# The application of the moving frame method to Integral Geometry in
Heisenberg group

HUNG-LIN CHIU, Yen-Chang Huang, SIN-HUA LAI  Department of Mathematics,
National Central University, Chung Li, 32054, Taiwan, R.O.C.
<EMAIL_ADDRESS>Department of Mathematics, National Central University,
Chung Li, 32054, Taiwan, R.O.C<EMAIL_ADDRESS>Department of
Mathematics, National Central University, Chung Li, 32054, Taiwan, R.O.C.
<EMAIL_ADDRESS>
###### Abstract.

We study the local equivalence problems of curves and surfaces in
3-dimensional Heisenberg group $H_{1}$ via Cartan’s method of moving frames
and Lie groups theory. We show the fundamental theorem of the curves and the
surfaces in $H_{1}$ and find a complete set of invariants for curves and
surfaces respectively. As the application of the main theorem for curves, we
show Crofton’s formula in terms of p-area, which makes the connection between
the CR Geometry and the Integral Geometry.

###### Contents

  1. 1 Introduction
  2. 2 Calculus on the Lie groups
  3. 3 The group of pseudo-hermitian transformations on $H_{1}$
    1. 3.1 The pseudo-hermitian transformations on $H_{1}$
    2. 3.2 Representation of $PSH(1)$
    3. 3.3 The oriented frames on $H_{1}$
    4. 3.4 Moving frame formula
  4. 4 Differential invariants of horizontally regular curves in $H_{1}$
    1. 4.1 The Proof of Theorem 1.2
    2. 4.2 The derivation of the $p$-curvature and the contact normality
  5. 5 Differential invariants of parametrized surfaces in $H_{1}$
    1. 5.1 The proof of Theorem 1.8
    2. 5.2 Invariants of surfaces
    3. 5.3 A complete set of invariants for surfaces in $H_{1}$
  6. 6 The derivation of the integrability condition (1.13)
  7. 7 The proof of Theorem 1.10
  8. 8 Application: Crofton’s formula
  9. 9 Appendix:The second proof of Theorem 1.2

##  1\. Introduction

In Euclidean space, the fundamental theorem of curves states that any unit-
speed curve is completely determined by its curvature and torsion. More
precisely, given two functions $k(s)$ and $\tau(s)$ with $k(s)>0$, there
exists a unit-speed curve whose curvature and torsion are the functions $k$
and $\tau$, respectively, uniquely up to a Euclidean rigid motion. The main
theorems in the paper are to show the analogous theorems of curves and
surfaces in the $3$-dimensional Heisenberg group $H_{1}$.

We give a brief review of the Heisenberg group. All the details can be found
in [1]. The Heisenberg group $H_{1}$ is the space
$\operatorname{\mathbb{R}}^{3}$ associated with the group multiplication

(1.1)
$(x_{1},y_{1},z_{1})\circ(x_{2},y_{2},z_{2})=(x_{1}+x_{2},y_{1}+y_{2},z_{1}+z_{2}+y_{1}x_{2}-x_{1}y_{2}),$

which is also a $3$-dimensional Lie group. The standard left-invariant vector
fields in $H_{1}$

(1.2) $\mathring{e}_{1}=\frac{\partial}{\partial x}+y\frac{\partial}{\partial
z},\ \ \mathring{e}_{2}=\frac{\partial}{\partial y}-x\frac{\partial}{\partial
z},\ \ \text{and}\ \ T=\frac{\partial}{\partial z}$

form a basis of the vector space of all left-invariant vector fields, where
$(\frac{\partial}{\partial x},\frac{\partial}{\partial
y},\frac{\partial}{\partial z})$ denotes the standard basis in
$\operatorname{\mathbb{R}}^{3}$. The standard contact bundle on $H_{1}$ is
denoted by $\xi=span\\{\mathring{e}_{1},\mathring{e}_{2}\\}$ which is also a
subbundle of the tangent bundle $TH_{1}$. We also have

$\displaystyle\xi=\ker\Theta,$

where

(1.3) $\Theta=dz+xdy-ydx$

is the standard contact form. The CR structure on $H_{1}$ is the endomorphism
$J:\xi\rightarrow\xi$ defined by

(1.4) $J(\mathring{e}_{1})=\mathring{e}_{2}\ \ \text{and}\
J(\mathring{e}_{2})=-\mathring{e}_{1}.$

Heisenberg group $H_{1}$ can be regarded as a pseudo-hermitian manifold by
considering $H_{1}$ with the standard pseudo-hermitian structure $(J,\Theta)$.
For more information about pseudo-hermitian structure, we refer the readers to
[3][10][11][15]. Recall that a pseudo-hermitian transformation on $H_{1}$ is a
diffeomorphism on $H_{1}$ preserving the pseudo-hermitian structure
$(J,\Theta)$. Denote $PSH(1)$ be the group of all pseudo-hermitian
transformations on $H_{1}$, and we call the element in $PSH(1)$ a symmetry. A
symmetry is the Heisenberg rigid motion in $H_{1}$, which will be
characterized in Subsection 3.1.

Let $\gamma:I\rightarrow H_{1}$ be a parametrized curve. For each $t\in I$,
the velocity $\gamma^{{}^{\prime}}(t)$ has the natural decomposition

(1.5)
$\gamma^{{}^{\prime}}(t)=\gamma^{{}^{\prime}}_{\xi}(t)+\gamma^{{}^{\prime}}_{T}(t),$

where $\gamma^{{}^{\prime}}_{\xi}(t)$ and $\gamma^{{}^{\prime}}_{T}(t)$ are,
respectively, the orthogonal projection of $\gamma^{{}^{\prime}}(t)$ on $\xi$
along $T$ and the orthogonal projection of $\gamma^{{}^{\prime}}(t)$ on $T$
along $\xi$.

######  Definition 1.1.

A horizontally regular curve is a parametrized curve $\gamma(t)$ such that
$\gamma^{{}^{\prime}}_{\xi}(t)\neq 0$ for all $t\in I$. We say $\gamma(t)$ is
a horizontal curve if $\gamma^{{}^{\prime}}(t)=\gamma_{\xi}^{{}^{\prime}}(t)$
for all $t\in I$.

Proposition 4.1 shows that a horizontally regular curve can always be
reparametrized by the parameter $s$ satisfying
$|\gamma_{\xi}^{{}^{\prime}}(s)|=1$ for all $s$. Such the curve is called
being with horizontally unit-speed and the parameter $s$ is called the
horizontal arc-length for $\gamma(s)$, which is unique up to a constant.
Through the article the length of the vectors $|\cdot|$ and the inner product
$<\cdot,\cdot>$ are defined with respect to the Levi-metric, and the
orthonormality of the vectors $\mathring{e}_{1}$ and $\mathring{e}_{2}$ on the
contact plane $\xi$ is always in the sense of Levi-metric.

For a horizontally regular curve $\gamma(s)$ parameterized by the horizontal
arc-length $s$, we define the p-curvature $k(s)$ and contact normality
$\tau(s)$

(1.6) $\begin{split}k(s)&:=<\frac{dX(s)}{ds},Y(s)>,\\\
\tau(s)&:=<\gamma^{{}^{\prime}}(s),T>,\end{split}$

where $X(s)=\gamma^{{}^{\prime}}_{\xi}(s)$ and $Y(s)=JX(s)$. Notice that
$k(s)$ is analogous to the curvature of the curve in
$\operatorname{\mathbb{R}}^{n}$, while $\tau(s)$ measures how far the curve is
from being horizontal. Our first theorem says that horizontally regular curves
are completely prescribed by the quantities $k(s)$ and $\tau(s)$.

######  Theorem 1.2 (The fundamental theorem for curves in $H_{1}$).

Given $C^{1}$-functions $k(s),\tau(s)$, there exists a horizontally regular
curve $\gamma(s)$ with horizontally unit-speed having $k(s)$ and $\tau(s)$ as
its $p$-curvature and contact normality, respectively. In additional, any
regular curve $\widetilde{\gamma}(s)$ with horizontally unit-speed satisfying
the same $p$-curvature $k(s)$ and contact normality $\tau(s)$ differs from
$\gamma(s)$ by a pseudo-hermitian transformation $g\in PSH(1)$, namely,

(1.7) $\widetilde{\gamma}{(s)}=g\circ\gamma(s)$

for all $s$.

Since a curve $\gamma(s)$ is horizontal if and only if the contact normality
$\tau(s)=0$, and we immediately have the following corollary.

######  Corollary 1.3.

Given a $C^{1}$-function $k(s)$, there exists a horizontal curve $\gamma(s)$
with horizontally unit-speed having $k(s)$ as its $p$-curvature. In
additional, any horizontal curve $\widetilde{\gamma}(s)$ with horizontally
unit-speed satisfying the same $p$-curvature differs from $\gamma(s)$ by a
pseudo-hermitian transformation $g\in PSH(1)$, namely,

$\widetilde{\gamma}{(s)}=g\circ\gamma(s)$

for all $s$.

Even the horizontally regular curve $\gamma$ is not parameterized by
horizontal arc-length, in Subsection 4.2, we also show the explicit formulae
for the $p$-curvature and contact normality.

######  Theorem 1.4.

Let $\gamma(t)=\big{(}x(t),y(t),z(t)\big{)}\in H_{1}$ be a horizontally
regular curve, not necessarily with horizontally unit-speed. The $p$-curvature
$k(t)$ and contact normality $\tau(t)$ of $\gamma(s)$ are

(1.8)
$\begin{split}k(t)&=\frac{x^{{}^{\prime}}y^{{}^{\prime\prime}}-x^{{}^{\prime\prime}}y^{{}^{\prime}}}{\big{(}(x^{{}^{\prime}})^{2}+(y^{{}^{\prime}})^{2}\big{)}^{\frac{3}{2}}}(t),\\\
\tau(t)&=\frac{xy^{{}^{\prime}}-x^{{}^{\prime}}y+z^{{}^{\prime}}}{\big{(}(x^{{}^{\prime}})^{2}+(y^{{}^{\prime}})^{2}\big{)}^{\frac{1}{2}}}(t).\end{split}$

Notice that in (1.8) the $p$-curvature $k(t)$ depends only on $x(t)$, $y(t)$.
We observe that $k(t)$ is the signed curvature of the plane curve
$\alpha(t):=\pi\circ\gamma(t)=\big{(}x(t),y(t)\big{)}$, where $\pi$ is the
projection onto the $xy$-plane along the $z$-axis. Because the signed
curvature completely describes the plane curves, we have the corollary:

######  Corollary 1.5.

Suppose two horizontally regular curves in $H_{1}$ differ by a Heisenberg
rigid motion, then their projections onto the $xy$-plane along the $z$-axis
differ by a Euclidean rigid motion. In particular, two horizontal curves in
$H_{1}$ differ by a Heisenberg rigid motion if and only if they are congruent
in the Euclidean plane.

As an application, we calculate the $p$-curvature and contact normality for
the geodesics, and obtain the characteristic description of the geodesics.

######  Theorem 1.6.

In $H_{1}$, the geodesics are the horizontally regular curves with constant
$p$-curvature and zero contact normality.

We also establish the fundamental theorem for the surface in $H_{1}$. For a
embedded regular surface $\Sigma\subset H_{1}$, recall that a singular point
$p\in\Sigma$ is a point such that the tangent plane $T_{p}\Sigma$ coincides
with the contact plane $\xi_{(}p)$. Therefore outside the singular set (the
non-singular part of $\Sigma$) is integrated to be a one-dimensional foliation
for the intersection of $T\Sigma$ and $\xi$, which is called the
characteristic foliation.

######  Definition 1.7.

Let $F:U\rightarrow H_{1}$ be a parameterized surface with coordinates $(u,v)$
on $U\subset\operatorname{\mathbb{R}}^{2}$. We say $F$ is the normal
parametrization if

  1. (1)

$F(U)$ is a surface without singular points,

  2. (2)

$F_{u}:=\frac{\partial F}{\partial u}$ defines the characteristic foliation on
$F(U)$,

  3. (3)

$|F_{u}|=1$ for each point $(u,v)\in U$, where the norm is respect to the
Levi-metric.

We call $(u,v)$ a normal coordinates of the surface $F(U)$.

It is easy to see that the normal coordinates always exists locally near the
non-singular point $p\in\Sigma$. In addition, for a normal parameterized
surface $F$, denote $X=F_{u},\ Y=JX$ and $T=\frac{\partial}{\partial z}$. We
define the smooth functions $a,b,c,l$ and $m$ on $U$ by

(1.9) $\begin{array}[]{lll}a:=<F_{v},X>,&b:=<F_{v},Y>,&c=:<F_{v},T>,\\\
l:=<F_{uu},Y>,&m:=<F_{uv},Y>,&\end{array}$

and call $a,b$ and $c$ the coefficients of the first kind of $F$, and $l,m$
the coefficients of the second kind. All coefficients satisfy the
integrability conditions

(1.10) $\begin{split}a_{u}&=bl,\ \ b_{u}=-al+m,\ \ c_{u}=2b,\\\
l_{v}&-m_{u}=0,\end{split}$

where the subscripts denote the partial derivatives.

The following theorem states that those coefficients are the completely
differential invariants for the map $F$.

######  Theorem 1.8.

Let $U\subset\operatorname{\mathbb{R}}^{2}$ be a simply connected open set.
Suppose that $a,b,c,l$ and $m$ are functions on $U$ satisfying the
integrability condition (1.10). Then there exists a normal parameterized
surface $F:U\rightarrow H_{1}$ having $a,b,c$ and $l,m$ as the coefficients of
the first kind and the second kind of $F$, respectively. In addition, any
$\widetilde{F}:U\rightarrow H_{1}$ normal parameterized surface with the same
coefficients must differ from $F$ by a Heisenberg rigid motion, namely,
$\widetilde{F}(u,v)=g\circ F(u,v)$ for all $(u,v)\in U$ for some $g\in
PSH(1)$.

In (5.24), we will show that the function $l$, up to a sign, is independent of
the choices of the normal coordinates, and hence it is a differential
invariant of the surface $F(U)$. Actually $l$ is the $p$-mean curvature.
Particularly, $F(U)$ is a $p$-minimal surface when $l=0$. Such the
parametrization $F:U\rightarrow H_{1}$ is called a normal $p$-minimal
parameterized surface. In this case, the integrability condition (1.10)
becomes

(1.11) $\displaystyle a_{u}=0,\ \ b_{uu}$ $\displaystyle=0,\ \ c_{u}=2b,$
(1.12) $\displaystyle m$ $\displaystyle=b_{u},$

and the coefficients of first kind completely dominate the those of second
kind.

