and $\langle \mathbb{X}\left(\mathbf{u}\right),{\mathbb{X}}^n\left(\mathbf{u}\right)\rangle =-2,$ ${\mathcal{L}}_4$ is the unique Legendrian lift of $M=\mathbb{X}\left(U\right).$ We call ${\mathbb{X}}^n\left(\mathbf{u}\right)$ the nullcone normal vector field of Lorentzian hypersurface $M=\mathbb{X}\left(U\right)$ at $\mathbb{X}\left(\mathbf{u}\right).$ We consider the diffeomorphism ${\mathcal{L}}_{41}:{\Delta }_4^{-}\to {\Delta }_1,$ with ${\mathcal{L}}_{41}(\mathbf{v},\mathbf{w})=\left(\frac{\mathbf{v}+\mathbf{w}}{2},\frac{\mathbf{v}-\mathbf{w}}{2}\right).$ One can obtain a Legendrian submanifold ${\mathcal{L}}_1:U\to {\Delta }_1$ by ${\mathcal{L}}_1\left(\mathbf{u}\right)={\mathcal{L}}_{41}\circ {\mathcal{L}}_4\left(\mathbf{u}\right).$ In particular, let ${\mathcal{L}}_1\left(\mathbf{u}\right)=({\mathbb{X}}^t\left(\mathbf{u}\right),{\mathbb{X}}^s\left(\mathbf{u}\right)),$ then

$${\mathbb{X}}^t\left(\mathbf{u}\right)=\frac{\mathbb{X}\left(\mathbf{u}\right)+{\mathbb{X}}^n\left(\mathbf{u}\right)}{2},{\mathbb{X}}^s\left(\mathbf{u}\right)=\frac{\mathbb{X}\left(\mathbf{u}\right)-{\mathbb{X}}^n\left(\mathbf{u}\right)}{2}.$$

We call ${\mathbb{X}}^t\left(\mathbf{u}\right)$ and ${\mathbb{X}}^s\left(\mathbf{u}\right)$ anti-de Sitter normal vector field and pseudo-sphere normal vector field of Lorentzian hypersurface $M=\mathbb{X}\left(U\right)$ at $\mathbb{X}\left(\mathbf{u}\right)$, respectively. Since $\mathbb{X}\left(\mathbf{u}\right)$ and ${\mathbb{X}}^n\left(\mathbf{u}\right)$ are linearly independent null vectors and $\mathbb{X}\left(\mathbf{u}\right)$ is a Lorentzian hypersurface, we obtain a basis

$$\mathbb{X}\left(\mathbf{u}\right),{\mathbb{X}}^n\left(\mathbf{u}\right),{\mathbb{X}}_{u_1}\left(\mathbf{u}\right),\cdots ,{\mathbb{X}}_{u_{n-1}}\left(\mathbf{u}\right)$$

of $T_p{\mathbb{R}}_2^{n+1},$ where $\mathbf{p}=\mathbb{X}\left(\mathbf{u}\right).$ We call ${\mathbb{X}}^n:U\to {\Lambda }^n$ the nullcone Gaussian image, ${\mathbb{X}}^t:U\to H_1^n(-1)$ the anti-de Sitter Gaussian image, and ${\mathbb{X}}^s:U\to S_2^n\left(1\right)$ the pseudo-sphere Gaussian image of the Lorentzian hypersurface $M=\mathbb{X}\left(U\right)$, respectively. One can define the following three linear transformations, which are shape operators. We, respectively, call $S_p^n\left[\theta \right]\left(\mathbf{u}\right)=-d{\mathbb{X}}^n\left(\mathbf{u}\right):T_{p}M\to T_{p}M$ the nullcone shape operator, $S_p^t\left[\theta \right]\left(\mathbf{u}\right)=-d{\mathbb{X}}^t\left(\mathbf{u}\right):T_{p}M\to T_{p}M$ the anti-de Sitter shape operator and $S_p^s\left[\theta \right]\left(\mathbf{u}\right)=-d{\mathbb{X}}^n\left(\mathbf{u}\right):T_{p}M\to T_{p}M$ the pseudo-sphere shape operator, where $\mathbf{p}=\mathbb{X}\left(\mathbf{u}\right).$

