# A characterization of horizontally totally geodesic hypersurfaces in
Heisenberg groups

Andrea Pinamonti  Dipartimento di Matematica, Università di Trento, via
Sommarive 14, 38123 Povo (TN), Italy<EMAIL_ADDRESS>and  Simone
Verzellesi  Dipartimento di Matematica, Università di Trento, via Sommarive
14, 38123 Povo (TN), Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper we achieve a first concrete step towards a better understanding
of the so-called _Bernstein problem_ in higher dimensional Heisenberg groups.
Indeed, in the sub-Riemannian Heisenberg group $\mathbb{H}^{n}$, with
$n\geqslant 2$, we show that the only entire hypersurfaces with vanishing
horizontal symmetric second fundamental form are hyperplanes. This result
relies on a sub-Riemannian characterization of a higher dimensional ruling
property, as well as on the study of sub-Riemannian geodesics on Heisenberg
hypersurfaces.

Key words. Bernstein problem, Heisenberg group, ruled hypersurface, minimal
hypersurface, second fundamental form, totally geodesic.
MSC. 53C17, 49Q05

##  1\. Introduction

In the present paper we achieve a first concrete step towards a better
understanding of the so-called _Bernstein problem_ in higher dimensional sub-
Riemannian Heisenberg groups. Namely, we reduce the solution of the latter to
validity of suitable sub-Riemannian curvature estimates. The characterization
of entire minimal hypersurfaces in higher dimensional sub-Riemannian
Heisenberg groups is an intriguing open problem in the framework of sub-
Riemannian geometry. This issue, which is typically known as _Bernstein
problem_ in view of its Euclidean counterpart (cf. [28] for a complete survey
of the topic), fits into the broader context of minimal hypersurfaces in sub-
Riemannian structures (cf. [9, 10, 11, 16, 19, 25, 27, 29, 39, 40, 41, 48] and
references therein). The study of this and related issues is particularly
relevant in the sub-Riemannian Heisenberg group $\mathbb{H}^{n}$, since the
latter constitutes a prototypical model in the setting of Carnot groups (cf.
[21, 6]), sub-Riemannian manifolds (cf. [2]), CR manifolds (cf. [7]) and
Carnot-Carathéodory spaces (cf. [30]). We briefly recall that the $n$-th
Heisenberg group $(\mathbb{H}^{n},\cdot)$ is $\mathbb{R}^{2n+1}$ endowed with
the group law

$p\cdot
p^{\prime}=(\bar{x},\bar{y},t)\cdot(\bar{x}^{\prime},\bar{y}^{\prime},t^{\prime})=(\bar{x}+\bar{x}^{\prime},\bar{y}+\bar{y}^{\prime},t+t^{\prime}+Q((\bar{x},\bar{y}),(\bar{x}^{\prime},\bar{y}^{\prime}))),$

where

$Q((\bar{x},\bar{y}),(\bar{x}^{\prime},\bar{y}^{\prime}))=\sum_{j=1}^{n}\left(x_{j}^{\prime}y_{j}-x_{j}y_{j}^{\prime}\right)$

and where we denoted points $p\in\mathbb{R}^{2n+1}$ by
$p=(z,t)=(\bar{x},\bar{y},t)=(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n},t)$. With
this operation, $\mathbb{H}^{n}$ is a Carnot group, whose associated
_horizontal distribution_ , which we denote by ${\mathcal{H}}$, is generated
by the left-invariant vector fields

$X_{j}=\frac{\partial}{\partial x_{j}}+y_{j}\frac{\partial}{\partial
t}\qquad\text{and}\qquad Y_{j}=\frac{\partial}{\partial
x_{j}}-x_{j}\frac{\partial}{\partial t}$

for $j=1,\ldots,n$. The standard sub-Riemannian structure

$({\mathbb{H}}^{n},{\mathcal{H}},\langle\cdot,\cdot\rangle)$

is given by a suitable scalar product $\langle\cdot,\cdot\rangle$ on
${\mathcal{H}}$. One of the key differences between the Euclidean and the
Heisenberg setting is that, as pointed out in [3], the classical Federer’s
notion of rectifiability in metric spaces (cf. [20]) is not suitable for the
Heisenberg group. To solve this issue, the authors of [22] introduced the
intrinsic notion of _$\mathbb{H}$ -regular hypersurface_, together with a
related notion of _intrinsic rectifiability_. We recall that an
$\mathbb{H}$-regular hypersurface is a subset of $\mathbb{H}^{n}$ which can be
described locally as the zero locus of a $C^{1}_{\mathbb{H}}$-function (cf.
[22] for more precise definitions). A special class of $\mathbb{H}$-regular
hypersurfaces is that of _non-characteristic hypersurfaces_. In this setting,
given a hypersurface $S$ of class $C^{1}$, we say that a point $p\in S$ is
_characteristic_ as soon as

${\mathcal{H}}_{p}=T_{p}S,$

otherwise we say that $p$ is _non-characteristic_. In this last case, the
_horizontal tangent space_

${\mathcal{H}}T_{p}S={\mathcal{H}}_{p}\cap T_{p}S$

is a $(2n-1)$-dimensional vector space. The set of characteristic points of
$S$ is denoted by $S_{0}$ and is called the _characteristic set_ of $S$. After
[22], it was clear that the importance of $\mathbb{H}$-regular hypersurfaces
went beyond rectifiability, as proved for instance from the striking
differences between characteristic and non-characteristic setting in the
approach to the sub-Riemannian Bernstein problem in $\mathbb{H}^{1}$. A first
study of the latter was carried out by [9, 44] in the class of _$t$ -graphs_
of class $C^{2}$. We recall that a hypersurface $S$ is a _$t$ -graph_ whenever
there exists $u:\mathbb{R}^{2n}\longrightarrow\mathbb{R}$ such that

$\operatorname{graph}(u):=S=\\{(\bar{x},\bar{y},u(\bar{x},\bar{y}))\,:\,(\bar{x},\bar{y})\in\mathbb{R}^{2n}\\}.$

In the previous set of papers, the authors classified minimal $t$-graphs of
class $C^{2}$ in the first Heisenberg group $\mathbb{H}^{1}$, finding examples
of minimal $t$-graphs which are not planes. These results were generalized in
[33, 16, 17] to more general embedded $C^{2}$-hypersurfaces in
$\mathbb{H}^{1}$. Moreover, as pointed out in [41], if one consider
hypersurfaces with low regularity, the examples of minimal hypersurfaces which
are not hyperplanes increase considerably. However, the situation is different
when considering non-characteristic hypersurfaces. In this context, a
meaningful counterpart of hyperplanes in the Euclidean setting is the class of
_vertical hyperplanes_. Let us recall that a vertical hyperplane is a set $S$
of the form

$S=\\{p\in\mathbb{H}^{n}\,:\,\langle(\bar{x},\bar{y}),(\bar{a},\bar{b})\rangle=c\\},$

for some $0\neq(\bar{a},\bar{b})\in{\mathbb{R}}^{2n}$ and $c\in{\mathbb{R}}$.
An easy computation (cf. Section 4) shows that $S$ is non-characteristic.
Moreover, every hyperplane which is not vertical is characteristic (cf. again
Section 4). A first result in the understanding of minimal non-characteristic
hypersufaces was achieved in [5] in the class of _intrinsic graphs_ (cf. [5]).
Indeed, the authors showed that the only minimal intrinsic graphs defined by a
$C^{2}$ function in $\mathbb{H}^{1}$ are vertical hyperplanes. This result was
generalized in [23] to the class of non-characteristic minimal
$C^{1}$-hypersurfaces of $\mathbb{H}^{1}$, in [38] to the class of minimal
intrinsic graphs defined by an Euclidean Lipschitz function in
$\mathbb{H}^{1}$, and in [26] to the class of $(X,Y)$-Lipschitz surfaces in
the sub-Finsler Heisenberg group $\mathbb{H}^{1}$. We point out that, as shown
in [37], weakening the regularity of the defining function allows to find
examples of minimal hypersurfaces which are not vertical planes even in the
class of intrinsic graphs. While the Bernstein problem is well understood in
$\mathbb{H}^{1}$, very few results are known in higher dimensions. On one
hand, it has recently proved in ([47]) that there is no rigidity in the class
of smooth $t$-graphs in $\mathbb{H}^{n}$. On the other hand, when $n\geqslant
5$, there are counterexamples even in the class of smooth intrinsic graphs
(cf. [5, 16]). The Bernstein problem for non-characteristic hypersurfaces is
still open when $n=2,3,4$. In $\mathbb{H}^{1}$, a key step consists in
understanding that the non-characteristic part $S\setminus S_{0}$ of an area-
stationary surface $S$ is foliated by _horizontal line_ segments in the
following sense.

######  Ruling Property.

[10, 23] Let $S$ be an area-stationary surface of class $C^{1}$ in
$\mathbb{H}^{1}$. Then, $S$ is foliated by horizontal line segments with
endpoints in $S_{0}$.

Here, by horizontal line, we mean an Euclidean line $\gamma$ such that

$\dot{\gamma}(t)=\sum_{j=1}^{n}a_{j}X_{j}(\gamma(t))+\sum_{j=1}^{n}b_{j}Y_{j}(\gamma(t)),$

for some $a_{1},\ldots,a_{n},b_{1},\ldots,b_{n}\in\mathbb{R}$ and for any
$t\in\mathbb{R}$. The importance of this ruling property became even more
evident in [49], where the author showed a Bernstein Theorem in the class of
those minimal intrinsic graphs which present the aforementioned ruling
property, thus without assuming any a priori regularity on the surfaces. The
importance of this merely differential property can be appreciated even by a
sub-Riemannian viewpoint. Indeed, in analogy with the Riemannian setting, the
sub-Riemannian structure of $\mathbb{H}^{n}$ allows to associate to the non-
characteristic part of $S$ various notions of _horizontal second fundamental
forms_ , which act on the horizontal tangent distribution ${\mathcal{H}}TS$.
Despite these forms have been introduced and studied by many authors in
different ways (cf. [42, 32, 15, 12, 43, 32]), and differently from the
Riemannian framework, it is possible to distinguish a _symmetric_ form
$\tilde{h}$ and a _non-symmetric_ form $h$. These forms has shown to be
important in various settings, for instance to introduce the so-called
_horizontal mean curvature_ $H$ (cf. [9, 39, 15]), a suitable notion of
_horizontal umbilicity_ (cf. [8]) and for the study of rigid motions (cf.
[12]). In the particular case of $\mathbb{H}^{1}$, the vanishing of the form
$h$, which coincides both with the symmetric form $\tilde{h}$ and with the
horizontal mean curvature $H$, is equivalent to the aforementioned ruling
property. In the higher dimensional case, however, $h$ and $\tilde{h}$ may
differ, although it is in general true that the norm of $\tilde{h}$ is
controlled by the norm of $h$ (cf. Section 5).