######  Theorem 1.9.

Let $U\subset\operatorname{\mathbb{R}}^{2}$ be a simply connected open set.
Suppose that $a,b$ and $c$ are smooth functions on $U$ satisfying the
integrability condition (1.11). Then there exists a normal $p$-minimal
parameterized surface $F:U\rightarrow H_{1}$ having $a,b$ and $c$ as the
coefficients of first kind of $F$, which also determines the coefficient $b$
of the second kind as in (1.12). In addition, if any normal $p$-minimal
parameterized surface $\widetilde{F}:U\rightarrow H_{1}$ with the same
conditions differs from $F$ by a Heisenberg rigid motion, namely,
$\widetilde{F}(u,v)=g\circ F(u,v)$ in $U$ for some $g\in PSH(1)$.

In Section 5, we will obtain the other invariants on $F(U)$:
$\alpha:=\frac{b}{c}$ (up to a sign, called the $p$-variation), and the
restricted adapted metric $g_{\Theta}|_{\Sigma}$ on the surface $\Sigma$.
Actually $\alpha$ is the function such that the vector field $\alpha e_{2}+T$
is tangent to the surface, where $e_{2}=Je_{1}$ and $e_{1}$ is a unit vector
field tangnet to the characteristic foliation. Let $e_{\Sigma}$ be the unit
vector field tangent to the surface

$e_{\Sigma}=\frac{\alpha e_{2}+T}{\sqrt{1+\alpha^{2}}},$

then we observe that these invariants $\alpha,l,e_{\Sigma}$ satisfy the
integrability condition:

(1.13)
$\begin{split}(1+\alpha^{2})^{\frac{3}{2}}(e_{\Sigma}l)&=(1+\alpha^{2})(e_{1}e_{1}\alpha)-\alpha(e_{1}\alpha)^{2}+4\alpha(1+\alpha^{2})(e_{1}\alpha)\\\
&\hskip 28.45274pt+\alpha(1+\alpha^{2})^{2}K+\alpha
l(1+\alpha^{2})^{\frac{1}{2}}(e_{\Sigma}\alpha)+\alpha(1+\alpha^{2})l^{2},\end{split}$

where $K$ is the Gaussian curvature with respect to $g_{\Theta}|_{\Sigma}$.

After studying the invariants in $H_{1}$, we show the second theorem which
says that the Riemannian metric, induced from the adapted metric, the $p$-mean
curvature, and $p$-variation comprise the complete set of invariants for the
surface without singular points.

######  Theorem 1.10 (The fundamental theorem for surfaces in $H_{1}$).

Let $(\Sigma,g)$ be a $2$-dimensional Riemannian manifold with Guassian
curvature $K$, and let $\alpha^{\prime},l^{\prime}$ be two real-valued
functions on $\Sigma$. Assume that $K$, $\alpha^{{}^{\prime}}$ and
$l^{{}^{\prime}}$ satisfy the integrability condition (1.13). Then for every
non-singular point $p\in\Sigma$, there exists an open neighborhood $U$
containing $p$ and an embedding $f:U\rightarrow H_{1}$ such that

$\displaystyle g$ $\displaystyle=f^{\ast}(g_{\Theta}),$
$\displaystyle\alpha^{{}^{\prime}}$ $\displaystyle=f^{\ast}\alpha,$
$\displaystyle l^{\prime}$ $\displaystyle=f^{\ast}l,$

where $\alpha,l$ are the induced $p$-variation and $p$-mean curvature on
$f(U)$. Moreover, $f$ is unique up to a Heisenberg rigid motion.

As the application for the motion equations and the structure equations, we
introduce the Crofton’s formmula in $H_{1}$. Crofton’s formula is a classic
result of Integral Geometry [12][13], relating the length of a fixed curve and
the number of intersections for the curve and randomly oriented lines passing
through it. In $\operatorname{\mathbb{R}}^{2}$, given a fixed piecewise
regular curve $\gamma$, the Crofton’s formula states that

$\int_{\l\cap\gamma\neq\emptyset}n(\l\cap\gamma)dL=4\cdot length(\gamma),$

where $dL$ is the kinematic density defined on the set of oriented lines in
$\operatorname{\mathbb{R}}^{2}$, $n(\l\cap\gamma)$ is the number of
intersections of the line $\l$ and $\gamma$. We also have the analogues
formula in $H_{1}$. Of particular interest is that the geometric quantity on
one side is the $p$-area which is naturally raised from the variation of the
volume for the domain in CR manifolds [4].

######  Theorem 1.11 (Croton’s formula in $H_{1}$).

Given a regular surface $\mathbb{X}:(u,v)\in\Omega\mapsto\Sigma\subset H_{1}$
for some domain $\Omega\subset\operatorname{\mathbb{R}}^{2}$. Let
$\mathcal{L}$ be the set of all oriented horizontal lines in $H_{1}$ and
$n(\l\cap\Sigma)$ be the number of intersections of the horizontal line
$\l\in\mathcal{L}$ and the surface $\Sigma$. Then we have the Crofton’s
formula in $H_{1}$

$\int_{\l\in\mathcal{L},\
\l\cap\Sigma\neq\emptyset}n(\l\cap\Sigma)dL=4\cdot\operatorname{p-area}(\Sigma),$

where $dL:=dp\wedge d\theta\wedge dt$ is the kinematic density on
$\mathcal{L}$.

We give the outline of the paper. In section $2$, we state two propositions
about the existence and uniqueness of mappings from a smooth manifold into a
Lie group $G$, which underlies our main theorems. In section $3$, we not only
express the representation of $\ PSH(1)$ but discuss how the matrix Lie group
$PSH(1)$ can be interpreted as the set of moving frames on the homogeneous
space $H_{1}=PSH(1)/SO(2)$; the moving frame formula in $H_{1}$ via the (left-
invariant) Maurer-Cartan form will also be derived. In section $4$, we compute
the Darboux derivative of the lift of a horizontally regular curve and give
the proof of the first main theorem; moreover, we calculate the $p$-curvature
and the contact normality of horizontally regular curves and geodesics. In
section $5$, we compute the Darboux derivative of the lift of normal
parameterized surfaces. By deriving the formula of change of the coordinates,
we achieve the complete set of differential invariants for a normal
parameterized surface. In section $6$, by calculating the Darboux derivative
of the lift of $f:\Sigma\rightarrow H_{1}$, we show the fundamental theorem
for surfaces $\Sigma$ in $H_{1}$. In section $7$, one of the applications for
the fundamental theorem of curves, the Crofton’s formula, will be shown there,
which connects the CR geometry and the Integral Geometry. In the last section,
we give the other proof of the Theorem 1.2 for the uniqueness by the method of
moving frame.

Acknowledgment. The first author’s research was supported in part by NCTS and
in part by NSC 100-2628-M-008-001-MY4. He would like to thank Prof. Jih-Hsin
Cheng for his teaching and talking with the author on this topic, Prof. Paul
Yang for his encouragement and advising in the research for the last few
years. The third author would like to express her thanks to Prof. Shu-Cheng
Chang for his teaching, constant encouragement and supports.

##  2\. Calculus on the Lie groups

We recall two basic theorems from Lie groups, which play the essential roles
in the proof of the main theorems. For the details we refer the readers to
[2][7][8][9][14].

Given a connected smooth manifold $M$. Let $G\subset GL(n,R)$ be the matrix
Lie group with Lie algebra $\mathfrak{g}$ and the (left-invariant) Maurer-
Cartan form $\omega$. We first introduce the theorem of uniqueness.

######  Theorem 2.1.

Given two maps $f,\widetilde{f}:M\rightarrow G$, then
$\widetilde{f}^{*}\omega=f^{*}\omega$ if and only if $\widetilde{f}=g\cdot f$
for some $g\in G$.

We call the Lie algebra-valued 1-form $f^{*}\omega$ the Darboux derivative of
the map $f:M\rightarrow G$.

The second result is the theorem of existence.

######  Theorem 2.2.

Suppose that $\phi$ is a $\mathfrak{g}$-valued 1-form on a simply connected
manifold $M$. Then there exists a map $f:M\rightarrow G$ satisfying
$f^{*}\omega=\phi$ if and only if $d\phi=-\phi\wedge\phi$. Moreover, the
resulting map $f$ is unique up to a group action.

We mention that the proof of Theorem 2.2 relies on the Frobenius theorem.

##  3\. The group of pseudo-hermitian transformations on $H_{1}$

###  3.1. The pseudo-hermitian transformations on $H_{1}$

A pseudo-hermitian transformation on $H_{1}$ is a diffeomorphism $\Phi$ on
$H_{1}$ preserving the CR structure $J$ and the contact form $\Theta$; it
satisfies

(3.1) $\Phi_{\ast}J=J\Phi_{\ast}\ \ \text{on}\ \xi\ \ \text{ and }\ \
\Phi^{\ast}\Theta=\Theta\text{ in }H_{1}.$

The trivial example of the pseudo-hermitian transformation is the left
translation $L_{p}$ in $H_{1}$; the other example is defined by
$\Phi_{R}:H_{1}\rightarrow H_{1}$

(3.2) $\Phi_{R}\left(\begin{array}[]{c}x\\\ y\\\
z\end{array}\right)\longrightarrow\left(\begin{array}[]{cc}R&0\\\
0&1\end{array}\right)\left(\begin{array}[]{c}x\\\ y\\\ z\end{array}\right),$

where $R\in SO(2)$ is a 2$\times$2 special orthogonal matrix.

Let $PSH(1)$ be the group of pseudo-hermitian transformations on $H_{1}$. We
shall show that the group $PSH(1)$ exactly consists of all the transformations
of the forms $\Phi_{p,R}:=L_{p}\circ\Phi_{R}$, a transformation $\Phi_{R}$
followed by a left translation $L_{p}$. More precisely, we have

(3.3) $\Phi_{p,R}\left(\begin{array}[]{c}x\\\ y\\\
z\end{array}\right)=\left(\begin{array}[]{c}ax+by+p_{1}\\\ cx+dy+p_{2}\\\
(ap_{2}-cp_{1})x+(bp_{2}-dp_{1})y+z+p_{3}\end{array}\right),$

where $p=(p_{1},p_{2},p_{3})^{t}\in H_{1}$ and
$R=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in SO(2)$.

######  Theorem 3.1.

Let $\Phi:H_{1}\rightarrow H_{1}$ be a pseudo-hermitian transformation. Then
$\Phi=L_{p}\circ\Phi_{R}$ for some $R\in SO(2)$ and $p\in H_{1}$.

###### Proof.

It suffices to consider the pseudo-hermitian transformation
$\Phi:H_{1}\rightarrow H_{1}$ such that $\Phi(0)=0$. Indeed, if $\Phi(0)=p$
for some $p\in H_{1}\setminus\\{0\\}$, then the composition
$L_{p^{-1}}\circ\Phi$ is a transformation fixing the origion. Therefore, we
reduce the proof of Theorem 3.1 to the following Lemma:

######  Lemma 3.2.

Let $\Phi$ be a pseudo-hermitian transformation on $H_{1}$ such that
$\Phi(0)=0$. Then, for any $p\in H_{1}$, the matrix representation of
$\Phi_{\ast}(p)$ with respect to the standard basis $(\frac{\partial}{\partial
x},\frac{\partial}{\partial y},\frac{\partial}{\partial z})$ of
$\mathbb{R}^{3}$ is

(3.4) $\Phi_{\ast}(p)=\begin{pmatrix}\cos\alpha_{0}&-\sin\alpha_{0}&0\\\
\sin\alpha_{0}&\cos\alpha_{0}&0\\\
0&0&1\end{pmatrix}_{\left(\frac{\partial}{\partial x},\frac{\partial}{\partial
y},\frac{\partial}{\partial z}\right)},$

for some real constant $\alpha_{0}$ which is independent of $p$. Thus
$\Phi_{\ast}$ is a constant matrix.

To prove Lemma 3.2, we calculate the matrix representation of $\Phi_{\ast}(p)$
with respect to the basis $(\mathring{e}_{1},\mathring{e}_{2},T)$. For
$i=1,2,$

$\Theta\left(\Phi_{\ast}\mathring{e}_{i}\right)=\left(\Phi^{\ast}\Theta\right)\left(\mathring{e}_{i}\right)=\Theta\left(\mathring{e}_{i}\right)=0,$

we see that the contact bundle $\xi$ is invariant under $\Phi_{\ast}$. In
addition, let $h$ be the Levi metric on $\xi$ defined by
$h(X,Y)=d\Theta(X,JY)$, then

$\begin{split}\Phi^{\ast}h(X,Y)&=h(\Phi_{\ast}X,\Phi_{\ast}Y)=d\Theta(\Phi_{\ast}X,J\Phi_{\ast}Y)\\\
&=d\Theta(\Phi_{\ast}X,\Phi_{\ast}JY)=\Phi^{\ast}(d\Theta)(X,JY)\\\
&=d(\Phi^{\ast}\Theta)(X,JY)=d\Theta(X,JY)=h(X,Y).\end{split}$

Hence $h\left(\Phi_{\ast}X,\Phi_{\ast}Y\right)=h\left(X,Y\right)$ for every
$X,Y\in\xi$. Thus $\Phi_{\ast}$ is orthogonal on $\xi$. On the other hand,

$\Theta(\Phi_{\ast}T)=\Theta\left(\Phi_{\ast}\frac{\partial}{\partial
z}\right)=\left(\Phi^{\ast}\Theta\right)\left(\frac{\partial}{\partial
z}\right)=\Theta\left(\frac{\partial}{\partial z}\right)=1,$

and, for all $X\in\xi$,

$\begin{split}d\Theta(X,\Phi_{\ast}T)&=d\Theta(\Phi_{\ast}\Phi_{\ast}^{-1}X,\Phi_{\ast}T)=(\Phi^{\ast}d\Theta)(\Phi_{\ast}^{-1}X,T)\\\
&=(d\Phi^{\ast}\Theta)(\Phi_{\ast}^{-1}X,T)=d\Theta(\Phi_{\ast}^{-1}X,T)=0.\end{split}$

By the uniqueness of the characteristic vector fields, we have
$\Phi_{\ast}T=T$. From the above argument, we conclude that

$\Phi_{\ast}(p)=\begin{pmatrix}\cos\alpha\left(p\right)&-\sin\alpha\left(p\right)&0\\\
\sin\alpha\left(p\right)&\cos\alpha\left(p\right)&0\\\
0&0&1\end{pmatrix}_{\left(\mathring{e}_{1},\mathring{e}_{2},\frac{\partial}{\partial
z}\right)},$

for some real-valued function $\alpha$ on $H_{1}$.