We, respectively, denote the eigenvalues of $S_p^n\left[\theta \right]\left(\mathbf{u}\right)$ by $k_n\left[\theta \right]\left(\mathbf{p}\right),$$S_p^t\left[\theta \right]\left(\mathbf{u}\right)$ by $k_t\left[\theta \right]\left(\mathbf{p}\right)$ and $S_p^s\left[\theta \right]\left(\mathbf{u}\right)$ by $k_s\left[\theta \right]\left(\mathbf{p}\right)$, which we call null principal curvature, anti-de Sitter principal curvature and pseudo-sphere principal curvature. One can check that

$$k_t\left[\theta \right]\left(\mathbf{p}\right)=\frac{k_n\left[\theta \right]\left(\mathbf{p}\right)-1}{2},k_s\left[\theta \right]\left(\mathbf{p}\right)=\frac{-k_n\left[\theta \right]\left(\mathbf{p}\right)-1}{2}.$$

We, respectively, define nullcone Gaussian curvature by $K_n\left[\theta \right]\left({\mathbf{u}}_0\right)=det{S}_p^n\left[\theta \right]\left({\mathbf{u}}_0\right),$ anti-de Sitter Gaussian curvature by $K_t\left[\theta \right]\left({\mathbf{u}}_0\right)=det{S}_p^t\left[\theta \right]\left({\mathbf{u}}_0\right),$ and pseudo-sphere Gaussian curvature by $K_s\left[\theta \right]\left({\mathbf{u}}_0\right)=det{S}_p^s\left[\theta \right]\left({\mathbf{u}}_0\right)$ at ${\mathbf{p}}_0=\mathbb{X}\left({\mathbf{u}}_0\right)$. If $K_n\left[\theta \right]\left(\mathbf{u}\right)=0$, then $\mathbf{p}=\mathbb{X}\left(\mathbf{u}\right)$ is called a nullcone parabolic point. If $S^n\left[\theta \right]\left(\mathbf{p}\right)=k_n\left[\theta \right]i{d}_{T_{p}M},$ we call $\mathbf{p}=\mathbb{X}\left(\mathbf{u}\right)$ a nullcone umbilic point. A hypersurface M is called a totally nullcone umbilic hypersurface if every point on M is nullcone umbilic. We define a hypersurface by $NH(\mathbf{n},c)={\Lambda }^n\cap HP(\mathbf{n},c)$, which can be taken as the model of the totally nullcone umbilic hypersurface in the nullcone. We summarize the classifications of totally nullcone umbilic hypersurfaces in the nullcone in

Table 2. Classifications of totally nullcone umbilic hypersurfaces in nullcone.

Conditions	Constant Normal Vector	Classifications
$k_n\left[\theta \right]<0$	$\mathbf{n}=\frac{-1}{2\sqrt{-k_n\left[\theta \right]}}(k_n\left[\theta \right]\mathbb{X}\left(\mathbf{u}\right)+{\mathbb{X}}^n\left(\mathbf{u}\right))\in S_2^n$	M is a subset of $NH(\mathbf{n},1/\sqrt{-k_n\left[\theta \right]})$
$k_n\left[\theta \right]=0$	$\mathbf{n}={\mathbb{X}}^n\left(\mathbf{u}\right)\in {\Lambda }^n$	M is a subset of $NH(\mathbf{n},-2)$
$k_n\left[\theta \right]>0$	$\mathbf{n}=\frac{1}{2\sqrt{k_n\left[\theta \right]}}(k_n\left[\theta \right]\mathbb{X}\left(\mathbf{u}\right)+{\mathbb{X}}^n\left(\mathbf{u}\right))\in H_1^n$	M is a subset of $NH(\mathbf{n},-1/\sqrt{k_n\left[\theta \right]})$

.

Proposition 2.