The aim of the present paper is twofold. On one hand, we propose a
generalization of the ruling property to higher dimensional Heisenberg groups,
relating this new notion with the vanishing of the symmetric form $\tilde{h}$.
More precisely, we will call _horizontally totally geodesic_ a hypersurface
such that $\tilde{h}\equiv 0$ on its non-characteristic part. We stress that
hypersurfaces for which $h\equiv 0$ are particular instances of horizontally
totally geodesic hypersurfaces. On the other hand, it is not always the case
that horizontally totally geodesic hypersurfaces satisfy $h\equiv 0$ (cf.
Section 5). In the Riemannian framework, this name is motivated by the fact
that every geodesic of a totally geodesic hypersurface is a geodesic of the
ambient manifold. This last characterization allows to deduce that, in
${\mathbb{R}}^{n}$, the only totally geodesic hypersurfaces are hyperplanes.
The second aim of this paper is to provide an analogous result in the
Heisenberg group. We stress that, at least in the non-characteristic case,
hypersurfaces with $h\equiv 0$ are easily vertical hyperplanes (cf. Section
5). Surprisingly, the same phenomenon continues to hold under the weaker
requirement $\tilde{h}\equiv 0$. The main achievements of this paper can be
then summarized in the following result.

######  Theorem 1.1 (Main Theorem).

Let $S\subseteq{\mathbb{H}}^{n}$ be a hypersurface without boundary of class
$C^{2}$. The following are equivalent:

  * $(i)$

$S$ is horizontally totally geodesic;

  * $(ii)$

$S$ is ruled.

If in addition $n\geqslant 2$ and $S$ is (topologically) closed, then $(i)$
and $(ii)$ hold if and only if $S$ is a hyperplane.

In particular, in the non-characteristic setting, Theorem 1.1 might constitute
an important tool in order to approach the resolution of the Bernstein problem
in the higher dimensional case.

######  Corollary 1.2.

Let $S\subseteq{\mathbb{H}}^{n}$ be a non-characteristic hypersurface without
boundary of class $C^{2}$. Assume that $n\geqslant 2$ and that $S$ is
(topologically) closed. If $S$ is horizontally totally geodesic, then $S$ is a
vertical hyperplane.

Corollary 1.2 allows to reduce the complexity of the problem to the estimate
of the norm of the horizontal second fundamental form $\tilde{h}$ associated
to a minimal hypersurface. We point out that an approach based on curvature
estimates for minimal hypersurfaces is already available in
${\mathbb{R}}^{n}$, in view of the celebrated paper [45]. Our approach to
Theorem 1.1 can be summarized in the following steps.

### Introduction of the higher dimensional ruling property

The starting point consists in generalizing the ruling property to the higher
dimensional case, which is done in Section 3 in two equivalent ways (cf.
Definition 3.1, 3.1 and Proposition 3.2). After discussing the connection
between the characteristic set and this new notion (cf. 3.2), we show that the
latter is well-behaved with respect to the intrinsic geometry of
$\mathbb{H}^{n}$. Namely, we prove that the class of ruled hypersurfaces is
closed under the action of intrinsic dilations (cf. 3.6), and the action of
the so-called _pseudohermitian transformations_ (cf. 3.10).

### Rigidity results for ruled hypersurfaces

Subsequently, we exploit the ruling property to provide rigidity results for
some classes of hypersurfaces. Basically, we show that under some constraints
on the size of the characteristic set, the higher dimensional ruling property
is more rigid than the corresponding one in $\mathbb{H}^{1}$.

######  Theorem 1.3.

Let $S$ be a hypersurface of class $C^{1}$. Assume that $n\geqslant 2$ and
that $S$ is closed and without boundary. Assume that $S_{0}$ is countable and
that $S$ is ruled. Then $S$ is a hyperplane.

This result constitutes a first remarkable difference with $\mathbb{H}^{1}$,
where there are instance of smooth ruled non-characteristic hypersurfaces
which are not planes (cf. Section 4).

### Introduction of the horizontal second fundamental forms

In Section 5 we begin building a bridge between the aforementioned result,
which is only differential in spirit, with the sub-Riemannian structure of
$\mathbb{H}^{n}$. To this aim, we formally introduce the two aforementioned
second fundamental forms $\tilde{h}$ and $h$. highlighting the main
differences between ${\mathbb{H}}^{1}$ and higher dimensional Heisenberg
groups. Moreover, we prove some simple formulas for the norms of $h$ and
$\tilde{h}$ (cf. Proposition 5.1 and Proposition 5.4), which allows to relate
in a quantitative way the two quantities.

### Ruled if and only if horizontally totally geodesic.

In view of Theorem 1.3, the main remaining obstacle to prove Theorem 1.1 is to
show the equivalence between the property of being horizontally totally
geodesic and the ruling property.

######  Theorem 1.4.

Let $S$ be a hypersurface without boundary of class $C^{2}$. Then $S$ is ruled
if and only if $S$ is horizontally totally geodesic.

This result strongly relies on a local existence and uniqueness result for a
particular geodesic-type initial value problem on the hypersurface (cf.
Theorem 6.5). Sub-Riemannian geodesics have been extensively studied in the
last years (cf. e.g. [35, 34, 1, 14, 36] and references therein). Although
local existence results for sub-Riemannian geodesics are available (cf. e.g.
[34]), it is not always the case that sub-Riemannian geodesics satisfies the
standard geodesic equation

$\nabla_{\dot{\Gamma}}\dot{\Gamma}=0,$

being $\nabla$ a suitable sub-Riemannian connection (cf. [44]). Therefore, we
devote Section 6 to the study of the initial value problem associated to this
kind of equations on hypersurfaces. The proof of Theorem 6.5 can be reduced to
the study of curves in domains of suitable _intrinsic graphs_ , and its main
difficulty lies in the fact that that the initial value problem that we need
to consider is _a priori_ overdetermined. Once Theorem 6.5 is achieved, we are
then in position to prove Theorem 1.4, and so, in view of the previous
considerations, to conclude the proof of Theorem 1.1.

We point out that, in view of Theorem 1.4, another striking difference with
the first Heisenberg group can be appreciated. Indeed, contrarily to what
happens in ${\mathbb{H}}^{1}$, it is easy to provide examples of minimal
smooth hypersurfaces, at least in the characteristic setting, which are not
horizontally totally geodesic.

######  Theorem 1.5.

Let $n\geqslant 2$ and let $S:=\operatorname{graph}(u)$, where
$u(\bar{x},\bar{y})=\frac{1}{2}x_{1}^{2}-\frac{1}{2}y_{1}^{2}.$ Then $S$ is a
minimal smooth hypersurface which is not horizontally totally geodesic.

This set of results and considerations both provides a direct way to approach
the Bernstein problem via curvature estimates, and highlights once more many
interesting differences between $\mathbb{H}^{1}$ and higher dimensional
Heisenberg groups. According to the authors’ hope, it may give a burst in the
grasp of such an interesting open problem as the Bernstein problem in this
anisotropic setting.

### Plan of the paper

In Section 2 we collect some basic preliminaries about the sub-Riemannian
Heisenberg group. In Section 3 we introduce the higher dimensional ruling
property and we study some of its properties. In Section 4 we prove Theorem
1.3. In Section 5 we recall the main definition and properties of the
horizontal forms $h$ and $\tilde{h}$, and we introduce the notion of
horizontally totally geodesic hypersurface. In Section 6 we introduce the
relevant geodesic-type initial value problem (cf. (6.1)) and we show a local
existence and uniqueness result (cf. Theorem 6.5). Moreover, we prove Theorem
1.4, Theorem 1.1 and Theorem 1.5.

### Acknowledgements

The authors thank G. Giovannardi, M. Ritoré, F. Serra Cassano, G. Tralli, D.
Vittone and R. Young for fruitful conversations about these topics. The
authors also thank D. Vittone for suggesting them the proof of Proposition
6.1.

##  2\. Preliminaries

###  2.1. The Heisenberg group $\mathbb{H}^{n}$

In the following we denote by $T$ the left-invariant vector field
$\frac{\partial}{\partial t}$. In this way
$X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n},T$ is a basis of left-invariant vector
fields. Moreover, the only nontrivial commutation relations are

$[X_{j},Y_{j}]=-[Y_{j},X_{j}]=-2T$

for any $j=1,\ldots,n$. A vector field which is tangent to ${\mathcal{H}}$ at
every point is called _horizontal_. For given $q\in\mathbb{H}^{n}$ and
$\lambda>0$, we define the _left-translation_
$\tau_{q}:\mathbb{H}^{n}\longrightarrow\mathbb{H}^{n}$ and the _intrinsic
dilation_ $\delta_{\lambda}:\mathbb{H}^{n}\longrightarrow\mathbb{H}^{n}$ by

$\tau_{q}(p):=q\cdot p\qquad\text{and}\qquad\delta_{\lambda}(p):=(\lambda
z,\lambda^{2}t)$

for any $p=(z,t)\in\mathbb{H}^{n}$ respectively. It is well known that both
$\tau_{q}$ and $\delta_{\lambda}$ are global diffeomorphisms, and that
$\delta_{\lambda}$ is a Lie group isomorphism. Moreover, we define the
_complex structure_ $J$ by letting

$J(X_{i})=Y_{i},\qquad J(Y_{i})=-X_{i}\qquad\text{and}\qquad J(T)=0$

for any $i=1,\ldots,n$, and extending it by linearity for general vector
fields. Given $p\in{\mathbb{H}}^{n}$ we will often identify the vector
$\sum_{j=1}^{n}v_{j}X_{j}|_{p}+v_{n+j}Y_{j}|_{p}$ with the point
$(v_{1},\ldots,v_{2n},0)\in{\mathbb{H}}^{n}$. The Haar measure of
$\mathbb{H}^{n}$ coincides with the $(2n+1)$-dimensional Lebesgue measure
$\mathcal{L}^{2n+1}$. The homogeneity property
$\mathcal{L}^{2n+1}(\delta_{\lambda}(E))=\lambda^{Q}\mathcal{L}^{2n+1}(E)$
holds for any measurable set $E\subseteq\mathbb{H}^{n}$, where $Q=2n+2$ is the
_homogeneous dimension_ of $\mathbb{H}^{n}$ (cf. [46]).