Next, we rewrite the matrix representation of $\Phi_{\ast}(p)$ from the basis
$\left(\mathring{e}_{1},\mathring{e}_{2},\frac{\partial}{\partial z}\right)$
to the basis $\left(\frac{\partial}{\partial x},\frac{\partial}{\partial
y},\frac{\partial}{\partial z}\right)$. Let
$\Phi=(\Phi^{1},\Phi^{2},\Phi^{3})$, $p=\left(p_{1},p_{2},p_{3}\right)$,
$\mathring{e}_{1}\left(p\right)=\frac{\partial}{\partial
x}+p_{2}\frac{\partial}{\partial z}$ and
$\mathring{e}_{2}\left(p\right)=\frac{\partial}{\partial
y}-p_{1}\frac{\partial}{\partial z}$, then

$\begin{split}\Phi_{\ast}(p)\left(\frac{\partial}{\partial
x}\right)&=\Phi_{\ast}(p)\left[\mathring{e}_{1}\left(p\right)-p_{2}\frac{\partial}{\partial
z}\right]=\Phi_{\ast}(p)\left[\mathring{e}_{1}\left(p\right)\right]-p_{2}\frac{\partial}{\partial
z}\\\
&=\cos\alpha\left(p\right)\mathring{e}_{1}\left[\Phi(p)\right]+\sin\alpha\left(p\right)\mathring{e}_{2}\left[\Phi(p)\right]-p_{2}\frac{\partial}{\partial
z}\\\ &=\cos\alpha\left(p\right)\frac{\partial}{\partial
x}+\sin\alpha\left(p\right)\frac{\partial}{\partial y}\\\ &\ \ \
+\left[\cos\alpha\left(p\right)\Phi^{2}\left(p\right)-\sin\alpha\left(p\right)\Phi^{1}\left(p\right)-p_{2}\right]\frac{\partial}{\partial
z},\end{split}$

and

$\begin{split}\Phi_{\ast}(p)\left(\frac{\partial}{\partial
y}\right)&=\Phi_{\ast}(p)\left[\mathring{e}_{2}\left(p\right)+p_{1}\frac{\partial}{\partial
z}\right]=\Phi_{\ast}(p)\left[\mathring{e}_{2}\left(p\right)\right]+p_{1}\frac{\partial}{\partial
z}\\\
&=-\sin\alpha\left(p\right)\mathring{e}_{1}\left[\Phi(p)\right]+\cos\alpha\left(p\right)\mathring{e}_{2}\left[\Phi(p)\right]+p_{1}\frac{\partial}{\partial
z}\\\ &=-\sin\alpha\left(p\right)\frac{\partial}{\partial
x}+\cos\alpha\left(p\right)\frac{\partial}{\partial y}\\\ &\ \ \
+\left[-\sin\alpha\left(p\right)\Phi^{2}\left(p\right)-\cos\alpha\left(p\right)\Phi^{1}\left(p\right)+p_{1}\right]\frac{\partial}{\partial
z}.\end{split}$

Thus,

(3.5)
$\displaystyle\Phi_{\ast}(p)=\begin{pmatrix}\cos\alpha\left(p\right)&-\sin\alpha\left(p\right)&0\\\
\sin\alpha\left(p\right)&\cos\alpha\left(p\right)&0\\\
\Phi_{x}^{3}(p)&\Phi_{y}^{3}(p)&1\end{pmatrix}_{\left(\frac{\partial}{\partial
x},\frac{\partial}{\partial y},\frac{\partial}{\partial
z}\right),}:=\begin{pmatrix}\Phi^{1}_{x}&\Phi^{1}_{y}&\Phi^{1}_{z}\\\
\Phi^{2}_{x}&\Phi^{2}_{y}&\Phi^{2}_{z}\\\
\Phi^{3}_{x}&\Phi^{3}_{y}&\Phi^{3}_{z}\\\ \end{pmatrix},$

where

(3.6) $\begin{split}\Phi_{x}^{3}(p)&:=\frac{\partial\Phi_{x}^{3}}{\partial
x}=\cos\alpha\left(p\right)\Phi^{2}\left(p\right)-\sin\alpha\left(p\right)\Phi^{1}\left(p\right)-p_{2},\\\
\Phi_{y}^{3}(p)&:=\frac{\partial\Phi_{y}^{3}}{\partial
y}=-\sin\alpha\left(p\right)\Phi^{2}\left(p\right)-\cos\alpha\left(p\right)\Phi^{1}\left(p\right)+p_{1},\end{split}$

and denote the subscripts as the partial derivatives for all $\Phi^{i}$’s. By
(3.5) that $\Phi_{z}^{1}=\Phi_{z}^{2}=0$, it follows that the functions
$\Phi^{1}$ and $\Phi^{2}$ both depend only on $x$ and $y$, and so is $\alpha$.
Moreover, use (3.5) again and the facts $\Phi_{xy}^{1}=\Phi_{yx}^{1}$ and
$\Phi_{xy}^{2}=\Phi_{yx}^{2}$, we have

$\begin{pmatrix}\cos\alpha&-\sin\alpha\\\
\sin\alpha&\cos\alpha\end{pmatrix}\begin{pmatrix}\alpha_{x}\\\
\alpha_{y}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix},$

which implies that $\alpha_{x}=\alpha_{y}=0$. Thus $\alpha$ is a constant on
$H_{1}$, say $\alpha=\alpha_{0}$. From (3.5) and notice that $\Phi(0)=0$, we
finally get

$\begin{split}\Phi^{1}&=x\cos{\alpha_{0}}-y\sin{\alpha_{0}},\\\
\Phi^{2}&=x\sin{\alpha_{0}}+y\cos{\alpha_{0}},\end{split}$

which implies that $\Phi_{x}^{3}=\Phi_{y}^{3}=0$. Therefore

$\Phi_{\ast}(p)=\begin{pmatrix}\cos\alpha_{0}&-\sin\alpha_{0}&0\\\
\sin\alpha_{0}&\cos\alpha_{0}&0\\\
0&0&1\end{pmatrix}_{\left(\frac{\partial}{\partial x},\frac{\partial}{\partial
y},\frac{\partial}{\partial z}\right)}.$

and the result follows.

###  3.2. Representation of $PSH(1)$

The pseudo-hermitian transformation $\Phi_{p,R}$ and the points $(x,y,z)^{t}$
in $H_{1}$ can be respectively represented as

(3.7) $\Phi_{p,R}\leftrightarrow M=\left(\begin{array}[]{cccc}1&0&0&0\\\
p_{1}&a&b&0\\\ p_{2}&c&d&0\\\
p_{3}&ap_{2}-cp_{1}&bp_{2}-dp_{1}&1\end{array}\right),$

and

(3.8) $\left(\begin{array}[]{c}x\\\ y\\\ z\end{array}\right)\leftrightarrow
X=\left(\begin{array}[]{c}1\\\ x\\\ y\\\ z\end{array}\right)$

satisfying

(3.9) $MX=\left(\begin{array}[]{c}1\\\ \Phi_{p,R}\left(\begin{array}[]{c}x\\\
y\\\ z\end{array}\right)\end{array}\right).$

Therefore, $PSH(1)$ can be represented as a matrix group

(3.10) $PSH(1)=\left\\{M\in GL(4,R)\ \Big{|}\ \
M=\left(\begin{array}[]{cccc}1&0&0&0\\\ p_{1}&a&b&0\\\ p_{2}&c&d&0\\\
p_{3}&ap_{2}-cp_{1}&bp_{2}-dp_{1}&1\end{array}\right)\right\\}.$

Let $psh(1)$ be the Lie algebra of $PSH(1)$. It is easy to see that the
element of $psh(1)$ is of the form

(3.11) $\left(\begin{array}[]{cccc}0&0&0&0\\\ x_{1}&0&-x_{1}{}^{2}&0\\\
x_{2}&x_{1}{}^{2}&0&0\\\ x_{3}&x_{2}&-x_{1}&0\end{array}\right).$

and the corresponding Maurer-Cartan form of $PSH(1)$ is of the form

(3.12) $\omega=\left(\begin{array}[]{cccc}0&0&0&0\\\
\omega^{1}&0&-\omega_{1}{}^{2}&0\\\ \omega^{2}&\omega_{1}{}^{2}&0&0\\\
\omega^{3}&\omega^{2}&-\omega^{1}&0\end{array}\right),$

where $\omega_{1}^{2}$ and $\omega^{j},j=1,2,3$ are 1-forms on $PSH(1)$.

###  3.3. The oriented frames on $H_{1}$

The oriented frame $(p;X,Y,T)$ on $H_{1}$ consists of the point $p\in H_{1}$
and the orthonormal vector fields $X\in\xi_{p}$, $Y=JX$ with respect to the
Levi metric. We can identify $PSH(1)$ with the space of all oriented frames on
$H_{1}$ as follows:

(3.13) $PSH(1)\ni M=\left(\begin{array}[]{cccc}1&0&0&0\\\ p_{1}&a&b&0\\\
p_{2}&c&d&0\\\
p_{3}&ap_{2}-cp_{1}&bp_{2}-dp_{1}&1\end{array}\right)\leftrightarrow(p;X,Y,T),$

where

(3.14) $\begin{split}p&=(p_{1},p_{2},p_{3})^{t},\\\
X&=a\frac{\partial}{\partial x}+c\frac{\partial}{\partial
y}+(ap_{2}-cp_{1})\frac{\partial}{\partial t},\\\ Y&=b\frac{\partial}{\partial
x}+d\frac{\partial}{\partial y}+(bp_{2}-dp_{1})\frac{\partial}{\partial
t}.\end{split}$

Actually, we have that $X=a\mathring{e}_{1}(p)+c\mathring{e}_{2}(p)$ and
$Y=b\mathring{e}_{1}(p)+d\mathring{e}_{2}(p)$, hence $M$ is the unique
$4\times 4$ matrix such that

(3.15) $(p;X,Y,T)=(0;\mathring{e}_{1},\mathring{e}_{2},T)M.$

###  3.4. Moving frame formula

Since $PSH(1)$ is a matrix Lie group, the Maurer-Cartan form has to be
$\omega=M^{-1}dM$ or $dM=M\omega$ (see [5]). Immediately one has that

(3.16) $(dp;dX,dY,dT)=(p;X,Y,T)\left(\begin{array}[]{cccc}0&0&0&0\\\
\omega^{1}&0&-\omega_{1}{}^{2}&0\\\ \omega^{2}&\omega_{1}{}^{2}&0&0\\\
\omega^{3}&\omega^{2}&-\omega^{1}&0\end{array}\right).$

Thus, we have reached the moving frame formula:

(3.17) $\displaystyle\begin{split}dp&=\omega^{1}X+\omega^{2}Y+\omega^{3}T,\\\
dX&=\ \omega_{1}{}^{2}Y+\omega^{2}T,\\\ dY&=-\omega_{1}{}^{2}X-\omega^{1}T,\\\
dT&=0.\end{split}$

##  4\. Differential invariants of horizontally regular curves in $H_{1}$

######  Proposition 4.1.

Any horizontally regular curve $\gamma(t)$ can be reparametrized by its
horizontal arc-length $s$ such that $|\gamma_{\xi}(s)|=1$.

###### Proof.

Define $s(t)=\int_{0}^{t}|\gamma_{\xi}^{{}^{\prime}}(u)|du$. Then any
horizontal arc-length differs $s$ up to a constant. By the fundamental theorem
of calculus, we have $\frac{ds}{dt}=|\gamma_{\xi}^{{}^{\prime}}(t)|$. Since

(4.1)
$\frac{d\gamma}{ds}=\frac{d\gamma}{dt}\frac{dt}{ds}=\frac{\gamma^{{}^{\prime}}(t)}{|\gamma_{\xi}^{{}^{\prime}}(t)|},$

we have
$\gamma_{\xi}^{{}^{\prime}}(s)=\frac{\gamma_{\xi}^{{}^{\prime}}(t)}{|\gamma_{\xi}^{{}^{\prime}}(t)|}$,
that is $|\gamma_{\xi}^{{}^{\prime}}(s)|=1$.

######  Definition 4.2.

A lift of a mapping $f:M\rightarrow G/H$ is defined to be a map
$F:M\rightarrow G$ such that the following diagram commutes:

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{f}$$\textstyle{G/H}$

where $G$ is a Lie group, $H$ is a closed Lie subgroup and $G/H$ is a
homogeneous space. In additional, the other lift $\tilde{F}$ of $f$ has to
satisfy

$\begin{array}[]{ccc}\tilde{F}(x)&=&F(x)g(x)\end{array}$

for some map $g:M\rightarrow H$.

######  Remark 4.3.

In the next section, we shall set $G=PSH(1)$, $M=(a,b)\subset R$, $f=\gamma$,
$F=\tilde{\gamma}$, $G/H=PSH(1)/SO(2)$, and identify $PSH(1)/SO(2)$ with
$H_{1}$.