Suppose that $M=\mathbb{X}\left(U\right)$ is a totally nullcone umbilic hypersurface with constant $k_n\left[\theta \right]\left(\mathbf{p}\right)=k$ in the nullcone; then, one can obtain the following classifications in Table 2.

Proposition 1 indicates that there are three kinds of totally umbilic hypersurfaces in the nullcone. A natural question is how to establish the relations among these totally umbilic hypersurfaces. As the second application of extended Legendrian dualities, we try to solve this question. We define ${\Delta }_{43}^{-}\left[\theta \right]={\Delta }_{34}^{-}[\frac{\pi }{2}-\theta ]$,$K_{43}^{-}\left[\theta \right]=K_{34}^{-}[\frac{\pi }{2}-\theta ]$,$\pi {\left[\theta \right]}_{\left(43\right)i}^{-}=\pi {[\frac{\pi }{2}-\theta ]}_{\left(34\right)i}^{-},$ where $i=1,2.$ In particular, we consider the following double fibration:

$$\begin{array}{cc}\hfill (10*)\left(\alpha \right)& {\Lambda }^n\times S_2^n\left({sin}^2\theta \right)\supset {\Delta }_{43}^{-}\left(\theta \right)=\left\{(\mathbf{v},\mathbf{w})\right|\langle \mathbf{v},\mathbf{w}\rangle =-(cos\theta +1)\},\hfill \\ \hfill \left(\beta \right)& \pi {\left[\theta \right]}_{\left(43\right)1}^{-}:{\Delta }_{43}^{-}\left(\theta \right)⟶{\Lambda }^n,\pi {\left[\theta \right]}_{\left(43\right)2}^{-}:{\Delta }_{43}^{-}\left(\theta \right)⟶S_2^n\left({sin}^2\theta \right),\hfill \\ \hfill \left(\gamma \right)& \eta {\left[\theta \right]}_{\left(43\right)1}^{-}=\langle d\mathbf{v},\mathbf{w}\rangle {{|}_{{\Delta }_{43}^{-}\left(\theta \right)},\eta {\left[\theta \right]}_{\left(43\right)2}^{-}=\langle \mathbf{v},d\mathbf{w}\rangle |}_{{\Delta }_{43}^{-}\left(\theta \right)}.\hfill \end{array}$$

By Theorem 1, $({\Delta }_{43}^{-}\left[\theta \right],K{\left[\theta \right]}_{43}^{-})$ is a contact manifold. One can prove that

$${\mathcal{L}}_{4\left(43\right)}^{-}:{\Delta }_4^{-}\to {\Delta }_{43}^{-}\left[\theta \right],{\mathcal{L}}_{4\left(43\right)}^{-}(\mathbf{v},\mathbf{w})=\left(\mathbf{v},\frac{1}{2}((cos\theta -1)\mathbf{v}+(cos\theta +1)\mathbf{w})\right)$$

is a contact diffeomorphism. We define a map ${\mathbb{N}}_n^s\left[\theta \right]:U\to S_2^n\left({sin}^2\theta \right)$ by

$${\mathbb{N}}_n^s\left[\theta \right]\left(\mathbf{u}\right)=\frac{1}{2}\left((cos\theta -1)\mathbb{X}\left(\mathbf{u}\right)+(cos\theta +1){\mathbb{X}}^n\left(\mathbf{u}\right)\right).$$

Furthermore, we define

$${\mathcal{L}}_{43}\left[\theta \right]:U\to {\Delta }_{43}^{-}\left[\theta \right]\subset {\Lambda }^n\times S_2^n\left({sin}^2\theta \right)$$

by ${\mathcal{L}}_{43}\left[\theta \right]\left(\mathbf{u}\right)=(\mathbb{X}\left(\mathbf{u}\right),{\mathbb{N}}_n^s\left[\theta \right]\left(\mathbf{u}\right)).$ Since ${\mathcal{L}}_{43}\left[\theta \right]\left(\mathbf{u}\right)={\mathcal{L}}_{4\left(43\right)}^{-}\circ {\mathcal{L}}_4\left(\mathbf{u}\right),$ ${\mathcal{L}}_{43}\left[\theta \right]\left(\mathbf{u}\right)$ is a Legendrian embedding. Therefore,

$$\langle d\mathbb{X}\left(\mathbf{u}\right),{\mathbb{N}}_n^s\left[\theta \right]\left(\mathbf{u}\right)\rangle ={\mathcal{L}}_{43}\left[\theta \right]{\left(\mathbf{u}\right)}^{*}\eta {\left[\theta \right]}_{\left(43\right)1}^{-}=0.$$