###  2.2. Sub-Riemannian structure on $\mathbb{H}^{n}$

We let $g$ be the unique Riemannian metric on $\mathbb{H}^{n}$ which makes
$X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n},T$ orthonormal. For the sake of
notational simplicity, we let

$Z_{j}=X_{j},\qquad Z_{n+j}=Y_{j}\qquad\text{and}\qquad Z_{2n+1}=T$

for any $j=1,\ldots,n$, and we recall for the sake of completeness that the
horizontal distribution ${\mathcal{H}}$ is defined by

${\mathcal{H}}_{p}=\operatorname{span}\\{Z_{1}|_{p},\ldots,Z_{2n}|_{p}\\}$

for any $p\in\mathbb{H}^{n}$. If we restrict $g$ to the horizontal
distribution ${\mathcal{H}}$, and we denote this restriction by
$\langle\cdot,\cdot\rangle$, then $\mathbb{H}^{n}$ inherits a _sub-Riemannian
structure_ which realizes it as a _sub-Riemannian manifold._ We denote by
$|\cdot|$ the norm induced by $\langle\cdot,\cdot\rangle$. Moreover, we denote
by $\nabla$ the so-called _pseudohermitian connection_ (cf. e.g. [43]), i.e.
the unique _metric connection_ (cf. [18]) with torsion tensor given by

(2.1) $\nabla_{X}Y-\nabla_{Y}X-[X,Y]=2\langle J(X),Y\rangle T$

for any pair of vector fields $X$ and $Y$. The most relevant feature of
$\nabla$ (cf. [42]) is the following property:

(2.2) $\nabla_{Z_{i}}Z_{j}=0$

for any $i,j=1,\ldots,2n+1$, and so can be seen as a flat connection on
${\mathbb{H}}^{n}$.

###  2.3. Carnot-Carathéodory structure on $\mathbb{H}^{n}$

If $\Gamma:[a,b]\longrightarrow\mathbb{H}^{n}$ is an absolutely continuous
curve, we say that it is _horizontal_ whenever

(2.3) $\dot{\Gamma}(t)\in{\mathcal{H}}_{\Gamma(t)}$

for almost every $t\in[a,b]$, and we say that it is _sub-unit_ whenever it is
horizontal with $|\dot{\Gamma}(t)|=1$ for almost every $t\in[a,b]$. Moreover,
we define

$d(p,p^{\prime}):=\inf\\{T\,:\,\Gamma:[0,T]\longrightarrow\mathbb{H}^{n}\text{
is sub-unit, $\Gamma(0)=p$ and $\Gamma(T)=p^{\prime}$}\\}$

which, by the Chow-Rashevskii theorem (cf. [13]), defines a distance on
$\mathbb{H}^{n}$, called _Carnot-Carathéodory distance_. The metric space
$(\mathbb{H}^{n},d)$ is then a prototype of _Carnot-Carathéodory space_ (cf.
[30]).

###  2.4. Horizontal perimeter and horizontal gradient

If $\Omega\subseteq\mathbb{H}^{n}$ is open and $E\subseteq\mathbb{H}^{n}$ is
measurable with $\chi_{E}\in L^{1}_{loc}(\Omega)$, we recall (cf. e.g. [22,
24]) that the _$\mathbb{H}$ -perimeter_ of $E$ in $\Omega$ is defined by

$P_{\mathbb{H}}(E,\Omega):=\sup\left\\{\int_{E}\operatorname{div}_{\mathbb{H}}(\bar{\varphi})\,d\mathcal{L}^{2n+1}\,:\,\bar{\varphi}\in
C^{1}_{c}(\Omega,\mathcal{H}),\,|\bar{\varphi}|_{p}\leqslant 1\text{ for any
}p\in\Omega\right\\},$

where by $C^{1}_{c}(\Omega,{\mathcal{H}})$ we denote the class of compactly
supported $C^{1}$ sections of the horizontal distribution ${\mathcal{H}}$, and
$\operatorname{div}_{\mathbb{H}}$ is the so-called _horizontal divergence_ ,
defined by

$\operatorname{div}_{\mathbb{H}}\left(\sum_{j=1}^{n}(\varphi_{j}X_{j}+\varphi_{n+j}Y_{j})\right):=\sum_{j=1}^{n}(X_{j}\varphi_{j}+Y_{j}\varphi_{n+j})$

for any $\sum_{j=1}^{n}(\varphi_{j}X_{j}+\varphi_{n+j}Y_{j})\in
C^{1}(\Omega,{\mathcal{H}})$. Moreover, we say that a set $E$ as above is an
_$\mathbb{H}$ -Caccioppoli set_ whenever $P_{\mathbb{H}}(E,\Omega)<+\infty$
for any bounded open set $\Omega\subseteq\mathbb{H}^{n}$. Finally, we recall
(cf. e.g. [46]) that an $\mathbb{H}$-Caccioppoli set $E$ is an _$\mathbb{H}$
-perimeter minimizer_ whenever

$P_{\mathbb{H}}(E,\Omega)\leqslant P_{\mathbb{H}}(F,\Omega)$

for any $\Omega\Subset\mathbb{H}^{n}$ and for any $\mathbb{H}$-Caccioppoli set
$F$ such that $E\Delta F\Subset\Omega$. The sub-Riemannian structure of
$\mathbb{H}^{n}$ allows to define a distributional notion of _horizontal
gradient_ (cf. [22]). More precisely, if $f\in L^{1}_{loc}(\Omega)$, we let

$\langle\nabla_{\mathbb{H}}f,\varphi\rangle:=-\int_{\Omega}f\operatorname{div}_{\mathbb{H}}\varphi\,d\mathcal{L}^{2n+1}$

for any $\varphi\in C^{\infty}_{c}(\Omega,{\mathbb{R}}^{2n})$. When $f$ is
continuous and $\nabla_{\mathbb{H}}f$ is represented by a continuous vector
field, then we say that $f\in C^{1}_{\mathbb{H}}(\Omega)$. Moreover, in this
case,

$\nabla_{\mathbb{H}}f=\sum_{j=1}^{n}X_{j}fX_{j}+\sum_{j=1}^{n}Y_{j}fY_{j}.$

###  2.5. Hypersurfaces in $\mathbb{H}^{n}$

We say that $S\subseteq\mathbb{H}^{n}$ is an _$\mathbb{H}$ -regular
hypersurface_ if, for any $p\in S$, there exists an open neighborhood $U$ of
$p$ and a function $f\in C^{1}_{\mathbb{H}}(U)$ such that

$S\cap
U=\\{q\in\mathbb{H}^{n}\,:\,f(q)=0\\}\qquad\text{and}\qquad\nabla_{\mathbb{H}}f\neq
0\text{ on }U.$

Here and in the rest of the paper, whether not specified, a hypersurface will
be always at least of class $C^{1}$ and without boundary. If $S$ is a
hypersurface of class $C^{1}$, we define

$S_{0}:=\\{p\in S:{\mathcal{H}}_{p}=T_{p}S\\}$

and we call it the _characteristic set_ of $S$. Notice that, since $S$ is of
class $C^{1}$ and ${\mathcal{H}}$ is a smooth distribution, then $S_{0}$ is
closed in $S$. Moreover, let us define

${\mathcal{H}}T_{p}S:={\mathcal{H}}_{p}\cap T_{p}S.$

When $p\in S_{0}$, then $\dim({\mathcal{H}}T_{p}S)=2n$. On the contrary, when
$p\in S\setminus S_{0}$, we have $\dim({\mathcal{H}}T_{p}S)=2n-1$. In this
case, the _horizontal normal_ to $S$ at $p$ is defined by

(2.4) $v^{\mathbb{H}}(p):=\frac{N^{\mathbb{H}}(p)}{|N^{\mathbb{H}}(p)|_{p}}$

for any $p\in S\setminus S_{0}$, where $N_{\mathbb{H}}(p)$ is the a section of
the horizontal bundle defined by

$N^{\mathbb{H}}(p):=\sum_{j=1}^{n}(\langle
N(p),X_{j}|_{p}\rangle_{{\mathbb{R}}^{2n+1}}X_{j}|_{p}+\sum_{j=1^{n}}\langle
N(p),Y_{j}|_{p}\rangle_{{\mathbb{R}}^{2n+1}}Y_{j}|_{p},$

being $N(p)$ the Euclidean unit normal to $S$ at $p$. Roughly speaking,
$v^{\mathbb{H}}$ is a unit horizontal vector field which is orthogonal to the
horizontal part of the tangent space. It is clear that a hypersurface of class
$C^{1}$ with empty characteristic set is $\mathbb{H}$-regular (cf. e.g. [47]).
When $S$ is of class $C^{2}$, then