###  4.1. The Proof of Theorem 1.2

By Proposition 4.1, we may assume that the horizontally regular curve
$\gamma(s)$ is parametrized by the horizontal arc-length $s$. Each point on
$\gamma$ uniquely defines the oriented frame

(4.2) $(\gamma(s);X(s),Y(s),T),$

where $X(s)=\gamma_{\xi}^{{}^{\prime}}(s)$ is the horizontally tangent vector
of $\gamma(s)$ and $Y(s)=JX(s)$. By Remark 4.3, there exists the lift
$\tilde{\gamma}\in PSH(1)$ of $\gamma$, which is unique up to a $SO(2)$ group
action, and is still denoted by the same notation

$\widetilde{\gamma}(s)=(\gamma(s);X(s),Y(s),T).$

Let $\omega$ be the Maurer-Cartan form of $PSH(1)$. We shall derive the
Darboux derivative $\widetilde{\gamma}^{\ast}\omega$ of the lift
$\widetilde{\gamma}(s)$:

By the moving frame formula (3.17),

(4.3) $\begin{split}d\widetilde{\gamma}(s)&=\widetilde{\gamma}^{\ast}dp\\\
&=X(s)\widetilde{\gamma}^{\ast}\omega^{1}+Y(s)\widetilde{\gamma}^{\ast}\omega^{2}+T\widetilde{\gamma}^{\ast}\omega^{3}.\end{split}$

We also observe that all pull-back 1-forms by $\widetilde{\gamma}$ are the
multiples of $ds$,

(4.4)
$\begin{split}d\widetilde{\gamma}(s)&=\gamma_{\xi}^{{}^{\prime}}(s)ds+\gamma_{T}^{{}^{\prime}}(s)ds\\\
&=X(s)ds+\gamma_{T}^{{}^{\prime}}(s)ds.\end{split}$

Comparing (4.3) and (4.4), we have

(4.5) $\begin{split}\widetilde{\gamma}^{\ast}\omega^{1}&=ds,\\\
\widetilde{\gamma}^{\ast}\omega^{2}&=0,\\\
\widetilde{\gamma}^{\ast}\omega^{3}&=<\gamma^{{}^{\prime}}(s),T>ds=\tau(s)ds.\end{split}$

Insert $\widetilde{\gamma}^{\ast}\omega^{3}$ into (3.17),

(4.6)
$dX(s)=Y(s)\widetilde{\gamma}^{\ast}\omega_{1}{}^{2}+T\widetilde{\gamma}^{\ast}\omega^{2}=Y(s)\widetilde{\gamma}^{\ast}\omega_{1}{}^{2},$

we get

(4.7)
$\widetilde{\gamma}^{\ast}\omega_{1}{}^{2}=<\frac{dX(s)}{ds},Y(s)>ds=k(s)ds.$

As a consequence we obtain the Darboux derivative of $\widetilde{\gamma}$

(4.8) $\widetilde{\gamma}^{\ast}\omega=\left(\begin{array}[]{cccc}0&0&0&0\\\
1&0&-k(s)&0\\\ 0&k(s)&0&0\\\ \tau(s)&0&-1&0\end{array}\right)ds.$

For any functions $k(s)$ and $\tau(s)$ defined on an open interval $I$.
Suppose $\varphi$ is the $psh(1)$-valued 1-form defined by (4.8). It is easy
to check that $\varphi$ satisfies $d\varphi+\varphi\wedge\varphi=0$.
Therefore, by Theorem 2.2, there exists a curve

$\widetilde{\gamma}(s)=(\gamma(s);X(s),Y(s),T)\in PSH(1)$

such that $\widetilde{\gamma}^{\ast}\omega=\varphi$. By the moving frame
formula (3.17), we have

(4.9) $\begin{split}d\gamma(s)&=X(s)ds+\tau(s)Tds,\\\ dX(s)&=k(s)Y(s)ds,\\\
dY(s)&=-k(s)X(s)ds-Tds,\end{split}$

which means

(4.10) $\begin{split}X(s)&=\gamma_{\xi}^{{}^{\prime}}(s),\\\
k(s)&=<\frac{dX(s)}{ds},Y(s)>,\\\
\tau(s)&=<\frac{d\gamma(s)}{ds},T>.\end{split}$

This completes the proof of the existence.

To prove the uniqueness, suppose that two horizontally regular curves
$\gamma_{1}$ and $\gamma_{2}$ have the same $p$-curvature $k(s)$ and contact
normality $\tau(s)$. The identity (4.8) shows that they must have the same
Darboux derivatives

$\widetilde{\gamma}_{1}^{\ast}\omega=\widetilde{\gamma}_{2}^{\ast}\omega.$

Therefore, by Theorem 2.1, there exists $g\in PSH(1)$ such that
$\widetilde{\gamma}_{2}(s)=g\circ\widetilde{\gamma}_{1}(s)$, hence
$\gamma_{2}(s)=g\circ\gamma_{1}(s)$ for all $s$. This completes the proof of
the uniqueness up to a group action.

###  4.2. The derivation of the $p$-curvature and the contact normality

In the subsection, we will compute the $p$-curvature and the contact normality
for horizontally regular curves (Theorem 1.4) and for the geodesics in $H_{1}$
(Theorem 1.6).

###### Proof of Theorem 1.4.

Let $\gamma(t)=(x(t),y(t),z(t))$ be a horizontally regular curve. The
horizontal arc-length $s$ is defined by

(4.11) $s(t)=\int_{0}^{t}|\gamma_{\xi}^{{}^{\prime}}(u)|du.$

We first observe that there is the natural decomposition

(4.12)
$\begin{split}\gamma^{{}^{\prime}}(t)&=(x^{{}^{\prime}}(t),y^{{}^{\prime}}(t),z^{{}^{\prime}}(t))=x^{{}^{\prime}}(t)\frac{\partial}{\partial
x}+y^{{}^{\prime}}(t)\frac{\partial}{\partial
y}+z^{{}^{\prime}}(t)\frac{\partial}{\partial z}\\\
&=\underbrace{x^{{}^{\prime}}(t)\mathring{e}_{1}+y^{{}^{\prime}}(t)\mathring{e}_{2}}_{\gamma^{\prime}_{\xi}(t)}+\underbrace{(z^{{}^{\prime}}(t)+xy^{{}^{\prime}}(t)-yx^{{}^{\prime}}(t))\frac{\partial}{\partial
z}}_{\gamma^{\prime}_{T}(t)},\end{split}$

where we abuse the notations by $\frac{\partial}{\partial z}=T$. Let
$\bar{\gamma}(s)$ be the reparametrization of $\gamma(t)$ by the horizontal
arc-length $s$. Since
$\gamma^{{}^{\prime}}(t)=\bar{\gamma}^{{}^{\prime}}(s)\frac{ds}{dt}$, by
comparing with the decomposition (4.12), one has

(4.13)
$\begin{split}\bar{\gamma}_{\xi}^{{}^{\prime}}(s)&=\frac{dt}{ds}(x^{{}^{\prime}}(t)\mathring{e}_{1}+y^{{}^{\prime}}(\mathring{e}_{2})),\\\
\bar{\gamma}_{T}^{{}^{\prime}}(s)&=\frac{dt}{ds}\left((z^{{}^{\prime}}(t)+xy^{{}^{\prime}}(t)-yx^{{}^{\prime}}(t))T\right).\end{split}$

For the $p$-curvature, by (4.13), note that
$X(s)=\frac{dt}{ds}(x^{{}^{\prime}}(t)\mathring{e}_{1}+y^{{}^{\prime}}(t)\mathring{e}_{2})$,
and
$Y(s)=JX(s)=\frac{dt}{ds}(x^{{}^{\prime}}(t)\mathring{e}_{2}-y^{{}^{\prime}}(t)\mathring{e}_{1})$.
A straight-forward computation shows

(4.14)
$\begin{split}\frac{dX(s)}{ds}&=\frac{d}{ds}\left(\frac{dt}{ds}\left(x^{{}^{\prime}}(t),y^{{}^{\prime}}(t),x^{{}^{\prime}}y(t)-xy^{{}^{\prime}}(t)\right)\right)\\\
&=\left(\frac{dt}{ds}\right)^{2}\left(x^{{}^{\prime\prime}}(t),y^{{}^{\prime\prime}}(t),x^{{}^{\prime\prime}}y(t)-xy^{{}^{\prime\prime}}(t)\right)+\frac{d^{2}t}{ds^{2}}\left(x^{{}^{\prime}}(t),y^{{}^{\prime}}(t),x^{{}^{\prime}}y(t)-xy^{{}^{\prime}}(t)\right)\\\
&=\left(x^{{}^{\prime\prime}}(t)\left(\frac{dt}{ds}\right)^{2}+x^{{}^{\prime}}(t)\frac{d^{2}t}{ds^{2}}\right)\mathring{e}_{1}+\left(y^{{}^{\prime\prime}}(t)\left(\frac{dt}{ds}\right)^{2}+y^{{}^{\prime}}(t)\frac{d^{2}t}{ds^{2}}\right)\mathring{e}_{2},\end{split}$

so

(4.15) $\begin{split}k(s)&=<\frac{dX(s)}{ds},Y(s)>\\\
&=-\left(x^{{}^{\prime\prime}}(t)\left(\frac{dt}{ds}\right)^{2}+x^{{}^{\prime}}(t)\frac{d^{2}t}{ds^{2}}\right)y^{{}^{\prime}}(t)\frac{dt}{ds}+\left(y^{{}^{\prime\prime}}(t)\left(\frac{dt}{ds}\right)^{2}+y^{{}^{\prime}}(t)\frac{d^{2}t}{ds^{2}}\right)x^{{}^{\prime}}(t)\frac{dt}{ds}\\\
&=-\left(x^{{}^{\prime\prime}}(t)y^{{}^{\prime}}(t)-x^{{}^{\prime}}(t)y^{{}^{\prime\prime}}(t)\right)\left(\frac{dt}{ds}\right)^{3}\\\
&=\frac{x^{{}^{\prime}}y^{{}^{\prime\prime}}-x^{{}^{\prime\prime}}y^{{}^{\prime}}}{\big{(}(x^{{}^{\prime}})^{2}+(y^{{}^{\prime}})^{2}\big{)}^{\frac{3}{2}}}(t).\end{split}$

Again by (4.13), the contact normality has to be

(4.16)
$\begin{split}\tau(s)&=<\bar{\gamma}^{{}^{\prime}}(s),T>=<\bar{\gamma}_{T}^{{}^{\prime}}(s),T>\\\
&=\frac{dt}{ds}(z^{{}^{\prime}}(t)+xy^{{}^{\prime}}(t)-yx^{{}^{\prime}}(t))\\\
&=\frac{xy^{{}^{\prime}}-x^{{}^{\prime}}y+z^{{}^{\prime}}}{\big{(}(x^{{}^{\prime}})^{2}+(y^{{}^{\prime}})^{2}\big{)}^{\frac{1}{2}}}(t),\end{split}$

and the result follows.

Next we use (4.15) and (4.16) to compute the $p$-curvature and the contact
normality for the geodesics in $H_{1}$.

###### Proof of Theorem 1.6.

Recall [1] that the Hamiltonian system on $H_{1}$ for the geodesics is

(4.17)
$\begin{array}[]{lll}\dot{x}^{k}\left(t\right)&=&h^{kj}\left(x\left(t\right)\right)\xi_{j}\left(t\right)\\\
\dot{\xi}_{k}\left(t\right)&=&-\frac{1}{2}\sum\limits_{i,j=1}^{3}\frac{\partial
h^{ij}\left(x\right)}{\partial x^{k}}\xi_{i}\xi_{j},\text{
}k=1,2,3,\end{array}$

where

$h^{ij}\left(x^{1},x^{2},x^{3}\right)=\begin{pmatrix}1&0&x^{2}\\\
0&1&-x^{1}\\\
x^{2}&-x^{1}&\left(x^{1}\right)^{2}+\left(x^{2}\right)^{2}\end{pmatrix}.\qquad$

So the Hamiltonian system (4.17) can be expressed by

(4.18) $\begin{split}\dot{x}^{1}\left(t\right)&=\xi_{1}+x^{2}\xi_{3},\\\
\dot{x}^{2}\left(t\right)&=\xi_{2}-x^{1}\xi_{3},\\\
\dot{x}^{3}\left(t\right)&=x^{2}\xi_{1}-x^{1}\xi_{2}+\xi_{3}\left[\left(x^{1}\right)^{2}+\left(x^{2}\right)^{2}\right],\\\
\dot{\xi}_{1}\left(t\right)&=\xi_{2}\xi_{3}-x^{1}\xi_{3}^{2},\\\
\dot{\xi}_{2}\left(t\right)&=-\xi_{1}\xi_{3}-x^{2}\xi_{3}^{2},\\\
\dot{\xi}_{3}\left(t\right)&=0.\end{split}$

Since $\dot{\xi}_{3}\left(t\right)=0$, we have $\xi_{3}\left(t\right)=c_{3}$
for some constant $c_{3}$. When $c_{3}=0$, one has
$x\left(t\right)=\left(c_{1}t+d_{1},c_{2}t+d_{2},\left(c_{1}d_{2}-c_{2}d_{1}\right)t+d_{3}\right)$,
and this implies $k\left(t\right)=0$ and $\tau\left(t\right)=0;$ when
$c_{3}>0$, one has

(4.19) $\displaystyle x\left(t\right)$
$\displaystyle=\left(x^{1}\left(t\right),x^{2}\left(t\right),x^{3}\left(t\right)\right),$

where

$\displaystyle x^{1}\left(t\right)$
$\displaystyle=a_{1}\sin\left(2c_{3}t\right)+a_{2}\cos\left(2c_{3}t\right)+d_{1},$
$\displaystyle x^{2}\left(t\right)$
$\displaystyle=-a_{2}\sin\left(2c_{3}t\right)+a_{1}\cos\left(2c_{3}t\right)+d_{2},$
$\displaystyle x^{3}\left(t\right)$
$\displaystyle=\left(a_{2}d_{1}+a_{1}d_{2}\right)\sin\left(2c_{3}t\right)+\left(a_{2}d_{2}-a_{1}d_{1}\right)\cos\left(2c_{3}t\right)$
$\displaystyle\hskip
28.45274pt+2c_{3}\left(a_{1}^{2}+a_{2}^{2}\right)t+d_{3}.$

Hence
$k\left(t\right)=-\frac{1}{\left[\left(a_{1}^{2}+a_{2}^{2}\right)\right]^{\frac{1}{2}}}<0$
and $\tau\left(t\right)=0;$ finally, when $c_{3}<0$, one has

(4.20) $\displaystyle x\left(t\right)$
$\displaystyle=\left(x^{1}\left(t\right),x^{2}\left(t\right),x^{3}\left(t\right)\right),$

where

$\displaystyle x^{1}\left(t\right)$
$\displaystyle=a_{1}\sin\left(-2c_{3}t\right)+a_{2}\cos\left(-2c_{3}t\right)+d_{1}$
$\displaystyle x^{2}\left(t\right)$
$\displaystyle=a_{2}\sin\left(-2c_{3}t\right)-a_{1}\cos\left(-2c_{3}t\right)+d_{2}$
$\displaystyle x^{3}\left(t\right)$
$\displaystyle=\left(a_{1}d_{1}+a_{2}d_{2}\right)\sin\left(-2c_{3}t\right)-\left(a_{2}d_{1}-a_{1}d_{2}\right)\cos\left(-2c_{3}t\right)$
$\displaystyle\hskip
28.45274pt+2c_{3}\left(a_{1}^{2}+a_{2}^{2}\right)t+d_{3}.$

Hence
$k\left(t\right)=\frac{1}{\left[\left(a_{1}^{2}+a_{2}^{2}\right)\right]^{\frac{1}{2}}}>0$
and $\tau\left(t\right)=0.$

The calculations above show that a horizontal curve is congruent to a geodesic
if it has positive constant $p$-curvature. Conversely, it is easy to prove
that any geodesic acted by a symmetry is still a geodesic. Therefore we
complete the proof of Theorem 1.6.