This means that ${\mathbb{N}}_n^s\left[\theta \right]\left(\mathbf{u}\right)$ is a normal vector of $M=\mathbb{X}\left(U\right)$ at $\mathbf{p}=\mathbb{X}\left(\mathbf{u}\right)$. We call it a $\theta$-pseudo sphere Gaussian map. One can obtain the following relations among these Gaussian maps in

Table 3. The first relations among Gaussian maps of Lorentzian hypersurfaces in nullcone.

Conditions	Relations among Gaussian Maps
$\theta =0$	${\mathbb{N}}_n^s\left[0\right]\left(\mathbf{u}\right)={\mathbb{X}}^n\left(\mathbf{u}\right)$
$\theta \in (0,\frac{\pi }{2})$	${\mathbb{N}}_n^s\left[\theta \right]\left(\mathbf{u}\right)=\frac{1}{2}\left((cos\theta -1)\mathbb{X}\left(\mathbf{u}\right)+(cos\theta +1){\mathbb{X}}^n\left(\mathbf{u}\right)\right).$
$\theta =\frac{\pi }{2}$	${\mathbb{N}}_n^s\left[\frac{\pi }{2}\right]\left(\mathbf{u}\right)={\mathbb{X}}^s\left(\mathbf{u}\right)$

.

We define a new model hypersurface in the nullcone by

$$NH(\mathbf{n},-(cos\theta +1))={\Lambda }^n\cap HP(\mathbf{n},-(cos\theta +1)).$$

The following proposition indicates that there is a new kind of interesting geometry where $NH(\mathbf{n},-(cos\theta +1\left)\right)$ can be seen as a totally umbilic hypersurface in the nullcone.

Proposition 3.

Let $\mathbb{X}:U⟶{\Lambda }^n$ be a Lorentzian hypersurface, and then ${\mathbb{N}}_n^s\left[\theta \right]\left(\mathbf{u}\right)$ is a constant vector if and only if $\mathbb{X}\left(U\right)$ is a subset of $NH(\mathbf{n},-(cos\theta +1\left)\right)$, where $\mathbf{n}\in S_2^n\left({sin}^2\theta \right).$

Proof. 

If ${\mathbb{N}}_n^s\left[\theta \right]\left(\mathbf{u}\right)=\mathbf{n}$ is a constant vector, then

$$\langle \mathbb{X}\left(\mathbf{u}\right),\mathbf{n}\rangle =\langle \mathbb{X}\left(\mathbf{u}\right),{\mathbb{N}}_n^s\left[\theta \right]\left(\mathbf{u}\right)\rangle =-(cos\theta +1).$$

This means that

$$\mathbb{X}\left(U\right)\subset NH({\mathbb{N}}_n^s\left[\theta \right]\left(\mathbf{u}\right),-(cos\theta +1)).$$

Therefore, M is a subset of $NH({\mathbb{N}}_n^s\left[\theta \right]\left(\mathbf{u}\right),-(cos\theta +1))$. Conversely, if

$$M\subset NH({\mathbb{N}}_n^s\left[\theta \right]\left(\mathbf{u}\right),-(cos\theta +1)),$$

where $\mathbf{n}\in S_2^n\left({sin}^2\theta \right).$ Since $\mathbf{n}$ is a normal vector of M, there are real numbers a and b such that $\mathbf{n}=a\mathbb{X}\left(\mathbf{u}\right)+b{\mathbb{X}}^n\left(\mathbf{u}\right)$ and ${sin}^2\theta =-4ab.$ By definition, we obtain

$$-(cos\theta +1)=\langle \mathbb{X}\left(\mathbf{u}\right),\mathbf{n}\rangle =-2b.$$