(2.5) $\sum_{h=1}^{2n}v^{\mathbb{H}}_{h}Z_{k}(v^{\mathbb{H}}_{h})=0$

for any $k=1,\ldots,2n$, where by $v^{\mathbb{H}}$ we mean any $C^{2}$
extension of $v^{\mathbb{H}}|_{S}$ in a neighborhood of $S$ such that
$|v^{\mathbb{H}}|\equiv 1$. Indeed, (2.5) follows by taking derivatives of
$|v^{\mathbb{H}}|^{2}\equiv 1$. There is a particular choice of such an
extension which allows to derive further relations. Indeed, if we let
$d^{\mathbb{H}}$ be the signed Carnot-Carathéodory distance from $S$ with
respect to $(Z_{1},\ldots,Z_{2n})$, then it is well known (cf. [43]) that
$d^{\mathbb{H}}$ inherits the same regularity of $S$ in a neighborhood of any
non-characteristic point $p\in S$. Moreover, since $d^{\mathbb{H}}$ satisfies
the horizontal Eikonal equation in a neighborhood of $S$, meaning that
$|\nabla^{\mathbb{H}}d^{\mathbb{H}}|\equiv 1$ in a neighborood of any non-
characteristic point $p\in S$ (cf. [43]), then $v^{\mathbb{H}}|_{S}$ can be
extended by letting

(2.6)
$v^{\mathbb{H}}=\sum_{j=1}^{n}X_{j}d^{\mathbb{H}}X_{j}+\sum_{j=1}^{n}Y_{j}d^{\mathbb{H}}Y_{j}.$

With this particular extension,

(2.7) $Z_{k}(v^{\mathbb{H}}_{h})=Z_{h}(v^{\mathbb{H}}_{k})$

for any $h,k=1,\ldots,2n$ such that $|h-k|\neq n$. Moreover,

(2.8)
$X_{k}(v^{\mathbb{H}}_{n+k})=Y_{k}(v^{\mathbb{H}}_{k})-2Td^{\mathbb{H}}\qquad\text{and}\qquad
Y_{k}(v^{\mathbb{H}}_{k})=X_{k}(v^{\mathbb{H}}_{n+k})+2Td^{\mathbb{H}}$

for any $k=1,\ldots,n$. Finally, thanks to (2.5), (2.7) and (2.8), we see that

(2.9)
$\sum_{h=1}^{2n}v^{\mathbb{H}}_{h}Z_{h}(v^{\mathbb{H}}_{k})=-2Td^{\mathbb{H}}J(v^{\mathbb{H}})_{k}$

for any $k=1,\ldots,2n$. Moreover, a simple computation shows that

(2.10) $Td^{\mathbb{H}}=\frac{N_{2n+1}}{|N^{\mathbb{H}}|}$

According to [15], we define the _horizontal tangential derivatives_

$\delta_{i}\xi=Z_{i}\bar{\xi}-g_{\mathbb{H}}\left(\nabla_{\mathbb{H}},\bar{\xi}\right)v^{\mathbb{H}}_{i}$

for any $i=1,\ldots,2n$, where $\xi$ is a $C^{1}$ function on an open subset
of $S$ and $\bar{\xi}$ is any $C^{1}$ extension of $\xi$. As customary, the
horizontal tangential derivatives do not depend on the chosen extension (cf.
[15]). We recall that the _tangent pseudohermitian connection_ $\nabla^{S}$ is
defined in the non- characteristic part of $S$ by

$\nabla^{S}_{X}Y=\nabla_{X}Y-\langle\nabla_{X}Y,v^{\mathbb{H}}\rangle
v^{\mathbb{H}}$

for any pair of tangent horizontal vector fields $X$ and $Y$. Notice that
$\nabla^{S}$ is easily a metric connection with respect to the metric
$\langle\cdot,\cdot\rangle$ restricted to ${\mathcal{H}}TS$. We say that a
hypersurface of class $C^{1}$ is _${\mathbb{H}}$ -minimal_ whenever it
coincides with the boundary of an $\mathbb{H}$-perimeter minimizer.

###  2.6. Intrinsic graphs

We follow the notation and the approach of [4]. Let us denote points
$q\in{\mathbb{R}}^{2n}$ by
$q=(\xi_{1},\ldots,\xi_{n},\eta_{2},\ldots,\eta_{n},\tau)=(\bar{\xi},\tilde{\eta},\tau)$.
We wish to identify ${\mathbb{R}}^{2n}$ with
$\\{p\in\mathbb{H}^{n}\,:\,y_{1}=0\\}$. To this aim, we introduce the
immersion map $i:{\mathbb{R}}^{2n}\longrightarrow\mathbb{H}^{n}$ defined by

$i(\bar{\xi},\tilde{\eta},\tau)=(\bar{\xi},0,\tilde{\eta},\tau)$

for any $(\bar{\xi},\tilde{\eta},\tau)\in{\mathbb{R}}^{2n}$. Moreover, we
identify ${\mathbb{R}}$ with
$\\{(\bar{0},y_{1},\tilde{0},0)\,:\,y_{1}\in{\mathbb{R}}\\}$ by means of the
inclusion $j:{\mathbb{R}}\longrightarrow\mathbb{H}^{n}$ defined by

$j(y_{1})=(\bar{0},y_{1},\tilde{0},0)$

for any $y_{1}\in{\mathbb{R}}$. The maps $i$ and $j$ are clearly smooth,
injective and open. For a given open set $\Omega\subseteq{\mathbb{R}}^{2n}$
and a function $\varphi:\Omega\longrightarrow{\mathbb{R}}$, we recall that the
_$Y_{1}$ -graph of $\varphi$ on $\Omega$_ is defined by

$\operatorname{graph}_{Y_{1}}(\varphi,\Omega)=\\{i(w)\cdot
j(\varphi(w))\,:\,w\in\Omega\\}=\\{(\bar{\xi},\varphi(\bar{\xi},\tilde{\eta},\tau),\tilde{\eta},\tau-\xi_{1}\varphi(\bar{\xi},\tilde{\eta},\tau))\,:\,(\bar{\xi},\tilde{\eta},\tau)\in\Omega\\}.$

Moreover, we define its parametrization map
$\Psi:\Omega\longrightarrow{\mathbb{H}}^{n}$ by

$\Psi(\bar{\xi},\tilde{\eta},\tau)=(\bar{\xi},\varphi(\bar{\xi},\tilde{\eta},\tau),\tilde{\eta},\tau-\xi_{1}\varphi(\bar{\xi},\tilde{\eta},\tau))$

for any $(\bar{\xi},\tilde{\eta},\tau)\in\Omega$. We introduce also the
_intrinsic projection map_
$\Pi:{\mathbb{H}}^{n}\longrightarrow{\mathbb{R}}^{2n}$ by

$\Pi(\bar{x},\bar{y},t)=(\bar{x},\tilde{y},t+x_{1}y_{1})$

for any $(\bar{x},\bar{y},t)\in\mathbb{H}^{n}$. It is easy to check that

$\Pi(\Psi(q))=q\qquad\text{and}\qquad\Psi(\Pi(p))=p$

for any $q\in\Omega$ and any
$p\in\operatorname{graph}_{Y_{1}}(\varphi,\Omega)$. If $\varphi\in
C^{1}(\Omega)$ and $S=\operatorname{graph}_{Y_{1}}(\varphi,\Omega)$, then

$T_{\Psi(q)}S=\operatorname{span}\left(\frac{\partial\Psi}{\partial\xi_{1}}\Big{|}_{q},\ldots,\frac{\partial\Psi}{\partial\xi_{n}}\Big{|}_{q},\frac{\partial\Psi}{\partial\eta_{2}}\Big{|}_{q},\ldots,\frac{\partial\Psi}{\partial\eta_{n}}\Big{|}_{q},\frac{\partial\Psi}{\partial\tau}\Big{|}_{q}\right).$

Letting
$D\varphi=(\varphi_{\xi_{1}},\ldots\varphi_{\xi_{n}},\varphi_{\eta_{2}},\ldots,\varphi_{\eta_{n}},\varphi_{\tau})$,
an easy computation shows that

$\frac{\partial\Psi}{\partial\xi_{1}}\Big{|}_{q}=X_{1}|_{\Psi(q)}+\varphi_{\xi_{1}}(q)Y_{1}|_{\Psi(q)}-2\varphi(q)T|_{\Psi(q)},$
$\frac{\partial\Psi}{\partial\xi_{j}}\Big{|}_{q}=X_{j}|_{\Psi(q)}+\varphi_{\xi_{j}}(q)Y_{1}|_{\Psi(q)}-\eta_{j}T|_{\Psi(q)},\qquad\frac{\partial\Psi}{\partial\eta_{j}}\Big{|}_{q}=Y_{j}|_{\Psi(q)}+\varphi_{\eta_{j}}(q)Y_{1}|_{\Psi(q)}+\xi_{j}T|_{\Psi(q)}$

for any $j=2,\ldots,n$ and

$\frac{\partial\Psi}{\partial\tau}\Big{|}_{q}=\varphi_{\tau}(q)Y_{1}|_{\Psi(q)}+T|_{\Psi(q)}.$

It is easy to check that $(E_{1},\ldots,E_{n},F_{2},\ldots,F_{n})$ constitutes
a global basis of ${\mathcal{H}}TS$, where

$E_{1}=X_{1}+W^{\varphi}\varphi Y_{1},\quad E_{j}=X_{j}+\tilde{X}_{j}\varphi
Y_{1}\quad\text{and}\quad F_{j}=Y_{j}+\tilde{Y}_{j}\varphi Y_{1}$

for any $j=2,\ldots,n$, and where the family of vector fields
$\nabla^{\varphi}=(W^{\varphi}\varphi,\tilde{X}_{2},\ldots\tilde{X}_{n},\tilde{Y}_{2},\ldots,\tilde{Y}_{n})$
is defined by

$W^{\varphi}=\frac{\partial}{\partial\xi_{1}}+2\varphi\tilde{T},\quad\tilde{X}_{j}=\frac{\partial}{\partial\xi_{j}}+\eta_{j}\tilde{T}\quad\text{and}\quad\tilde{Y}_{j}=\frac{\partial}{\partial\eta_{j}}-\xi_{j}\tilde{T}$

for any $j=2,\ldots,n$, where we have set
$\tilde{T}=\frac{\partial}{\partial\tau}$. Therefore, a quick computation
implies that

(2.11) $v^{\mathbb{H}}=W^{-\frac{1}{2}}\left(W^{\varphi}\varphi
X_{1}+\sum_{j=2}^{n}\tilde{X}_{j}\varphi
X_{j}-Y_{1}+\sum_{j=2}^{n}\tilde{Y}_{j}\varphi Y_{j}\right),$

where we have set

$W=1+|\nabla^{\varphi}\varphi|^{2}.$

Notice that, since

$[\tilde{X}_{j},\tilde{Y}_{j}]=-2\tilde{T}$

for any $j=2,\ldots,n$, then $(\Omega,d_{\varphi})$ is a _Carnot-Carathéodory
space_ (cf. [30]) where $\Omega$ is any domain of ${\mathbb{R}}^{2n}$ and
$d_{\varphi}$ is the _Carnot-Carathéodory distance_ induced by
$\nabla^{\varphi}$.