######  Remark 4.4.

Actually, the geodesics (4.19) for $c_{3}>0$ and (4.20) for $c_{3}<0$ travel
along the same path with reverse direction.

##  5\. Differential invariants of parametrized surfaces in $H_{1}$

###  5.1. The proof of Theorem 1.8

Let $F:U\rightarrow H_{1}$ be a normal parametrized surface with $a,b,c,l$ and
$m$ as the coefficients in (1.9). Denote the unique lift $\widetilde{F}$ of
$F$ to $PSH(1)$ as

$\displaystyle\widetilde{F}=<F(u,v);X(u,v),Y(u,v),T>,$ $\displaystyle
X(u,v)=F_{u}(u,v),\ JX(u,v)=Y(u,v).$

For the convenience, henceforward we simplify $F(u,v)$ by $F$, and $X(u,v)$ by
$X$ and so on. We first derive the Darboux derivative
$\widetilde{F}^{\ast}\omega$ of $\widetilde{F}$:

By the moving frame formula (3.17),

(5.1)
$\begin{split}dF&=X(\widetilde{F}^{\ast}\omega^{1})+Y(\widetilde{F}^{\ast}\omega^{2})+T(\widetilde{F}^{\ast}\omega^{3})\\\
&=F_{u}du+F_{v}dv,\end{split}$

and apply on $\frac{\partial}{\partial u}$ to get

(5.2) $\begin{split}F_{u}&=dF(\frac{\partial}{\partial
u})=X(\widetilde{F}^{\ast}\omega^{1})(\frac{\partial}{\partial
u})+Y(\widetilde{F}^{\ast}\omega^{2})(\frac{\partial}{\partial
u})+T(\widetilde{F}^{\ast}\omega^{3})(\frac{\partial}{\partial u}.\end{split}$

Compare the coefficients in (5.1)(5.2) and note that $F_{u}=X$, we have

(5.3) $(\widetilde{F}^{\ast}\omega^{1})(\frac{\partial}{\partial u})=1,\ \
(\widetilde{F}^{\ast}\omega^{2})(\frac{\partial}{\partial
u})=(\widetilde{F}^{\ast}\omega^{3})(\frac{\partial}{\partial u})=0.$

Next we insert $\frac{\partial}{\partial v}$ into (5.1) and compare the
coefficients, one has

(5.4) $\begin{split}(\widetilde{F}^{\ast}\omega^{1})(\frac{\partial}{\partial
v})&=<F_{v},X>=a,\\\ (\widetilde{F}^{\ast}\omega^{2})(\frac{\partial}{\partial
v})&=<F_{v},Y>=b,\\\ (\widetilde{F}^{\ast}\omega^{3})(\frac{\partial}{\partial
v})&=<F_{v},T>=c.\end{split}$

Combine (5.3) and (5.4) to get

(5.5) $\begin{split}\widetilde{F}^{\ast}\omega^{1}&=du+adv,\\\
\widetilde{F}^{\ast}\omega^{2}&=bdv,\\\
\widetilde{F}^{\ast}\omega^{3}&=cdv.\end{split}$

To derive $\widetilde{F}^{*}\omega_{1}^{2}$, we use (3.17) again and repeat
the same process above. Since

(5.6)
$\begin{split}dX&=Y(\widetilde{F}^{\ast}\omega_{1}^{2})+T(\widetilde{F}^{\ast}\omega^{2})\\\
&=Y(\widetilde{F}^{\ast}\omega_{1}^{2})(\frac{\partial}{\partial
u})du+Y(\widetilde{F}^{\ast}\omega_{1}^{2})(\frac{\partial}{\partial
v})dv+bTdv,\end{split}$

and

(5.7) $dX=dF_{u}=F_{uu}du+F_{uv}dv.$

We obtain

(5.8)
$\begin{split}(\widetilde{F}^{\ast}\omega_{1}^{2})(\frac{\partial}{\partial
u})&=<F_{uu},Y>=l,\\\
(\widetilde{F}^{\ast}\omega_{1}^{2})(\frac{\partial}{\partial
v})&=<F_{uv},Y>=m,\\\ \omega_{1}^{2}&=ldu+mdv,\\\ b&=<F_{uv},T>,\\\
0&=<F_{uv},X>=<F_{uu},X>=<F_{uu},T>.\end{split}$

Therefore, by (5.5) and (5.8) we have reached the Darboux derivative

(5.9) $\widetilde{F}^{\ast}\omega=\left(\begin{array}[]{cccc}0&0&0&0\\\
du+adv&0&-ldu-mdv&0\\\ bdv&ldu+mdv&0&0\\\ cdv&bdv&-du-
adv&0\end{array}\right).$

By (5.9), the coefficients $a,b,c,l,m$ uniquely determine the Darboux
derivative, and it completes the proof of the uniqueness.

For the existence, suppose $a,b,c$ and $m,l$ are functions defined on $U$.
Suppose $\phi$ is the $psh(1)$-valued 1-form defined by (5.9). Then we have

(5.10) $d\phi=\left(\begin{array}[]{cccc}0&0&0&0\\\ \frac{\partial a}{\partial
u}&0&\frac{\partial l}{\partial v}-\frac{\partial m}{\partial u}&0\\\
\frac{\partial b}{\partial u}&-\frac{\partial l}{\partial v}+\frac{\partial
m}{\partial u}&0&0\\\ \frac{\partial c}{\partial u}&\frac{\partial b}{\partial
u}&-\frac{\partial a}{\partial u}&0\end{array}\right)du\wedge dv,$

and

(5.11) $\phi\wedge\phi=\left(\begin{array}[]{cccc}0&0&0&0\\\ -lb&0&0&0\\\
al-m&0&0&0\\\ -2b&-m+al&bl&0\end{array}\right)du\wedge dv.$

Thus, $\phi$ satisfies the integrability condition $d\phi=-\phi\wedge\phi$ if
and only lf the coefficients $a,b,c,l$ and $m$ satisfy the integrability
condition (1.10). Therefore Theorem 2.2 implies there exists a map

$\widetilde{F}^{\ast}(u,v)=\left(F(u,v);X(u,v),Y(u,v),T\right)$

such that $\widetilde{F}^{\ast}\omega=\phi$. Finally, the moving frame formula
(3.17) implies that $F:U\rightarrow H_{1}$ is a map with $a,b,c,l$ and $m$ as
the coefficients of first kind and second kind respectively.

###  5.2. Invariants of surfaces

Let $\Sigma\hookrightarrow H_{1}$ be a surface such that all points on
$\Sigma$ are non-singular. For each point $p\in\Sigma$, one can choose a
normal parametrization $F:(u,v)\in U\rightarrow\Sigma$ around $p$ such that

(5.12) $F_{u}=\frac{\partial F}{\partial u}=X,$

where $X$ is an unit vector field defining the characteristic foliation. The
following lemma characterizes the normal coordinates.

######  Lemma 5.1.

The normal coordinates is determined up to a transformation of the form

(5.13) $\begin{split}\widetilde{u}&=\pm u+g(v)\\\
\widetilde{v}&=h(v),\end{split}$

for some smooth functions $g(v),h(v)$ with $\frac{\partial h}{\partial v}\neq
0$.

###### Proof.

Suppose that $(\widetilde{u},\widetilde{v})$ is any normal coordinates around
$p$, i.e.,

(5.14) $F_{\widetilde{u}}=\widetilde{X},$

where $\widetilde{X}=\pm X$. We have the formula for the change of the
coordinates

(5.15)
$\begin{split}F_{u}&=F_{\widetilde{u}}\frac{\partial\widetilde{u}}{\partial
u}+F_{\widetilde{v}}\frac{\partial\widetilde{v}}{\partial u},\\\
F_{v}&=F_{\widetilde{u}}\frac{\partial\widetilde{u}}{\partial
v}+F_{\widetilde{v}}\frac{\partial\widetilde{v}}{\partial v}.\end{split}$

Expand
$F_{\widetilde{v}}=\widetilde{a}\widetilde{X}+\widetilde{b}\widetilde{Y}+\widetilde{c}\widetilde{T}$
by the orthonormal basis $\\{\widetilde{X},\widetilde{Y},\widetilde{T}\\}$.
The first identity of (5.15) implies

(5.16)
$\begin{split}X&=F_{u}=\widetilde{X}\frac{\partial\widetilde{u}}{\partial
u}+\left(\widetilde{a}\frac{\partial\widetilde{v}}{\partial
u}\widetilde{X}+\widetilde{b}\frac{\partial\widetilde{v}}{\partial
u}\widetilde{Y}+\widetilde{c}\frac{\partial\widetilde{v}}{\partial
u}\widetilde{T}\right)\\\ &=\left(\frac{\partial\widetilde{u}}{\partial
u}+\widetilde{a}\frac{\partial\widetilde{v}}{\partial
u}\right)\widetilde{X}+\widetilde{b}\frac{\partial\widetilde{v}}{\partial
u}\widetilde{Y}+\widetilde{c}\frac{\partial\widetilde{v}}{\partial
u}\widetilde{T}.\end{split}$

Since $p$ is a non-singular point, we see that $\widetilde{c}\neq 0$ around
$p$, so

(5.17) $\frac{\partial\widetilde{v}}{\partial u}=0,$

namely, $\widetilde{v}=h(v)$ for some function $h(v)$. In addition, comparing
the coefficient of $X$, we have

(5.18) $\pm 1=\frac{\partial\widetilde{u}}{\partial
u}+\widetilde{a}\frac{\partial\widetilde{v}}{\partial
u}=\frac{\partial\widetilde{u}}{\partial u},$

hence $\widetilde{u}=\pm u+g(v)$ for some function $g(v)$. Finally we compute

(5.19) $\det{\left(\begin{array}[]{cc}\frac{\partial\widetilde{u}}{\partial
u}&\frac{\partial\widetilde{u}}{\partial v}\\\
\frac{\partial\widetilde{v}}{\partial u}&\frac{\partial\widetilde{v}}{\partial
v}\end{array}\right)}=\det{\left(\begin{array}[]{cc}\pm 1&\frac{\partial
g}{\partial v}\\\ 0&\frac{\partial h}{\partial
v}\end{array}\right)}=\pm\frac{\partial h}{\partial v}\neq 0,$

and the result follows.

As what did in the previous section (5.9), we can also derive the the Darboux
derivative $\widetilde{F}^{*}\omega$ for the normal parametrization as. One
obtains four 1-forms defined on $\Sigma$ locally as follows:

(5.20)
$\begin{array}[]{ccl}I=\widetilde{F}^{*}\omega^{1}=du+adv,&II=\widetilde{F}^{*}\omega^{2}=bdv,&III=\widetilde{F}^{*}\omega^{3}=cdv\\\
IV=\widetilde{F}^{*}\omega_{1}^{2}=ldu+mdv,&&\end{array}$

where functions $a,b,c,m$ and $l$ are defined as (1.9). We will show that
those four 1-forms are invariants under the change of coordinates.

######  Proposition 5.2.

Suppose $\widetilde{I},\widetilde{II},\widetilde{III},\widetilde{IV}$ are
those defined as (5.20) with respect to the other normal coordinates
$(\widetilde{u},\widetilde{v})$. Then we have

(5.21) $\displaystyle\widetilde{I}=\pm I,\ \widetilde{II}=\pm II,\
\widetilde{III}=III,\ \widetilde{IV}=IV.$

###### Proof.

Suppose
$\widetilde{a},\widetilde{b},\widetilde{c},\widetilde{l},\widetilde{m}$ are
the coefficients of first and second kinds with respect to the normal
coordinates $(\widetilde{u},\widetilde{v})$. We point out that all such the
coefficients have the same expression as in (1.9) w.r.t. the new coordinates
except for $\widetilde{X}=\pm X$ and $\widetilde{Y}=J\widetilde{X}=\pm Y$.

By lemma 5.1, there exists the functions $g(v)$ and $h(v)$ such that

$\begin{split}\widetilde{u}&=\pm u+g(v),\\\ \widetilde{v}&=h(v).\end{split}$

We compute the transformation laws of the coefficients of the fundamental
forms

(5.22)
$\begin{split}a&=<F_{v},X>=<F_{\widetilde{u}}\frac{\partial\widetilde{u}}{\partial
v}+F_{\widetilde{v}}\frac{\partial\widetilde{v}}{\partial v},X>\\\ &=<\pm
X\frac{\partial g}{\partial v}+F_{\widetilde{v}}\frac{\partial h}{\partial
v},X>\\\ &=\pm\left(\frac{\partial g}{\partial v}+\frac{\partial h}{\partial
v}\widetilde{a}\right).\end{split}$

Similarly, we have

(5.23) $b=\pm\frac{\partial h}{\partial v}\widetilde{b},\ c=\frac{\partial
h}{\partial v}\widetilde{c},$

so $F_{u}=\pm F_{\widetilde{u}}$ and
$F_{uu}=\pm(F_{\widetilde{u}\widetilde{u}}\frac{\partial\widetilde{u}}{\partial
u}+F_{\widetilde{u}\widetilde{v}}\frac{\partial\widetilde{v}}{\partial
u})=F_{\widetilde{u}\widetilde{u}}$. Thus

(5.24) $l=\pm\widetilde{l}.$

Similarly

(5.25) $m=\frac{\partial g}{\partial v}\widetilde{l}+\frac{\partial
h}{\partial v}\widetilde{m}.$

From the transformation laws (5.22), (5.23), (5.24) and (5.25), the result
(5.21) follows.