Therefore, $b=\frac{1}{2}(cos\theta +1)$ and $a=\frac{1}{2}(cos\theta -1).$ This indicates that $\mathbf{n}={\mathbb{N}}_n^s\left[\theta \right]\left(\mathbf{u}\right).$ □

We define ${\Delta }_{42}^{-}\left[\theta \right]={\Delta }_{24}^{-}[\frac{\pi }{2}-\theta ]$,$K_{42}^{-}\left[\theta \right]=K_{24}^{-}[\frac{\pi }{2}-\theta ]$,$\pi {\left[\theta \right]}_{\left(42\right)i}^{-}=\pi {[\frac{\pi }{2}-\theta ]}_{\left(24\right)i}^{-},$ where $i=1,2.$ In particular, we consider the following double fibration:

$$\begin{array}{cc}\hfill (9*)\left(\alpha \right)& H_1^n(-{sin}^2\theta )\times {\Lambda }^n\supset {\Delta }_{42}^{-}\left(\theta \right)=\left\{(\mathbf{v},\mathbf{w})\right|\langle \mathbf{v},\mathbf{w}\rangle =-(cos\theta +1)\},\hfill \\ \hfill \left(\beta \right)& \pi {\left[\theta \right]}_{\left(42\right)1}^{-}:{\Delta }_{42}^{-}\left(\theta \right)⟶H_1^n(-{sin}^2\theta ),\pi {\left[\theta \right]}_{\left(42\right)2}^{-}:{\Delta }_{42}^{-}\left(\theta \right)⟶{\Lambda }^n,\hfill \\ \hfill \left(\gamma \right)& \eta {\left[\theta \right]}_{\left(42\right)1}^{-}=\langle d\mathbf{v},\mathbf{w}\rangle {{|}_{{\Delta }_{42}^{-}\left(\theta \right)},\eta {\left[\theta \right]}_{\left(42\right)2}^{-}=\langle \mathbf{v},d\mathbf{w}\rangle |}_{{\Delta }_{42}^{-}\left(\theta \right)}.\hfill \end{array}$$

By Theorem 1, $({\Delta }_{42}^{-}\left[\theta \right],K{\left[\theta \right]}_{42}^{-})$ is a contact manifold. One can check that

$${\mathcal{L}}_{4\left(42\right)}^{-}:{\Delta }_4^{-}\to {\Delta }_{42}^{-}\left[\theta \right],{\mathcal{L}}_{4\left(42\right)}^{-}(\mathbf{v},\mathbf{w})=\left(\frac{1}{2}((1-cos\theta )\mathbf{v}+(1+cos\theta )\mathbf{w}),\mathbf{w}\right)$$

is a contact diffeomorphism. We define a map ${\mathbb{N}}_n^t\left[\theta \right]:U\to H_1^n(-{sin}^2\theta )$ by

$${\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right)=\frac{1}{2}((1-cos\theta )\mathbb{X}\left(\mathbf{u}\right)+(1+cos\theta ){\mathbb{X}}^n\left(\mathbf{u}\right)).$$

Furthermore, we define ${\mathcal{L}}_{42}\left[\theta \right]:U\to {\Delta }_{42}^{-}\left[\theta \right]\subset H_1^n(-{sin}^2\theta )\times {\Lambda }^n$ by

$${\mathcal{L}}_{42}\left[\theta \right]\left(\mathbf{u}\right)=({\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right),\mathbb{X}\left(\mathbf{u}\right)).$$

Since ${\mathcal{L}}_{42}\left[\theta \right]\left(\mathbf{u}\right)={\mathcal{L}}_{4\left(42\right)}^{-}\circ {\mathcal{L}}_4\left(\mathbf{u}\right),$ ${\mathcal{L}}_{42}\left[\theta \right]\left(\mathbf{u}\right)$ is a Legendrian embedding. Therefore,

$$\langle d{\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right),\mathbb{X}\left(\mathbf{u}\right)\rangle ={\mathcal{L}}_{42}\left[\theta \right]{\left(\mathbf{u}\right)}^{*}\eta {\left[\theta \right]}_{\left(42\right)1}^{-}=0.$$