##  3\. Higher dimensional ruled hypersurfaces

As already mentioned, a key step in the study of minimal surfaces in
${\mathbb{H}}^{1}$ consists in showing that the non-characteristic part of an
area-stationary surface is foliated by horizontal line segments. This property
extends to the higher dimensional case as follows.

######  Definition 3.1 (Local ruling property).

Let $S$ be a hypersurface of class $C^{1}$. We say that $S$ is _locally ruled
at $p\in S\setminus S_{0}$_ if there exists a neighborhood $U$ of $p$ such
that

$p\cdot{\mathcal{H}}T_{p}S\cap U\subseteq S.$

Moreover, we say that $S$ is _locally ruled_ if it is locally ruled at $p\in
S\setminus S_{0}$ for any $p\in S\setminus S_{0}$.

Beside this local definition, we propose a global one, which will be useful in
the following.

######  Definition 3.1 (Global ruling property).

Let $S$ be a hypersurface of class $C^{1}$. We say that $S$ is _ruled_ if for
any $p\in S\setminus S_{0}$, for any $v\in{\mathcal{H}}T_{p}S$ and for any
$s\geqslant 0$, the following property holds. If $s$ is maximal with the
property that

$p\cdot\delta_{\tau}(v)\in S$

for any $\tau\in[0,s]$, then

$p\cdot\delta_{s}(v)\in S_{0}.$

The previous two definitions are actually equivalent.

######  Proposition 3.2.

Let $S$ be a hypersurface of class $C^{1}$. Then the following are equivalent.

  * $(i)$

$S$ is locally ruled.

  * $(ii)$

$S$ is ruled.

###### Proof.

Assume that $S$ is ruled. Assume by contradiction that there exists $p\in
S\setminus S_{0}$ and a sequence $(p_{h})_{h}\subseteq
p\cdot{\mathcal{H}}T_{p}S\setminus S$ converging to $p$ as $h\to+\infty$.
Then, for any $h\in\mathbb{N}$, there exists $\lambda_{h}>0$ and
$v_{h}\in{\mathcal{H}}T_{p}S$ such that
$p_{h}=p\cdot\delta_{\lambda_{h}}(v_{h})$. If, up to a subsequence, for any
$h$ there exists $0<\mu_{h}\leqslant\lambda_{h}$ such that
$p\cdot\delta_{\mu_{h}}(v_{h})$ belongs to the manifold boundary of $S$, then,
being the latter closed, so does $p$, a contradiction with $p\in S$.
Therefore, since $p_{h}\notin S$ and up to a subsequence, we can assume that
for any $h$ there exists $s_{h}\geqslant 0$ maximal such that
$p\cdot\delta_{\tau}(v_{h})\in S$ for any $\tau\in[0,s_{h}]$. Clearly
$s_{h}\leqslant\lambda_{h}$. Therefore, being $S$ ruled, then
$q_{h}:=p\cdot\delta_{s_{h}}(v_{h})\in S_{0}$. But then, by construction,
$(q_{h})_{h}$ converges to $p$ as $h\to+\infty$, and so, being $S_{0}$ closed,
we conclude that $p\in S_{0}$, a contradiction. On the contrary, assume that
$S$ is locally ruled. Assume by contradiction that that there exists $p\in
S\setminus S_{0}$, $w\in{\mathcal{H}}T_{p}S$ and $s$ maximal with the property
that

(3.1) $p\cdot\delta_{\tau}(w)\in S$

for any $\tau\in[0,s]$ and

$p\cdot(sw,0)\notin S_{0}.$

Set $\bar{p}:=p\cdot(sw,0)$. Consider the left-invariant vector field

$W=\sum_{j=1}^{2n}w_{j}Z_{j}.$

Recalling that left-translations preserve the horizontal distribution, and
being $W$ left invariant, we conclude that

$d\tau_{(sw,0)}|_{p}(W_{p})=W|_{\bar{p}}\in{\mathcal{H}}_{\bar{p}}.$

Moreover, by construction, $W$ is clearly tangent to $S$ at $\bar{p}$. We
conclude that $w\in{\mathcal{H}}T_{\bar{p}}S$. Since $S$ is locally ruled and
$\bar{p}\in S\setminus S_{0}$, there exists $\tilde{s}>0$ such that
$\bar{p}\cdot(\tilde{s}w,0)\in S$, which implies, recalling the definition of
$\bar{p}$, that

$\bar{p}\cdot(\tilde{s}w,0)=p\cdot(\bar{s}w,0)\cdot(\tilde{s}w,0)=p\cdot((\bar{s}+\tilde{s})w,0)\in
S,$

a contradiction with (3.1). ∎

######  Proposition 3.2.

Let $S$ be a ruled hypersurface of class $C^{1}$. Assume that $S$ is
(topologically) closed. Let $p\in S\setminus S_{0}$ and
$v\in{\mathcal{H}}T_{p}S$ be such that

$\left\\{p\cdot\delta_{s}(v,0)\,:\,s\geqslant 0\right\\}\cap S_{0}=\emptyset.$

Then

$\left\\{p\cdot\delta_{s}(v,0)\,:\,s\geqslant 0\right\\}\subseteq S.$

In particular, if

$p\cdot{\mathcal{H}}T_{p}S\cap S_{0}=\emptyset,$

then

(3.2) $p\cdot{\mathcal{H}}T_{p}S\subseteq S.$

###### Proof.

Let $p\in S\setminus S_{0}$ and $v\in{\mathcal{H}}T_{p}S$ be as in the
statement, and assume by contradiction that there exists $\lambda>0$ such that
$q=p\cdot\delta_{\lambda}(v)\notin S$. Being $S$ closed, there exists
$s\geqslant 0$ maximal as in 3.1. Then we can argue as in the proof of
Proposition 3.2 to find $s\geqslant 0$ such that $p\cdot\delta_{s}(v)\in
S_{0}$, which is a contradiction. The second claim clearly follows from the
first one. ∎

Notice that, in view of Proposition 3.2, the notion of ruled hypersurface
becomes much more simpler in the case of non-characteristic hypersurfaces.
Indeed, if $S$ is a closed, non-characteristic ruled hypersurface of class
$C^{1}$ and $p\in S$, then clearly $p\cdot{\mathcal{H}}T_{p}S\cap
S_{0}=\emptyset$. Therefore a closed non-characteristic hypersurface of class
$C^{1}$ is ruled if and only if it satisfies (3.2). Now let us discuss some
instances of ruled hypersurfaces. We begin with the simplest non-
characteristic smooth hypersurface.

######  Example 3.3 (Vertical Hyperplanes).

Let $S$ be a vertical hyperplane of the form

$S=\\{p\in\mathbb{H}^{n}\,:\,\langle(\bar{x},\bar{y}),(\bar{a},\bar{b})\rangle=c\\}$

for some $0\neq(\bar{a},\bar{b})\in\mathbb{R}^{2n}$ and $c\in\mathbb{R}$.
Without loss of generality, we assume that $a_{1}\neq 0$. It is easy to see
that

$T_{p}S=\operatorname{span}\\{(a_{2},-a_{1},0,\ldots,0),(a_{3},0,-a_{1},0,\ldots,0),\ldots,(b_{n},0,\ldots,0,-a_{1},0),T\\}$

for any $p\in S$. Notice that $S_{0}=\emptyset$. We show that $S$ is ruled.
Indeed, noticing that $T\in T_{p}S$ for any $p\in S$, it follows that

${\mathcal{H}}T_{p}S=\operatorname{span}\\{Z_{2}|_{p},\ldots,Z_{n}|_{p},W_{1}|_{p},\ldots,W_{n}|_{p}\\}$

for any $p\in S$, where

(3.3) $Z_{i}=a_{i}X_{1}-a_{1}X_{i}\qquad\text{and}\qquad
W_{j}=b_{j}X_{1}-a_{1}Y_{j}$

for any $i=2,\ldots,n$ and $j=1,\ldots,n$. Let now $p=(\bar{x},\bar{y},t)\in
S$, and let $w=(\bar{x}^{\prime},\bar{y}^{\prime},0)\in{\mathcal{H}}T_{p}S$.
Then there exists $\alpha_{j},\beta_{j}\in\mathbb{R}$ such that

$w=\left(\sum_{j=2}^{n}\alpha_{j}a_{j}+\sum_{j=1}^{n}\beta_{j}b_{j},-\alpha_{2}a_{1},\ldots,-\beta_{n}a_{1},0\right).$

We conclude noticing that

$\langle(\bar{x}^{\prime},\bar{y}^{\prime}),(\bar{a},\bar{b})\rangle=a_{1}\sum_{j=2}^{n}\alpha_{j}a_{j}+a_{1}\sum_{j=1}^{n}\beta_{j}b_{j}-\sum_{j=2}^{n}\alpha_{j}a_{1}a_{j}-\sum_{j=1}^{n}\beta_{j}a_{1}b_{j}=0.$

Next we consider an instance in the characteristic case.