######  Remark 5.3.

In the previous proof (5.23), denote

(5.26) $\displaystyle\alpha=\frac{b}{c},\hskip
28.45274pt\widetilde{\alpha}=\frac{\widetilde{b}}{\widetilde{c}},$

then we have $\alpha=\pm\widetilde{\alpha}$. Actually, $\alpha$ is a function
defined on the non-singular part of $\Sigma$, independent of the chooses of
the normal coordinates up to a sign, such that $\alpha e_{2}+T\in T\Sigma$,
and hence an invariant of $\Sigma$ on the non-singular part. Similarly, from
(5.24), so is for $l$, which actually is the $p$-mean curvature.

######  Remark 5.4.

We point out that the signs appearing for $\alpha$ and $l$ are due to the
different choices of the orientations. Indeed, if one chooses the normal
coordinates with respect to a fixed orientation of the characteristic
foliation, then we will have $\alpha=\widetilde{\alpha}$ and
$l=\widetilde{l}$.

Besides the invariants $\alpha$ and $l$, we now proceed the other invariant of
$\Sigma$, which is globally defined on $\Sigma$, not just on the non-singular
part. From Proposition 5.2, we have

(5.27) $I\otimes I+II\otimes II+III\otimes
III=\widetilde{I}\otimes\widetilde{I}+\widetilde{II}\otimes\widetilde{II}+\widetilde{III}\otimes\widetilde{III}.$

Therefore the differential form $I\otimes I+II\otimes II+III\otimes III$ again
is independent of the choices of the normal coordinates, and hence also an
invariant of $\Sigma$. Next we characterize the invariant.

######  Lemma 5.5.

Let $g_{\Theta}$ be the adapted metric on $H_{1}$. Then we have

(5.28) $g_{\Theta}|_{\Sigma}=I\otimes I+II\otimes II+III\otimes III,$

on the non-singular part of $\Sigma$.

###### Proof.

This lemma is a direct consequence of the moving frame formula (3.17).

###  5.3. A complete set of invariants for surfaces in $H_{1}$

In this section, we will obtain the last invariant
$IV=\widetilde{F}^{\ast}\omega_{1}^{2}$ which is completely determined by the
invariants $\alpha,g_{\Theta},l$. We therefore have a complete set of
invariants for the nonsingular part of the surfaces in $H_{1}$.

Let $\Sigma$ be an oriented surface and suppose $f:\Sigma\rightarrow H_{1}$ be
an embedding. For the convenience, we will not distinguish the surfaces
$\Sigma$ and $f(\Sigma)$. For each non-singular point $p\in\Sigma$, we specify
an orthonormal frame $(p;e_{1},e_{2},T)$, where $e_{1}$ is tangent to the
characteristic foliation and $e_{2}=Je_{1}$. A Darboux frame is a moving frame
which is smoothly defined on $\Sigma$ except for the singular points, and
hence there exists a lifting of $f$ to $PSH(1)$ defined by $F$. Now we would
like to compute the Darboux derivative $F^{\ast}\omega$ of $F$. In the
following, instead of $F^{\ast}\omega$, we still use

(5.29) $\omega=\left(\begin{array}[]{cccc}0&0&0&0\\\
\omega^{1}&0&-\omega_{1}{}^{2}&0\\\ \omega^{2}&\omega_{1}{}^{2}&0&0\\\
\omega^{3}&\omega^{2}&-\omega^{1}&0\end{array}\right),$

to express the Darboux derivative. It satisfies the integrability condition
$d\omega+\omega\wedge\omega=0$, that is,

(5.30) $\begin{split}d\omega^{1}&=\omega_{1}^{2}\wedge\omega^{2},\\\
d\omega^{2}&=-\omega_{1}^{2}\wedge\omega^{1},\\\ d\omega^{3}&=2\
\omega^{1}\wedge\omega^{2},\\\ d\omega_{1}^{2}&=0.\end{split}$

Let $g_{\Theta}=h+\Theta^{2}$ be the adapted metric. From Section 5, we have
$\omega^{2}=\alpha\omega^{3}$ on the non-singular part of $\Sigma$, it is easy
to see that

$\begin{split}g_{\Theta}|_{\Sigma}&=\omega^{1}\otimes\omega^{1}+\omega^{2}\otimes\omega^{2}+\omega^{3}\otimes\omega^{3},\\\
&=\omega^{1}\otimes\omega^{1}+(1+\alpha^{2})\omega^{3}\otimes\omega^{3}.\end{split}$

Define

(5.31) $\begin{split}\hat{\omega}^{1}&=\omega^{1},\\\
\hat{\omega}^{2}&=\sqrt{1+\alpha^{2}}\omega^{3}.\end{split}$

This is an orthonormal coframe of $g_{\Theta}|_{\Sigma}$ and the corresponding
dual frame is

(5.32) $\begin{split}\hat{e}_{1}&=e_{1},\\\
\hat{e}_{2}&=e_{\Sigma}=\frac{\alpha
e_{2}+T}{\sqrt{1+\alpha^{2}}}.\end{split}$

Let $\hat{\omega}_{1}^{2}$ be the Levi-Civita connection of
$g_{\Theta}|_{\Sigma}$ with respect to the coframe
$\hat{\omega}^{1},\hat{\omega}^{2}$. By the fundamental theorem in Riemannian
geometry, we have the structure equation

(5.33)
$\begin{split}d\hat{\omega}^{1}&=-\hat{\omega}_{2}^{1}\wedge\hat{\omega}^{2},\\\
d\hat{\omega}^{2}&=-\hat{\omega}_{1}^{2}\wedge\hat{\omega}^{1},\\\
\hat{\omega}_{1}^{2}&=-\hat{\omega}_{2}^{1}.\end{split}$

The following Proposition shows that $\omega_{1}^{2}$ is completely determined
by the induced fundamental form $g_{\Theta}|_{\Sigma}$ and the functions
$\alpha$ and $l$ defined in (5.26).

######  Proposition 5.6.

We have

(5.34)
$\begin{split}\omega_{1}^{2}&=\frac{\alpha}{\sqrt{1+\alpha^{2}}}\hat{\omega_{1}^{2}}+\frac{l}{1+\alpha^{2}}\hat{\omega}^{1}+\frac{e_{1}\alpha}{(1+\alpha^{2})^{\frac{3}{2}}}\hat{\omega}^{2}\\\
&=l\hat{\omega}^{1}+\frac{2\alpha^{2}+(e_{1}\alpha)}{\sqrt{1+\alpha^{2}}}\hat{\omega}^{2},\\\
&\\\
\hat{\omega}_{1}^{2}&=\frac{\alpha}{\sqrt{1+\alpha^{2}}}\omega_{1}^{2}+\frac{2\alpha}{1+\alpha^{2}}\hat{\omega}^{2}\\\
&=\frac{l\alpha}{\sqrt{1+\alpha^{2}}}\hat{\omega}^{1}+\left(2\alpha+\frac{\alpha(e_{1}\alpha)}{1+\alpha^{2}}\right)\hat{\omega}^{2}.\end{split}$

###### Proof.

By $\omega^{2}=\alpha\omega^{3}$ and the second identity of (5.31), we have

$\begin{split}d\omega^{2}&=d\left(\frac{\alpha}{(1+\alpha^{2})^{\frac{1}{2}}}\hat{\omega}^{2}\right)\\\
&=d\left(\frac{\alpha}{(1+\alpha^{2})^{\frac{1}{2}}}\right)\wedge\hat{\omega}^{2}+\frac{\alpha}{(1+\alpha^{2})^{\frac{1}{2}}}d\hat{\omega}^{2}\\\
&=e_{1}\left(\frac{\alpha}{(1+\alpha^{2})^{\frac{1}{2}}}\right)\hat{\omega}^{1}\wedge\hat{\omega}^{2}-\frac{\alpha}{(1+\alpha^{2})^{\frac{1}{2}}}\hat{\omega}_{1}^{2}\wedge\hat{\omega}^{1}\\\
&=\hat{\omega}^{1}\wedge\left(e_{1}\left(\frac{\alpha}{(1+\alpha^{2})^{\frac{1}{2}}}\right)\hat{\omega}^{2}+\frac{\alpha}{(1+\alpha^{2})^{\frac{1}{2}}}\hat{\omega}_{1}^{2}\right),\end{split}$

where we have used the second formula of the structure equation (5.33) at the
third equality above. On the other hand, from the Maurer-Cartan structure
equation (5.30)

$d\omega^{2}=-\omega_{1}^{2}\wedge\omega^{1}=\hat{\omega}^{1}\wedge\omega_{1}^{2}.$

Combine two identities above and use the Cartan lemma, we see that there
exists a function $D$ such that

(5.35)
$\begin{split}\omega_{1}^{2}&=\frac{e_{1}\alpha}{(1+\alpha^{2})^{\frac{3}{2}}}\hat{\omega}^{2}+\frac{\alpha}{(1+\alpha^{2})^{\frac{1}{2}}}\hat{\omega}_{1}^{2}+D\hat{\omega}^{1}.\end{split}$

Similarly,

(5.36)
$\begin{split}-\hat{\omega}_{2}^{1}\wedge\hat{\omega}^{2}&=d\hat{\omega}^{1}=d\omega^{1}=\omega_{1}^{2}\wedge\omega^{2}\\\
&=\frac{\alpha}{\sqrt{1+\alpha^{2}}}\omega_{1}^{2}\wedge\hat{\omega}^{2}.\end{split}$

Again, by Cartan lemma, there exists a function $A$ such that

(5.37)
$-\hat{\omega}_{2}^{1}=\frac{\alpha}{\sqrt{1+\alpha^{2}}}\omega_{1}^{2}+A\hat{\omega}^{2}.$

Finally, use (5.33) again

(5.38)
$\begin{split}-\hat{\omega}_{1}^{2}\wedge\hat{\omega}^{1}&=d\hat{\omega}^{2}=d\left((1+\alpha^{2})^{\frac{1}{2}}\omega^{3}\right)\\\
&=(1+\alpha^{2})^{\frac{1}{2}}d\omega^{3}+d(1+\alpha^{2})^{\frac{1}{2}}\wedge\omega^{3}\\\
&=2\alpha(1+\alpha^{2})^{\frac{1}{2}}\hat{\omega}^{1}\wedge\omega^{3}+\frac{\alpha}{(1+\alpha^{2})^{\frac{1}{2}}}d\alpha\wedge\omega^{3}\\\
&=\left(2\alpha+\frac{\alpha(e_{1}\alpha)}{1+\alpha^{2}}\right)\hat{\omega}^{1}\wedge\hat{\omega}^{2},\end{split}$

where we have used the third formula of (5.30) and
$\hat{\omega}^{2}\wedge\omega^{3}=0$. Therefore, there exists a function $B$
such that

(5.39)
$\hat{\omega}_{1}^{2}=\left(2\alpha+\frac{\alpha(e_{1}\alpha)}{1+\alpha^{2}}\right)\hat{\omega}^{2}+B\hat{\omega}^{1}.$

By (5.35), (5.37), we get

$\begin{split}D&=\omega_{1}^{2}(e_{1})-\frac{\alpha}{\sqrt{1+\alpha^{2}}}\hat{\omega}_{1}^{2}(e_{1})\\\
&=\frac{\omega_{1}^{2}(e_{1})}{1+\alpha^{2}}=\frac{l}{1+\alpha^{2}}.\end{split}$

Similarly, by (5.35), (5.37), (5.39), we obtain

(5.40) $\begin{split}A&=\frac{2\alpha}{1+\alpha^{2}},\\\
B&=\frac{l\alpha}{\sqrt{1+\alpha^{2}}}.\end{split}$

These complete the proof.

##  6\. The derivation of the integrability condition (1.13)

###### Proof.

We compute

(6.1) $\begin{split}0&=d\omega_{1}^{2}\\\
&=d\left(\frac{\alpha}{\sqrt{1+\alpha^{2}}}\hat{\omega_{1}^{2}}+\frac{l}{1+\alpha^{2}}\hat{\omega}^{1}+\frac{e_{1}\alpha}{(1+\alpha^{2})^{\frac{3}{2}}}\hat{\omega}^{2}\right)\\\
&=\Big{\\{}-(1+\alpha^{2})^{\frac{3}{2}}(e_{\Sigma}l)+(1+\alpha^{2})(e_{1}e_{1}\alpha)-\alpha(e_{1}\alpha)^{2}+4\alpha(1+\alpha^{2})(e_{1}\alpha)\\\
&\hskip 42.67912pt+\alpha(1+\alpha^{2})^{2}K+\alpha
l(1+\alpha^{2})^{\frac{1}{2}}(e_{\Sigma}\alpha)+\alpha(1+\alpha^{2})l^{2}\Big{\\}}\frac{\hat{\omega}^{1}\wedge\hat{\omega}^{2}}{(1+\alpha^{2})^{\frac{5}{2}}}.\end{split}$

Therefore the integrability condition (1.13) is equivalent to
$d\omega_{1}^{2}=0$.