This indicates that ${\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right)$ can be seen as a normal vector of $M=\mathbb{X}\left(U\right)$ at $\mathbf{p}=\mathbb{X}\left(\mathbf{u}\right).$ We call ${\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right)$ a $\theta$-hyperbolic pseudo-sphere Gaussian map. One can also obtain the following relations among Gaussian maps in

Table 4. The second relations among Gaussian maps of Lorentzian hypersurfaces in nullcone.

Conditions	Relations among Gaussian Maps
$\theta =0$	${\mathbb{N}}_n^t\left[0\right]\left(\mathbf{u}\right)={\mathbb{X}}^n\left(\mathbf{u}\right)$
$\theta \in (0,\frac{\pi }{2})$	${\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right)=\frac{1}{2}((1-cos\theta )\mathbb{X}\left(\mathbf{u}\right)+(1+cos\theta ){\mathbb{X}}^n\left(\mathbf{u}\right))$
$\theta =\frac{\pi }{2}$	${\mathbb{N}}_n^t\left[\frac{\pi }{2}\right]\left(\mathbf{u}\right)={\mathbb{X}}^t\left(\mathbf{u}\right)$

.

Accodding to the results in Table 3 and Table 4, we establish the relations among different kinds of geometries of Lorentzian hypersurfaces in the nullcone.

Proposition 4.

Let $\mathbb{X}:U⟶{\Lambda }^n$ be a Lorentzian hypersurface; then, ${\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right)$ is a constant vector if and only if $\mathbb{X}\left(U\right)$ is a subset of $NH(\mathbf{n},-(cos\theta +1\left)\right)$, where $\mathbf{n}\in H_1^n(-{sin}^2\theta ).$

Proof. 

If ${\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right)=\mathbf{n}$ is a constant vector, then

$$\langle \mathbb{X}\left(\mathbf{u}\right),\mathbf{n}\rangle =\langle \mathbb{X}\left(\mathbf{u}\right),{\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right)\rangle =-(cos\theta +1).$$

This means that

$$\mathbb{X}\left(U\right)\subset NH({\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right),-(cos\theta +1)).$$

Therefore, M is a subset of $NH({\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right),-(cos\theta +1))$. Conversely, if

$$M\subset NH({\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right),-(cos\theta +1)),$$

where $\mathbf{n}\in H_1^n(-{sin}^2\theta ).$ Since $\mathbf{n}$ is a normal vector of M, there are real numbers c and d such that $\mathbf{n}=c\mathbb{X}\left(\mathbf{u}\right)+d{\mathbb{X}}^n\left(\mathbf{u}\right)$ and $-4cd=-{sin}^2\theta .$ By definition, we obtain

$$-(cos\theta +1)=\langle \mathbb{X}\left(\mathbf{u}\right),\mathbf{n}\rangle =-2d.$$

Therefore, $d=\frac{1}{2}(1+cos\theta )$ and $c=\frac{1}{2}(1-cos\theta ).$ This indicates that $\mathbf{n}={\mathbb{N}}_n^t\left[\theta \right]\left(\mathbf{u}\right).$ □

5. Conclusions

This paper deals with an interesting question of Legendrian dualities for continuous families of pseudo-spheres in semi-Euclidean space. We construct all contact diffeomorphisms among the contact manifolds and display them in a table of Legendrian dualities. We also extend the theorem of Legendrian dualities to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space. Finally, we give two applications of the extended Legendrian duality theorem.

As a future work, we plan to proceed to study some applications of Legendrian dualities combined with singularity theory and submanifold theory, etc., in [13,14,15,16,17,26,27,28,29,30,31,32,33,34], to obtain new results and theorems. Furthermore, we will explore some new geometric properties of Lie groups based on the results in [35,36,37,38].