######  Example 3.4 (Horizontal Hyperplane).

Let $S$ be the horizontal hyperplane ${\mathcal{H}}_{0}$. Notice that

$T_{p}S=\operatorname{span}\left\\{\frac{\partial}{\partial
x_{1}},\ldots,\frac{\partial}{\partial
y_{n}}\right\\}=\operatorname{span}\\{X_{1}-y_{1}T,\ldots,X_{n}-y_{n}T,Y_{1}+x_{1}T,\ldots,Y_{n}+x_{n}T\\}$

for any $p\in S$. This in particular implies that $S_{0}=\\{0\\}$. Therefore,
let $p=(\bar{x},\bar{y},t)\neq 0$, and assume without loss of generality that
$y_{1}\neq 0$. This implies that

${\mathcal{H}}T_{p}S=\operatorname{span}\\{y_{2}X_{1}-y_{1}X_{2},\ldots,y_{n}X_{1}-y_{1}X_{n},x_{1}X_{1}+y_{1}Y_{1},\ldots,x_{n}X_{1}+y_{1}Y_{n}\\}.$

Therefore, let $w=(z,0)\in{\mathcal{H}}T_{p}S$, and let
$\alpha_{j},\beta_{j}\in\mathbb{R}$ be such that

$z=\left(\sum_{j=2}^{n}\alpha_{j}y_{j}+\sum_{j=1}^{n}\beta_{j}x_{j},-\alpha_{2}y_{1},\ldots,-\alpha_{n}y_{1},\beta_{1}y_{1},\ldots,\beta_{n}y_{1}\right).$

Hence it follows that

$Q((\bar{x},\bar{y}),z)=y_{1}\sum_{j=2}^{n}\alpha_{j}y_{j}+y_{1}\sum_{j=1}^{n}\beta_{j}x_{j}-\sum_{j=2}^{n}\alpha_{j}y_{1}y_{j}-\sum_{j=1}^{n}\beta_{j}y_{1}x_{j}=0.$

With the next couple of propositions we show that the class of ruled
$C^{1}$-hypersurfaces is closed under the action of left translations and
intrinsic dilations.

######  Proposition 3.5.

Let $S$ be a ruled hypersurface of class $C^{1}$. Then $\tau_{q}(S)$ is ruled
for any $q\in\mathbb{H}^{n}$.

###### Proof.

Fix $q=(\bar{x}^{q},\bar{y}^{q},t)\in\mathbb{H}^{n}$, define
$\tilde{S}:=\tau_{q}(S)$ and, given a point
$\tilde{p}\in\tilde{S}\setminus\tilde{S}_{0}$, let $p\in S$ be such that
$\tilde{p}=\tau_{q}(p)$. Being $\tau_{q}:S\longrightarrow\tilde{S}$ a
diffeomorphism, then $d\tau_{q}|_{p}:T_{p}S\longrightarrow
T_{\tilde{p}}\tilde{S}$ is an isomorphism. Therefore we have that

$d\tau_{q}|_{p}(T_{p}S)=T_{\tilde{p}}\tilde{S}.$

Moreover, by definition of ${\mathcal{H}}$, it is also the case that

$d\tau_{q}|_{p}({\mathcal{H}}_{p})={\mathcal{H}}_{\tilde{p}}.$

Hence we infer that

$d\tau_{q}|_{p}({\mathcal{H}}T_{p}S)=d\tau_{q}|_{p}({\mathcal{H}}_{p}\cap
T_{p}S)=d\tau_{q}|_{p}({\mathcal{H}}_{p})\cap
d\tau_{q}|_{p}(T_{p}S)={\mathcal{H}}_{\tilde{p}}\cap
T_{\tilde{p}}\tilde{S}={\mathcal{H}}T_{\tilde{p}}\tilde{S}.$

In particular, notice that $p\in S\setminus S_{0}$. Let
$w\in\tilde{p}\cdot{\mathcal{H}}T_{\tilde{p}}S$ and assume that there exists
$s\geqslant 0$ maximal with the property that
$\tilde{p}\cdot\delta_{\tau}(w)\in\tilde{S}$ for any $\tau\in[0,s].$ We claim
that $\tilde{p}\cdot\delta_{s}(w)\in\tilde{S}_{0}$. Let
$v=(\bar{a},\bar{b},0)\in{\mathcal{H}}T_{p}S$ be such that
$d\tau_{q}|_{p}(v)=w$. By the left-invariance of the horizontal distribution,
it follows that $w=(\bar{a},\bar{b},0)$. Therefore $s$ is maximal with the
property that $p\cdot\delta_{\tau}(v)\in S$ for any $\tau\in[0,s].$ Hence
$p\cdot\delta_{s}(v)\in S_{0}$, and so, since

$\tilde{p}\cdot\delta_{s}(w)=\tilde{p}\cdot(s\bar{a},s\bar{b},0)=q\cdot
p\cdot(s\bar{a},\bar{b},0)=q\cdot(p\cdot\delta_{s}(v))$

and observing that $\tau_{q}(S_{0})=\tilde{S}_{0}$, we conclude that
$\tilde{p}\cdot\delta_{s}(w)\in\tilde{S}_{0}$. ∎

######  Proposition 3.6.

Let $S$ be a ruled hypersurface. Then $\delta_{\lambda}(S)$ is ruled for any
$\lambda>0$.

###### Proof.

Fix $\lambda>0$, define $\tilde{S}:=\delta_{\lambda}(S)$ and, given a point
$\tilde{p}\in\tilde{S}\setminus\tilde{S}_{0}$, let $p=(\bar{x},\bar{y},t)\in
S$ be such that $\tilde{p}=\delta_{\lambda}(p)$. Arguing as in the proof of
Proposition 3.5, we get that

(3.4)
$d\delta_{\lambda}|_{p}({\mathcal{H}}T_{p}S)={\mathcal{H}}T_{\tilde{p}}\tilde{S}.$

Therefore, again, $p\in S\setminus S_{0}$. Let
$w\in\tilde{p}\cdot{\mathcal{H}}T_{\tilde{p}}S$ and assume that there exists
$s\geqslant 0$ maximal with the property that
$\tilde{p}\cdot\delta_{\tau}(w)\in\tilde{S}$ for any $\tau\in[0,s].$ We claim
that $\tilde{p}\cdot\delta_{s}(w)\in\tilde{S}_{0}$. Let
$v=(\bar{a},\bar{b},0)\in{\mathcal{H}}T_{p}S$ be such that
$d\delta_{\lambda}|_{p}(v)=w$. We claim that that $w=\delta_{\lambda}(v)$.
Indeed, recalling that the Jacobian matrix of $\delta_{\lambda}$ is a diagonal
matrix with diagonal $(\lambda,\ldots,\lambda,\lambda^{2})$, then

(3.5)
$\begin{split}w(f)(q)&=\sum_{j=1}^{n}a_{j}\frac{\partial(f\circ\delta_{\lambda})}{\partial
x_{j}}(p)+\sum_{j=1}^{n}b_{j}\frac{\partial(f\circ\delta_{\lambda})}{\partial
x_{j}}(p)+\sum_{j=1}^{n}\left(a_{j}y_{j}-b_{j}x_{j}\right)T(f\circ\delta_{\lambda})(p)\\\
&=\sum_{j=1}^{n}\lambda a_{j}\frac{\partial f}{\partial
x_{j}}(\tilde{p})+\sum_{j=1}^{n}\lambda b_{j}\frac{\partial f}{\partial
x_{j}}(\tilde{p})+\sum_{j=1}^{n}((\lambda a_{j})(\lambda y_{j})-(\lambda
b_{j})(\lambda x_{j}))Tf(\tilde{p}).\end{split}$

The conclusion then follows as in the previous proof, just noticing that

$\delta_{\lambda}(p\cdot\delta_{\tau}(v))=\delta_{\lambda}(p)\cdot\delta_{\lambda}(\delta_{\tau}(v))=\tilde{p}\cdot\delta_{\lambda\tau}(v)=\tilde{p}\cdot\delta_{\tau}(\delta_{\lambda}(v))=\tilde{p}\cdot\delta_{\tau}(w)$

for any $\tau\in\mathbb{R}$, and that $\delta_{\lambda}(S_{0})=\tilde{S}_{0}$.
∎

In view of Proposition 3.5, we can enlarge the class of examples of ruled
hypersurfaces.

######  Example 3.7 (Non-Vertical Hyperplanes).

We already know that ${\mathcal{H}}_{0}$ is a characteristic ruled smooth
hypersurface. For any fixed
$q=(\bar{x}_{q},\bar{y}_{q},t_{q})\in\mathbb{H}^{n}$, we know from Proposition
3.5 that $\tau_{q}({\mathcal{H}}_{0})$ is a characteristic ruled smooth
hypersurface. Moreover, an easy computation shows that

$\tau_{q}({\mathcal{H}}_{0})=\\{(\bar{x},\bar{y},t)\in\mathbb{H}^{n}\,:\,\langle(\bar{a},\bar{b}),(\bar{x},\bar{y})\rangle+t+d=0\\},$

where $(\bar{a},\bar{b})=(-\bar{y}_{q},\bar{x}_{q})$ and $d=-t_{q}$. Finally,
notice that any hyperplane which is not vertical can be obtained as left-
translation of the horizontal hyperplane ${\mathcal{H}}_{0}$. Hence we
conclude that every hyperplane of $\mathbb{H}^{n}$ is ruled, and it is non-
characteristic if and only if it is vertical. Finally, notice that we cannot
exploit Proposition 3.6 to obtain more ruled hypersurfaces, since dilations of
hyperplanes are hyperplanes.

To conclude this section, we show that the class of ruled hypersurfaces is
closed under the action of the so-called _pseudohermitian transformations_ of
$\mathbb{H}^{n}$. To introduce this notion, we define the map
$\mathcal{J}:\mathbb{H}^{n}\longrightarrow\mathbb{H}^{n}$ by

$\mathcal{J}(\bar{x},\bar{y},t):=(-\bar{y},\bar{x},t)$

for any $p=(\bar{x},\bar{y},t)\in\mathbb{H}^{n}$. The map $\mathcal{J}$ is a
global diffeomorphism which preserves the horizontal distribution, related to
the CR structure $J$ by

$d\mathcal{J}|_{{\mathcal{H}}}=J|_{\mathcal{H}}.$

A global diffeomorphism $\varphi:\mathbb{H}^{n}\longrightarrow\mathbb{H}^{n}$
is said to be a pseudohermitian transformation of $\mathbb{H}^{n}$ if it
preserves the horizontal distribution and it commutes with $\mathcal{J}$, that
is

$d\varphi({\mathcal{H}})\subseteq{\mathcal{H}}\qquad\text{and}\qquad\varphi\circ\mathcal{J}=\mathcal{J}\circ\varphi.$

Let us begin by considering a special subclass of pseudohermitian
transformations. To this aim, let us define the map
$\varphi_{R}:\mathbb{H}^{n}\longrightarrow\mathbb{H}^{n}$ by

(3.6) $\varphi_{R}(\bar{x},\bar{y},t):=(R(\bar{x},\bar{y}),t),$

where $R$ is an orthogonal matrix of the form

$R=\left[\begin{array}[]{cc}A&B\\\ -B&A\end{array}\right],$

where $A$ and $B$ are real-valued $n\times n$ matrices.