##  7\. The proof of Theorem 1.10

###### Proof.

First we show the existence. Define an $psh(1)$-valued 1-form $\phi$ on the
non-singular part of $\Sigma$ by

(7.1) $\phi=\left(\begin{array}[]{cccc}0&0&0&0\\\
\hat{\omega}^{1}&0&-\omega_{1}^{2}&0\\\
\frac{\alpha^{{}^{\prime}}}{\sqrt{1+(\alpha^{{}^{\prime}})^{2}}}\hat{\omega}^{2}&\omega_{1}^{2}&0&0\\\
\frac{1}{\sqrt{1+(\alpha^{{}^{\prime}})^{2}}}\hat{\omega}^{2}&\frac{\alpha^{{}^{\prime}}}{\sqrt{1+(\alpha^{{}^{\prime}})^{2}}}\hat{\omega}^{2}&-\hat{\omega}^{1}&0\end{array}\right),$

where

(7.2)
$\omega_{1}^{2}=\frac{\alpha^{{}^{\prime}}}{\sqrt{1+(\alpha^{{}^{\prime}})^{2}}}\hat{\omega_{1}^{2}}+\frac{l^{{}^{\prime}}}{1+(\alpha^{{}^{\prime}})^{2}}\hat{\omega}^{1}+\frac{e_{1}\alpha^{{}^{\prime}}}{(1+(\alpha^{{}^{\prime}})^{2})^{\frac{3}{2}}}\hat{\omega}^{2}.$

It is easy to check that $\phi$ satisfies $d\phi+\phi\wedge\phi=0$ if and only
if the integrability condition (1.13) holds. Therefore, by Theorem 2.2, for
each point $p\in\Sigma$ there exists an open set $U$ containing $p$ and an
embedding $f:U\rightarrow H_{1}$ such that
$g=f^{*}(g_{\Theta}),\alpha^{{}^{\prime}}=f^{*}\alpha$ and
$l^{{}^{\prime}}=f^{*}l$. For the uniqueness, by Proposition 5.6, the Darboux
derivative is completely determined by the induced metric
$g_{\Theta}|_{\Sigma}$, the $p$-variation $\alpha$ and the $p$-mean curvature
$l$. Therefore, by Theorem 2.1, the embedding into $H_{1}$ is unique up to a
Heisenberg rigid motion.

##  8\. Application: Crofton’s formula

Since the singular set in $H_{1}$ consists of isolated points and the integral
over the set of isolated points has zero measure, we can always assume that
there are no singular points in the context.

######  Definition 8.1.

An oriented horizontal line $\l$ in $H_{1}$ is a straightly oriented line such
that any point $p\in\l$ the tangent vector of the line at $p$ lies on the
contact plane $\xi_{p}$. For the convenience we sometimes call a horizontal
line or a line. Denote $\mathcal{L}$ by the set of all oriented horizontal
lines in $H_{1}$.

######  Proposition 8.2.

Any horizontal line $l\in\mathcal{L}$ can be coordinatized by the triple
$(p,\theta,t)\in\operatorname{\mathbb{R}}\times
S^{1}\times\operatorname{\mathbb{R}}$, and can also be parameterized by a base
point $B=(p\cos\theta,p\sin\theta,t)$ with a horizontally unit-speed vector
$U=(\sin\theta,-\cos\theta,p)$, namely,

(8.1)
$\displaystyle\l(s):(p\cos\theta,p\sin\theta,t)+s(\sin\theta,-\cos\theta,p),\forall
s\in\operatorname{\mathbb{R}}.$

###### Proof.

We consider the projection $\pi(\l)$ of the line $l\in\mathcal{L}$ on the
$xy$-plane. Since $\pi(\l)$ can be uniquely determined by the pair
$(p,\theta)$ where $p\in\operatorname{\mathbb{R}}$ is the oriented distance
from the origin to the line $\pi(l)$ (see [6] or the Remark below) and
$\theta\in[0,2\pi)$ is the angle from the positive $x$-axis to the normal
(Fig. 1.1), the points $(x,y)\in\pi(\l)$ satisfy the equation

(8.2) $\displaystyle x\cos\theta+y\sin\theta=p.$

Fig. 1.1

On the projection $\pi(\l)$, denote the foot point

$b=(p\cos\theta,p\sin\theta),$

and the unit tangent vector of $\pi(l)$ along the projection

(8.3) $\displaystyle u$
$\displaystyle=(\sin\theta,-\cos\theta),|u|_{\operatorname{\mathbb{R}}^{2}}=1,$

where $|u|_{\operatorname{\mathbb{R}}^{2}}$ is the Euclidean length of $u$ on
the $xy$-plane; on the line $\l\in H_{1}$, we have the lifting of the foot
point $b$, called the base point

$B=(p\cos\theta,p\sin\theta,t)\text{ for some }t\in\operatorname{\mathbb{R}}.$

Denote the tangent vector of $\l$ at point $B$ by $T(B)$. Since $\l$ is
horizontal, which implies that
$T(B)\in\xi_{B}:=span\\{\mathring{e}_{1}(B),\mathring{e}_{2}(B)\\}$, so

$\displaystyle T(B)$
$\displaystyle=\alpha\mathring{e}_{1}(B)+\beta\mathring{e}_{2}(B)$
$\displaystyle=\alpha(1,0,p\sin\theta)+\beta(0,1,-p\cos\theta)$ (8.4)
$\displaystyle=(\alpha,\beta,\alpha p\sin\theta-\beta p\cos\theta)$

for some $\alpha,\beta\in\mathbb{R}$. Notice that the projection $\pi(T(B))$
is exactly the unit tangent vector $u$ along the projection $\pi(\l)$. Hence
by comparing the first two components of (8.4) with (8.3) we have

$\displaystyle\alpha$ $\displaystyle=\sin\theta,$ $\displaystyle\beta$
$\displaystyle=-\cos\theta,$

and

$\displaystyle T(B)=(\sin\theta,-\cos\theta,p).$

Therefore by defining the horizontal vector

$U:=T(B)=\sin\theta\mathring{e}_{1}(B)-\cos\theta\mathring{e}_{2}(B),$

we have $|U|_{\xi(B)}=1$, the horizontally unit-speed, and conclude that the
line $\l$ can be uniquely determined by the triple $(p,\theta,t)$(i.e. the
base point $B$) and can also be parameterized by $B+sU$ for any
$s\in\operatorname{\mathbb{R}}$ as shown in (8.1).

######  Remark 8.3.

For our purpose of doing the integration later, the issue of the orientations
are naturally involved. Actually the orientations of lines in $H_{1}$ are
raised from the orientations of lines in $\operatorname{\mathbb{R}}^{2}$: In
[6], denote

$\mathcal{X}_{nonoritned}:=\\{\text{all lines in
}\operatorname{\mathbb{R}}^{2}\\},$

and consider the mapping

$\operatorname{\mathbb{R}}\times
S^{1}\xrightarrow{\phi}\mathcal{X}_{nonoritned},$

which carries $(p,\theta)$ to the line having the equation
$(\ref{projformula})$. It can be checked that $\mathcal{X}_{nonoriented}$ is a
two dimensional nonoriented smooth manifold equipping the two-folds covering
spaces, more precisely one has

$\displaystyle\phi(p,\theta)=\phi(p^{\prime},\theta^{\prime})$

if and only if either

$\displaystyle p=p^{\prime},\theta=\theta^{\prime}\text{ or
}p=-p^{\prime},\theta=\theta^{\prime}+\pi.$

To our purpose, we consider the larger space

$\mathcal{X}_{oritned}:=\\{(p,\theta)\in\mathbb{R}\times S^{1}\\}$

equipping two orientations. (Fig. 1.2)

Fig. 1.2 Two orientations for the line $l$ on $\mathbb{R}^{2}$.

Therefore, in $H_{1}$, instead of using the nonoriented coordinates for the
set

$\\{(p,\theta,t)|p\geq 0,\theta\in[0,2\pi),t\in\operatorname{\mathbb{R}}\\},$

we henceforth consider the set of all horizontally oriented lines

$\mathcal{L}:=\\{(p,\theta,t)\in\operatorname{\mathbb{R}}\times
S^{1}\times\operatorname{\mathbb{R}}\\}$

with two orientations.

Next, we consider the intersections of lines and a fixed regular surface
$\mathbb{X}:(u,v)\in\Omega\rightarrow(x(u,v),y(u,v),z(u,v))\in\Sigma$ embedded
in $H_{1}$ for some domain $\Omega\subset\operatorname{\mathbb{R}}^{2}$. To
describe the position of the intersection in $\operatorname{\mathbb{R}}^{3}$,
one needs exact three variables. We have already known, by Proposition 8.2, a
line can be represented by the triple $(p,\theta,t)$. Hence if we regard lines
and surfaces as a whole system (the configuration space) and use five
variables $\\{(p,\theta,t,u,v)\\}$ to describe the behavior of the
intersections, two additional constraints are necessarily required to make the
number of the freedoms be three. Those constraints can be obtained from the
following Proposition.

######  Proposition 8.4.

Let $\mathbb{X}(u,v)=(x(u,v),y(u,v),z(u,v))\in\Sigma$ be the parameterized
regular surface in $H_{1}$. Then the configuration space $D$ which describes
the horizonal oriented lines intersecting $\Sigma$ should be

$\displaystyle D$
$\displaystyle=\\{(p,\theta,t,u,v)\in\operatorname{\mathbb{R}}\times
S^{1}\times\operatorname{\mathbb{R}}\times\Omega$ $\displaystyle\hskip
28.45274pt|\text{ the lines }(p,\theta,t)\in\mathcal{L}\text{ passing through
the point }\mathbb{X}(u,v)\text{ on }\Sigma\\}$
$\displaystyle=\\{(p,\theta,t,u,v)\in\operatorname{\mathbb{R}}\times
S^{1}\times\operatorname{\mathbb{R}}\times\Omega$ $\displaystyle\hskip
85.35826pt|\text{ the variables }p,\theta,t,u,v\text{ satisfy
}(\ref{constraint1})\text{ and }(\ref{constraint2})\\},$

where

(8.5) $\displaystyle x(u,v)\cos\theta+y(u,v)\sin\theta=p,$ (8.6)
$\displaystyle z(u,v)=t+(x(u,v)\sin\theta-y(u,v)\cos\theta)p.$

###### Proof.

Suppose the line $\l(s)$ parameterized by (8.1) intersects the surface
$\Sigma$ at the point $q$. (Fig. 1.3)

Fig. 1.3

At the point $q$, by Proposition 8.2, we have

(8.7) $\displaystyle x(u,v)$ $\displaystyle=p\cos\theta+s\cdot\sin\theta,$
(8.8) $\displaystyle y(u,v)$ $\displaystyle=p\sin\theta-s\cdot\cos\theta,$
(8.9) $\displaystyle z(u,v)$ $\displaystyle=t+s\cdot p,$

for some $s\in\operatorname{\mathbb{R}}$. By (8.7), (8.8), one has

$\displaystyle x(u,v)\cos\theta+y(u,v)\sin\theta=p,$

which is compatible with $(\ref{projformula})$ and we obtain the first
constraint (8.5). Finally, use (8.7),(8.8) again to solve for the parameter
$s$, and substitute $s$ into $(\ref{z(uv)})$. It is easy to have the second
constraint (8.6)

######  Remark 8.5.

By a simple calculation and (8.5), we observe that

$\displaystyle U(B)$
$\displaystyle=\sin\theta\mathring{e}_{1}(B)-\cos\theta\mathring{e}_{2}(B)$
$\displaystyle=\sin\theta\mathring{e}_{1}(\mathbb{X}(u,v))-\cos\theta\mathring{e}_{2}(\mathbb{X}(u,v))=U(\mathbb{X}(u,v)),$

$i.e.$ the horizontally unit-speed vector field $U$ along the line have the
same vector-value when being evaluated at the based point $B$ and at the
intersection $q=\mathbb{X}(u,v)$.

Actually, the coordinates $(u,v)$ determine where the intersections should be
located on the surface, and the angle $\theta$ decides how those lines
penetrate through the surface. Thus, instead of using $(p,\theta,t)$ as the
coordinates for the configuration space, we can also adopt the triple
$\\{(u,v,\theta)\in\Omega\times S^{1}\\}$ as the coordinates. Since the
intersection $q$ is not only on the line but on the surface, we can derive the
change of the coordinates for those coordinates.

By Remark 8.5 we choose the frame
$\\{\mathbb{X}(u,v);e_{1}(\theta),e_{2}(\theta),T\\}$ on $D$ where

(8.13)
$\displaystyle\left\\{\begin{array}[]{ll}e_{1}&:=\sin\theta\mathring{e}_{1}-\cos\theta\mathring{e}_{2},\\\
e_{2}&:=Je_{1}=\cos\theta\mathring{e}_{1}+\sin\theta\mathring{e}_{2},\\\
T&:=(0,0,1),\end{array}\right.$

(Fig. 1.3) and denote the corresponding coframes
$\\{\mathbb{X}(u,v);\omega^{1},\omega^{2},\Theta\\}$ with the connection
$1$-form $\omega^{2}_{1}$.

The first formula connects the coframes and the coordinates $(p,\theta,t)$ of
the line.

######  Proposition 8.6.

If we choose the frames $\\{\mathbb{X}(u,v);e_{1}(\theta),e_{2}(\theta),T\\},$
defined by $(\ref{frames})$ and the corresponding coframes
$\\{\mathbb{X}(u,v);\omega^{1},\omega^{2},\Theta\\}$ with the connection
$1$-form $\omega^{2}_{1}$, then we have

(8.14) $\displaystyle\omega^{2}$ $\displaystyle=dp+<\mathbb{X},e_{1}>d\theta,$
(8.15) $\displaystyle\omega^{2}_{1}$ $\displaystyle=d\theta,$ (8.16)
$\displaystyle\Theta$ $\displaystyle=dt\text{, mod }d\theta,dp.$

One concludes that

(8.17) $\displaystyle\omega^{2}\wedge\omega^{2}_{1}$ $\displaystyle=dp\wedge
d\theta,$ (8.18) $\displaystyle\omega^{2}\wedge\omega_{1}^{2}\wedge\Theta$
$\displaystyle=dp\wedge d\theta\wedge dt=\pi^{*}dL,$

where $\pi$ is the projection from $D$ to $\mathcal{L}$, and $<,>$ is the
Levi-metric.