######  Proposition 3.8.

Let $\varphi_{R}$ be as in (3.6). Then $\varphi_{R}$ is a pseudohermitian
transformation. Moreover, it holds that

$d\varphi_{R}|_{p}(\bar{a},\bar{b},0)=(R(\bar{a},\bar{b}),0)$

for any $p\in\mathbb{H}^{n}$ and any $(\bar{a},\bar{b},0)\in H_{p}$.

###### Proof.

Let $p=(\bar{x},\bar{y},t)$ and $(\bar{a},\bar{b},0)$ as in the statement, and
let $\tilde{p}:=\varphi_{R}(p)=(\bar{\tilde{x}},\bar{\tilde{y}},t)$. We first
claim that

$d\varphi_{R}|_{p}(X_{j}|_{p})=\sum_{k=1}^{n}\left(R_{kj}X_{k}|_{\tilde{p}}+R_{(n+k)j}Y_{k}|_{\tilde{p}}\right)$

and

$d\varphi_{R}|_{p}(Y_{j}|_{p})=\sum_{k=1}^{n}\left(R_{k(n+j)}X_{k}|_{\tilde{p}}+R_{n+k)(n+j)}Y_{k}|_{\tilde{p}}\right)$

for any $j=1,\ldots,n$. Indeed, let $\psi$ be a $C^{1}$ function defined in a
neighborhood of $\tilde{p}$. Let us recall that, since
$(\bar{\tilde{x}},\bar{\tilde{y}})=R(\bar{x},\bar{y})$ and $R$ is orthogonal,
then $(\bar{x},\bar{y})=R^{T}(\bar{\tilde{x}},\bar{\tilde{y}})$, which means,
recalling also the special block shape of $R$, that

$-x_{j}=\sum_{k=1}^{n}\left(-R_{kj}\tilde{x}_{k}-R_{(n+k)j}\tilde{y}_{k}\right)=\sum_{k=1}^{n}\left(-R_{(n+k)(n+j)}\tilde{x}_{k}+R_{k(n+j)}\tilde{y}_{k}\right)$

and

$y_{j}=\sum_{k=1}^{n}\left(R_{k(n+j)}\tilde{x}_{k}+R_{(n+k)(n+j)}\tilde{y}_{k}\right)=\sum_{k=1}^{n}\left(-R_{(n+k)j}\tilde{x}_{k}+R_{kj}\tilde{y}_{k}\right).$

for any $j=1,\ldots,n$. Then it holds that

$\begin{split}d\varphi_{R}|_{p}&(X_{j}|_{p})(\psi)(\tilde{p})=X_{j}|_{p}(\psi\circ\varphi_{R})(p)\\\
&=\frac{\partial}{\partial
x_{j}}(\psi\circ\varphi_{R})(p)+y_{j}T(\psi\circ\varphi_{R})(p)\\\
&=\sum_{k=1}^{n}\left(R_{kj}\frac{\partial\psi}{\partial
x_{k}}(\tilde{p})+R_{(n+k)j}\frac{\partial\psi}{\partial
y_{k}}(\tilde{p})\right)+y_{j}T(\psi)(\tilde{p})\\\
&=\sum_{k=1}^{n}\left(R_{kj}\left(\frac{\partial\psi}{\partial
x_{k}}(\tilde{p})+\tilde{y}_{k}T(\psi(\tilde{p})\right)+R_{(n+k)j}\left(\frac{\partial\psi}{\partial
y_{k}}(\tilde{p})-\tilde{x}_{k}T(\psi)(\tilde{p})\right)\right)\\\
&=\sum_{k=1}^{n}\left(R_{kj}X_{k}|_{\tilde{p}}(\psi)(\tilde{p})+R_{(n+k)j}Y_{k}|_{\tilde{p}}(\psi)(\tilde{p})\right)\end{split}$

and, similarly,

$d\varphi_{R}|_{p}(Y_{j}|_{p})(\psi)(\tilde{p})=\sum_{k=1}^{n}\left(R_{k(n+j)}X_{k}|_{\tilde{p}}(\psi)(\tilde{p})+R_{(n+k)(n+j)}Y_{k}|_{\tilde{p}}(\psi)(\tilde{p})\right)$

for any $j=1,\ldots,n$. Hence we conclude that

$\begin{split}d\varphi_{R}|_{p}&(\bar{a},\bar{b},0)=\sum_{j=1}^{n}\left(a_{j}d\varphi_{R}|_{p}(X_{j}|_{p})+b_{j}d\varphi_{R}|_{p}(Y_{j}|_{p})\right)\\\
&=\sum_{j,k=1}^{n}\left(a_{j}\left(R_{kj}X_{k}|_{\tilde{p}}+R_{(n+k)j}Y_{k}|_{\tilde{p}}\right)+b_{j}\left(R_{k(n+j)}X_{k}|_{\tilde{p}}+R_{n+k)(n+j)}Y_{k}|_{\tilde{p}}\right)\right)\\\
&=\sum_{k=1}^{n}\left(\sum_{j=1}^{n}\left(R_{kj}a_{j}+R_{k(n+j)}b_{j}\right)X_{k}|_{\tilde{p}}+\sum_{j=1}^{n}\left(R_{(n+k)j}a_{j}+R_{(n+k)(n+j)}b_{j}\right)Y_{k}|_{\tilde{p}}\right)\\\
&=(R(\bar{a},\bar{b}),0).\end{split}$

∎

As a consequence of the previous result, it is easy to see that the class of
ruled hypersurfaces is closed under the action of maps of the form (3.6).

######  Proposition 3.9.

Let $S$ be a ruled hypersurface. Then $\varphi_{R}(S)$ is ruled for any
$\varphi_{R}$ as in (3.6).

###### Proof.

The proof of this result, with the help of Proposition 3.8, follows as the
proof of Proposition 3.5 and Proposition 3.6, noticing that
$\varphi_{R}(S_{0})=(\varphi_{R}(S))_{0}$ and that, for a given $p=(z,t)\in
S\setminus S_{0}$, $(v,0)\in{\mathcal{H}}T_{p}S$ and $s\in\mathbb{R}$, it
holds that

$\begin{split}\varphi_{R}(p\cdot\delta_{s}(v,0))&=\varphi_{R}(z+sv,t+Q(z,sv))\\\
&=(R(z+sv),t+sQ(z,v))\\\ &=(Rz+sRv,t+sQ(Rz,Rv))\\\ &=(Rz,t)\cdot(sRv,0))\\\
&=\varphi_{R}(p)\cdot\delta_{s}(Rv,0).\end{split}$

∎

As a corollary of Proposition 3.8, we can conclude our initial statement.

######  Theorem 3.10.

If $S$ is ruled, then $\varphi(S)$ is ruled for any pseudohermitian
transformation $\varphi$.

###### Proof.

It follows combining Proposition 3.5, Proposition 3.9 and [CL, Theorem 4.1]. ∎

##  4\. Ruled hypersurfaces with countable characteristic set

The aim of this section is to characterise ruled hypersurfaces of
$\mathbb{H}^{n}$ with countable characteristic set, when $n\geqslant 2$. In
the first Heisenberg group $\mathbb{H}^{1}$ there are examples of ruled, non-
characteristic, smooth surfaces which are not vertical planes. As an instance,
let us consider the surface $S$ parametrized by the map
$\varphi:\mathbb{R}^{2}\longrightarrow\mathbb{H}^{1}$ defined by

$\varphi(t,\theta):=(t\cos{\theta},t\sin{\theta},\theta).$

Notice that $\varphi$ is smooth and injective. Moreover,

$\frac{\partial\varphi}{\partial
t}(t,\theta)=\cos{\theta}\frac{\partial}{\partial
x}+\sin{\theta}\frac{\partial}{\partial
y}=\cos{\theta}X|_{\varphi(t,\theta)}+\sin{\theta}Y|_{\varphi(t,\theta)}$

and

$\frac{\partial\varphi}{\partial\theta}(t,\theta)=-t\sin{\theta}\frac{\partial}{\partial
x}+t\cos{\theta}\frac{\partial}{\partial
y}+T=-t\sin{\theta}X|_{\varphi(t,\theta)}+t\cos{\theta}Y|_{\varphi(t,\theta)}+(1+t^{2})T.$

This implies that $S$ is a smooth, non-characteristic surface, and moreover

${\mathcal{H}}T_{\varphi(t,\theta)}S=\operatorname{span}\left\\{\frac{\partial\varphi}{\partial
t}(t,\theta)\right\\}$

for any $(t,\theta)\in\mathbb{R}^{2}$. Finally, for given
$t,\theta,s\in\mathbb{R}$, it holds that

$(t\cos{\theta},t\sin{\theta},\theta)\cdot(s\cos{\theta},s\sin{\theta},0)=((t+s)\cos{\theta},(t+s)\sin{\theta},\theta)\in
S,$

and so $S$ is ruled. However, the situation in higher dimensional Heisenberg
groups is quite different, and the ruling condition turns out to be more
restrictive. Indeed, we are going to prove that the only closed, ruled
hypersurfaces with countable characteristic set in $\mathbb{H}^{n}$, with
$n\geqslant 2$, are hyperplanes. To this aim, we already know that vertical
hyperplanes are non-characteristic and ruled, and that every non-vertical
hyperplane

$P:=\left\\{(\bar{x},\bar{y},t)\in\mathbb{H}^{n}\,:\,\sum_{j=1}^{n}a_{j}x_{j}+\sum_{j=1}^{n}b_{j}y_{j}+ct+d=0\right\\},$

where clearly $c\neq 0$, is ruled and satisfies

$P_{0}=\left\\{\left(\frac{b_{1}}{c},\ldots,\frac{b_{n}}{c},-\frac{a_{1}}{c},\ldots,-\frac{a_{n}}{c},-\frac{d}{c}\right)\right\\}.$

Before proving Theorem 1.3 we establish some preliminary results.