###### Proof.

On the surface since
$\mathbb{X}=(x,y,z)=x(1,0,y)+y(0,1,-x)+(0,0,z)=x\mathring{e}_{1}+y\mathring{e}_{2}+zT$,
we have

$<\mathbb{X},e_{1}>=<x\mathring{e}_{1}+y\mathring{e}_{2}+zT,\sin\theta\mathring{e}_{1}-\cos\theta\mathring{e}_{2}>=x\sin\theta-y\cos\theta.$

Thus, by the moving frame formula (3.17) and the first constraint (8.5)

$\displaystyle\omega^{2}$ $\displaystyle=<d\mathbb{X},e_{2}>=<dx\
\mathring{e}_{1}+dy\ \mathring{e}_{2}+\Theta\ \frac{\partial}{\partial
z},e_{2}>$ $\displaystyle=\cos\theta dx+\sin\theta dy$
$\displaystyle=dp+(x\sin\theta-y\cos\theta)d\theta$
$\displaystyle=dp+<\mathbb{X},e_{1}>d\theta;$ $\displaystyle\omega^{2}_{1}$
$\displaystyle=-\omega^{1}_{2}$ $\displaystyle=-<de_{2},e_{1}>$
$\displaystyle=<\sin\theta d\theta\ \mathring{e}_{1}+\cos\theta d\theta\
\mathring{e}_{2},\sin\theta\mathring{e}_{1}+\cos\theta\mathring{e}_{2}>$
$\displaystyle=\sin^{2}\theta\ d\theta+\cos^{2}\theta\ d\theta$
$\displaystyle=d\theta.$

By the second constraint (8.6) and the parameterization of the line (8.1)

$\displaystyle\Theta$ $\displaystyle=dz+xdy-ydx$
$\displaystyle=\left(dt+(x\sin\theta-y\cos\theta)dp+pd(x\sin\theta-y\cos\theta)\right)+xdy-
ydx$ $\displaystyle=dt+(p\sin\theta-y)dx-(p\ \cos\theta-x)dy,\text{ mod
}d\theta,dp$ $\displaystyle=dt+s(\cos\theta dx+\sin\theta dy)\text{, mod
}d\theta,dp,\text{ for some }s\in\operatorname{\mathbb{R}}$
$\displaystyle=dt\text{ , mod }d\theta,dp,$

and the result follows.

The next lemma characterize the $1$-dimension foliation.

######  Lemma 8.7.

Let $E=\alpha\mathbb{X}_{u}+\beta\mathbb{X}_{v}$ be the tangent vector field
defined on the regular surface $\sum=\mathbb{X}(u,v)$. Then the vector $E$ is
also on the contact bundle $\xi$ (and hence in $TH_{1}\cap\xi$) if and only if
pointwisely the coefficients $\alpha$ and $\beta$ satisfy

(8.19) $\displaystyle\alpha t_{u}+\beta t_{v}+x(\alpha y_{u}+\beta
y_{v})-y(\alpha x_{u}+\beta x_{v})=0,$

equivalently,

(8.20)
$\displaystyle\alpha(t_{u}+xy_{u}-yx_{u})+\beta(t_{v}+xy_{v}-yx_{v})=0.$

###### Proof.

First, we assume that
$E=\alpha\mathbb{X}_{u}+\beta\mathbb{X}_{v}=\alpha(x_{u},y_{u},z_{u})+\beta(x_{v},y_{v},z_{v})=c\mathring{e}_{1}+d\mathring{e}_{2}=(c,d,cy-
dx)$ for some constants $c$ and $d$. Compare each component of $E$ to have

$\displaystyle\alpha x_{u}+\beta x_{v}$ $\displaystyle=c,$
$\displaystyle\alpha y_{u}+\beta y_{v}$ $\displaystyle=d,$
$\displaystyle\alpha z_{u}+\beta z_{v}$ $\displaystyle=cy-dx.$

Substitute the last equation by the first two, we get the necessary condition
$\alpha z_{u}+\beta z_{v}=(\alpha x_{u}+\beta x_{v})y-(\alpha y_{u}+\beta
y_{v})x$.

The reverse part can be obtained by the direct computation

$\displaystyle E$ $\displaystyle=(\alpha x_{u}+\beta x_{v},\alpha y_{u}+\beta
y_{v},\alpha z_{u}+\beta z_{v})$ $\displaystyle=(\alpha x_{u}+\beta
x_{v})(1,0,y)+(\alpha y_{u}+\beta y_{v})(0,1,-x)$ $\displaystyle\hskip
28.45274pt+\big{(}0,0,\alpha z_{u}+\beta z_{v}-y(\alpha x_{u}+\beta
x_{v})+x(\alpha y_{u}+\beta y_{v})\big{)}$ $\displaystyle=(\alpha x_{u}+\beta
x_{v})\mathring{e}_{1}+(\alpha y_{u}+\beta y_{v})\mathring{e}_{2}.$

We have used the condition (8.19) in the last equality.

The second formula for the change of coordinates connects the coframes and the
coordinates of the surface.

######  Proposition 8.8.

Suppose we choose the frames
$\\{\mathbb{X}(u,v);e_{1}(\theta),e_{2}(\theta),T\\}$ on $D$ and the coframes
with the connection $1$-form defined by above $(\ref{frames})$. We have the
identity

(8.21)
$\displaystyle\Theta\wedge\omega^{2}\wedge\omega_{1}^{2}=<E,e_{2}>du\wedge
dv\wedge d\theta,$

where the singular foliation

(8.22) $\displaystyle
E:=(z_{u}+xy_{u}-yx_{u})\mathbb{X}_{v}-(z_{v}+xy_{v}-yx_{v})\mathbb{X}_{u}$

defines the characteristic foliation of $\Sigma$, which is induced from the
contact plane $\xi$.

###### Proof.

By Proposition 8.6 and the moving frame formula (3.17)

$\displaystyle\Theta\wedge\omega^{2}\wedge\omega_{1}^{2}$ $\displaystyle=$
$\displaystyle(dz+xdy-ydx)\wedge<d\mathbb{X},e_{2}>\wedge d\theta$
$\displaystyle=$
$\displaystyle\big{(}(z_{u}+xy_{u}-yx_{u})du+(z_{v}+xy_{v}-yx_{v})dv\big{)}\wedge\big{(}<\mathbb{X}_{u},e_{2}>du\wedge
d\theta+<\mathbb{X}_{v},e_{2}>dv\wedge d\theta\big{)}$ $\displaystyle=$
$\displaystyle<(z_{u}+xy_{u}-yx_{u})\mathbb{X}_{v}-(z_{v}+xy_{v}-yx_{v})\mathbb{X}_{u},e_{2}>du\wedge
dv\wedge d\theta$ $\displaystyle=$ $\displaystyle<E,e_{2}>du\wedge dv\wedge
d\theta.$

To prove the vector $E\in TM\cap\xi$, it suffices to show that the
coefficients $\alpha:=(z_{u}+xy_{u}-yx_{u})$ and
$\beta:=-(z_{v}+xy_{v}-yx_{v})$ satisfy the condition $(\ref{belonginboth2})$,
and we complete the proof by the previous Lemma 8.7.

######  Remark 8.9.

In classical Integral Geometry [6] [13], the quantity $dL:=dp\wedge
d\theta\wedge dt$ is called the (kinematic) density of the line
$(p,\theta,t)\in\mathbb{R}^{3}$, which is always chosen to be positive
depending the orientation. Hence, according to (8.18) and (8.21), in the
following proof we have to consider the orientation of $\\{(u,v,\theta)\\}$ to
ensure the positivity of the quantity $<E,e_{2}>$.

###### Proof of Theorem 1.11.

By Remark 8.9, we choose $du\wedge dv\wedge d\theta$ as the orientation of
$D$. Let $D=D^{+}\cup D^{-}$, where

$\displaystyle D^{+}$ $\displaystyle:=\\{(p,\theta,t,u,v)|<E,e_{2}>\geq 0\\},$
$\displaystyle D^{-}$ $\displaystyle:=\\{(p,\theta,t,u,v)|<E,e_{2}>\leq 0\\},$
$\displaystyle\Gamma$ $\displaystyle:=D^{+}\cap D^{-}.$

By the structure equation (5.30),

(8.23) $\displaystyle d(\Theta\wedge\omega^{1})$
$\displaystyle=d\Theta\wedge\omega^{1}-\Theta\wedge d\omega^{1}$
$\displaystyle=(2\omega^{1}\wedge\omega^{2})\wedge\omega^{1}-\Theta\wedge(\omega^{2}_{1}\wedge\omega^{2})$
$\displaystyle=\Theta\wedge\omega^{2}\wedge\omega^{2}_{1}.$

We also have

(8.24) $\displaystyle\Theta\wedge\omega^{1}$ $\displaystyle=(dz+xdy-
ydx)\wedge<d\mathbb{X},e_{1}>$
$\displaystyle=\left((z_{u}+xy_{u}-yx_{u})du+(z_{v}+xy_{v}-yx_{v})dv\right)\wedge\left(<\mathbb{X}_{u},e_{1}>du+<\mathbb{X}_{v},e_{1}>dv\right)$
$\displaystyle=<E,e_{1}>du\wedge dv.$

Thus, integrating the kinematic density $dL$ over the set $\mathcal{L}$, and
use (8.18), (8.23), the Stock’s theorem, and (8.24), which imply

(8.25) $\displaystyle\begin{split}\int_{\l\in\mathcal{L},\
\l\cap\Sigma\neq\emptyset}n(\l\cap\Sigma)dL&=2\Big{(}\int_{D^{+}}\pi^{*}dL-\int_{D^{-}}\pi^{*}dL\Big{)}\\\
&=2\Big{(}\int_{D^{+}}\omega^{2}\wedge\omega^{2}_{1}\wedge\Theta-\int_{D^{-}}\omega^{2}\wedge\omega^{2}_{1}\wedge\Theta\Big{)}\\\
&=2\Big{(}\int_{\partial D^{+}}\Theta\wedge\omega^{1}-\int_{\partial
D^{-}}\Theta\wedge\omega^{1}\Big{)}\\\
&=2\Big{(}\int_{\Gamma^{+}\cup\Gamma}\Theta\wedge\omega^{1}-\int_{\Gamma^{-}\cup\Gamma}\Theta\wedge\omega^{1}\Big{)},\end{split}$

where $\Gamma^{\pm}:=\partial D^{\pm}\setminus\Gamma.$ We point out that for
each line, there are two orientations passing through the surface, so we put
double in front of the integral.
Next, we show that $du\wedge dv=0$ on $\Gamma^{\pm}$. Indeed, by using the
coordinates $\\{(u,v,\theta)\\}$ for the configuration space $D$, any vector
field defined on $\Gamma^{+}$ can be represented by
$A\wedge\frac{\partial}{\partial\theta}\in\partial\Sigma\times S^{1}$ for some
vector $A$ defined on the tangent bundle $T\partial\Sigma$. The value
$du\wedge dv$ evaluated on $\Gamma^{+}$ has to be

$\displaystyle du\wedge
dv(A\wedge\frac{\partial}{\partial\theta})=du(A){\cancel{dv(\frac{\partial}{\partial\theta})}}^{\
=0}-dv(A){\cancel{du(\frac{\partial}{\partial\theta})}}^{\ =0}=0.$

Therefore, (8.25) becomes

(8.26) $\displaystyle\begin{split}\int_{\l\in\mathcal{L},\
\l\cap\Sigma\neq\emptyset}n(\l\cap\Sigma)dL&=2\Big{(}2\int_{\Gamma}\Theta\wedge\omega^{1}+\int_{\Gamma^{+}}\Theta\wedge\omega^{1}-\int_{\Gamma^{-}}\Theta\wedge\omega^{1}\Big{)}\\\
&=4\int_{\Gamma}\Theta\wedge\omega^{1}\\\ &=4\int_{\Gamma}|E|du\wedge dv\\\
&=4\cdot\operatorname{p-area}(\Sigma),\end{split}$

we have used (8.24) and $E$ is parallel to $e_{1}$ on $\Gamma$ at the third
equality.

##  9\. Appendix:The second proof of Theorem 1.2

In this Appendix, we give the alternative proof of Theorem 1.2 for the
uniqueness.

Let $\gamma(s)$ be a horizontally regular curve $\gamma(s)$ parametrized by
horizontal arc-length $s$. Suppose the moving frame $(\gamma(s);X(s),Y(s),T)$
chosen as defined on (4.2). Then we have the moving frame formula

(9.1) $\begin{split}X^{{}^{\prime}}(s)&=k(s)Y(s),\\\
Y^{{}^{\prime}}(s)&=-k(s)X(s)-T,\\\ T^{{}^{\prime}}(s)&=0.\end{split}$

Notice that

(9.2) $\gamma^{{}^{\prime}}(s)=X(s)+\tau(s)T.$

Suppose that two curves $\gamma(s)$ and $\widetilde{\gamma}(s)$ satisfy the
conditions

(9.3) $k(s)=\widetilde{k}(s)\ \ \text{and}\ \ \tau(s)=\widetilde{\tau}(s),\
s\in I.$

Without loss of generality, we may assume that at some point $s_{0}\in I$

(9.4) $\gamma(s_{0})=\widetilde{\gamma}(s_{0}),\
X(s_{0})=\widetilde{X}(s_{0}),\ Y(s_{0})=\widetilde{Y}(s_{0}).$

Define the function $A(s)=<X(s),\widetilde{X}(s)>+<Y(s),\widetilde{Y}(s)>$.
Take the derivative and use formula (9.1),

(9.5)
$\begin{split}A^{{}^{\prime}}&=<X^{{}^{\prime}},\widetilde{X}>+<X,\widetilde{X}^{{}^{\prime}}>+<Y^{{}^{\prime}},\widetilde{Y}>+<Y,\widetilde{Y}^{{}^{\prime}}>\\\
&=k<Y,\widetilde{X}>+\widetilde{k}<X,\widetilde{Y}>+<-kX-T,\widetilde{Y}>+<Y,-\widetilde{k}\widetilde{X}-\widetilde{T}>\\\
&=0.\end{split}$

Since $A(s_{0})=2$, one has $A(s)=2$, and hence that $X(s)=\widetilde{X}(s)$
and $Y(s)=\widetilde{Y}(s)$ for each $s\in I$. Thus, we have
$\gamma_{\xi}^{{}^{\prime}}(s)=\widetilde{\gamma}_{\xi}^{{}^{\prime}}(s)$.
Moreover, by (9.2) and the assumption $\tau(s)=\widetilde{\tau}(s)$, we also
have $\gamma_{T}^{{}^{\prime}}(s)=\widetilde{\gamma}_{T}^{{}^{\prime}}(s)$.
Therefore $\gamma^{{}^{\prime}}(s)=\widetilde{\gamma}^{{}^{\prime}}(s)$, which
implies $\gamma(s)=\widetilde{\gamma}(s)+C$ for some constant $C$. Finally,
since $\gamma(s)=\widetilde{\gamma}(s)$ at $s=s_{0}$, we conclude that $C=0$
and complete the proof.

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