######  Proposition 4.1.

Let $S$ be a hypersurface of class $C^{1}$. Assume that $S$ is closed and
without boundary. Assume that $S$ is ruled and that $S_{0}$ is countable. Then

$p\cdot{\mathcal{H}}T_{p}S\subseteq S$

for any $p\in S\setminus S_{0}$.

###### Proof.

Let $S$ and $p$ as in the statement. Assume by contradiction that there exists
$q\in p\cdot{\mathcal{H}}T_{p}S\setminus S$. Combining 3.2 with the fact that
$S_{0}$ is countable and that $S$ is ruled, it is easy to construct a sequence
$(q_{h})_{h}\subseteq S$ converging to $q$ as $h\to\infty$. Being $S$ closed,
then $q\in S$, a contradiction. ∎

######  Proposition 4.2.

Let $S$ be a hypersurface of class $C^{1}$. Assume that $S$ is closed and
without boundary. Assume that $S$ is ruled and that $S_{0}$ is countable.
Assume that $0\in S\setminus S_{0}$. Then

$S\cap{\mathcal{H}}_{0}={\mathcal{H}}T_{0}S.$

###### Proof.

First, since $0\in S\setminus S_{0}$ and in view of Proposition 4.1, then
${\mathcal{H}}T_{0}S\subseteq S\cap{\mathcal{H}}_{0}$. Assume by contradiction
that there exists
$q=(z_{q},0)\in(S\cap{\mathcal{H}}_{0})\setminus{\mathcal{H}}T_{0}S$. If $S$
is tangent to ${\mathcal{H}}_{0}$ at $q$, then $q\in S\setminus S_{0}$.
Otherwise, since ${\mathcal{H}}T_{0}S$ is closed and $S_{0}$ is countable, it
is possible to find another point in $(S\setminus
S_{0})\setminus{\mathcal{H}}T_{0}S$. In the end, we can assume that $q\in
S\setminus S_{0}$. Again, thanks to Proposition 4.1,
$q\cdot{\mathcal{H}}T_{q}S\subseteq S$, and so
$q\cdot{\mathcal{H}}T_{q}S\cap{\mathcal{H}}_{0}\subseteq
S\cap{\mathcal{H}}_{0}$. Note that both ${\mathcal{H}}T_{0}S$ and
$q\cdot{\mathcal{H}}T_{q}S\cap{\mathcal{H}}_{0}$ are affine subspaces of
${\mathcal{H}}_{0}$. Moreover, $\dim({\mathcal{H}}T_{0}S)=2n-1$ and
$\dim(q\cdot{\mathcal{H}}T_{q}S\cap{\mathcal{H}}_{0})\geqslant 2n-2$.
Therefore we conclude that

$\dim({\mathcal{H}}T_{0}S\cap(q\cdot{\mathcal{H}}T_{q}S\cap{\mathcal{H}}_{0}))\geqslant\dim({\mathcal{H}}T_{0}S)+\dim(q\cdot{\mathcal{H}}T_{q}S\cap{\mathcal{H}}_{0})-2n=2n-3\geqslant
1,$

since $n\geqslant 2$. Therefore
$({\mathcal{H}}T_{0}S)\cap(q\cdot{\mathcal{H}}T_{q}S\cap H_{0})$ contains a
one-dimensional affine subspace of $H_{0}$. In particular, being $S_{0}$
countable, there exists
$p=(z_{p},0)\in{\mathcal{H}}T_{0}S\cap(q\cdot{\mathcal{H}}T_{q}S\cap{\mathcal{H}}_{0})\cap(S\setminus
S_{0})$. Let $v\in{\mathcal{H}}T_{p}S$ be such that $p\cdot tv=q$ for some
$t\in{\mathbb{R}}$, and let $\gamma_{p}(t):=(tz_{p},0)$. Notice that, by
construction, then $\gamma_{p}(t)\in S$ for any $t\in{\mathbb{R}}$. Moreover,
$\dot{\gamma}_{p}(1)=(z_{p},0)\in{\mathcal{H}}_{p}$, and so
$w:=(z_{p},0)\in{\mathcal{H}}T_{p}S$. Again, since $p\in S\setminus S_{0}$ and
in view of Proposition 4.1, then $p\cdot{\mathcal{H}}T_{p}S\subseteq S$.
Therefore, in particular, it holds that

$p\cdot(\alpha v+\beta w)\in S$

for any $\alpha,\beta\in{\mathbb{R}}$. Hence, if we let
$\gamma_{q}(t):=(tz_{q},0)$, we conclude that $\gamma(t)\in
S\cap{\mathcal{H}}_{0}$ for any $t\in{\mathbb{R}},$ and so
$\dot{\gamma}_{q}(0)=(z_{q},0)\in T_{0}S$. Since clearly
$(z_{q},0)\in{\mathcal{H}}_{0}$, then $q\in{\mathcal{H}}T_{0}S$, which is a
contradiction. ∎

######  Proposition 4.3.

Let $S$ be a hypersurface of class $C^{1}$. Assume that $S$ is closed and
without boundary. Assume that $S$ is ruled and that $S_{0}$ is countable. Then
either $S$ is a $t$-graph or $S$ is a vertical hyperplane.

###### Proof.

If $S$ is a $t$-graph we are done. If $S$ is not a $t$-graph, being $S_{0}$
countable, there exists $p\in S\setminus S_{0}$ such that $T|_{p}\in T_{p}S$.
Up to a left-translation, recalling 3.5, we assume that $p=0$. We show that
$S$ is a vertical hyperplane, dividing the proof into some steps.
Step 1. Thanks to Proposition 4.2, we know that there exists
$0\neq(\bar{a},\bar{b})\in{\mathbb{R}}^{2n}$ such that

${\mathcal{H}}T_{0}S={\mathcal{H}}_{0}\cap
S=\\{(\bar{x},\bar{y},0)\in\mathbb{H}^{n}\,:\,\langle(\bar{a},\bar{b}),(\bar{x},\bar{y})\rangle=0\\}.$

We assume without loss of generality that $a_{1}\neq 0$, and we let
$f(\bar{x},\bar{y}):=\langle(\bar{a},\bar{b}),(\bar{x},\bar{y})\rangle$. We
claim that

$\pi(p\cdot{\mathcal{H}}T_{p}S)\subseteq\pi({\mathcal{H}}T_{0}S)$

for any $p\in{\mathcal{H}}T_{0}S\cap(S\setminus S_{0})$, where here and in the
following the map $\pi:\mathbb{H}^{n}\longrightarrow{\mathbb{R}}^{2n}$ is
defined by

$\pi(\bar{x},\bar{y},t):=(\bar{x},\bar{y}).$

Assume by contradiction that there exists
$p=(z_{p},0)\in{\mathcal{H}}T_{0}S\cap(S\setminus S_{0})$ and
$v=(v,0)\in{\mathcal{H}}T_{p}S$ such that
$z_{p}+v\notin\pi({\mathcal{H}}T_{0}S)$. This is equivalent to say that
$f(z_{p}+v)\neq 0$. Let us define $q:=p\cdot v=(z_{p}+v,Q(z_{p},v))$. Since
$p\in S\setminus S_{0}$ and by Proposition 4.1, then $q\in S$. Moreover,
$Q(z_{p},v)\neq 0$, since otherwise $q\in{\mathcal{H}}T_{0}S$ and consequently
$f(z_{p}+v)=0$. Moreover, since $z_{p}\in{\mathcal{H}}T_{0}S$, then, letting
$\gamma(t):=(tz_{p},0)$, it holds that $\gamma(t)\in S$ for any
$t\in{\mathbb{R}}$, and so $(z_{p},0)\in{\mathcal{H}}T_{p}S$. Hence, since
$p\cdot{\mathcal{H}}T_{p}S\subseteq S$, we conclude in particular that

$P:=\\{(z_{p},0)+\alpha(z_{p},0)+\beta(v,Q(z_{p},v))\,:\,\alpha,\beta\in{\mathbb{R}}\\}\subseteq
S.$

Notice that $P$ is a vector subspace of ${\mathbb{R}}^{2n+1}$. Then in
particular $0\in P$ and $(v,Q(z_{p},v))\in T_{0}S$. Therefore, as $T\in
T_{0}S$, then $(v,0)\in T_{0}S$, and so, since $(v,0)\in{\mathcal{H}}_{0}$, we
conclude that $(v,0)\in{\mathcal{H}}T_{0}S$. Then $f(v)=0$, and so, as
$p\in{\mathcal{H}}T_{0}S$, $f(z_{p}+v)=f(z_{p})+f(v)=0$, a contradiction.
Step 2. Let $p=(z_{p},0)\in{\mathcal{H}}T_{0}S\cap(S\setminus S_{0})$. Thanks
to Step 1, we know that
$\pi(p\cdot{\mathcal{H}}T_{p}S)\subseteq\pi({\mathcal{H}}T_{0}S)$. Therefore,
if $v\in{\mathcal{H}}T_{p}S$, then $f(z_{p}+v)=0$. Since $f(z_{p})=0,$ we
conclude that $f(v)=0$, which implies that

(4.1) ${\mathcal{H}}T_{p}S={\mathcal{H}}T_{0}S$

for any $p\in{\mathcal{H}}T_{0}S$. Moreover, an easy computation shows that

${\mathcal{H}}T_{0}S=\operatorname{span}\\{Z_{2}|_{0},\ldots,Z_{n}|_{0},W_{1}|_{0},\ldots,W_{n}|_{0}\\},$

where

(4.2) $Z_{i}=a_{i}X_{1}-a_{1}X_{i}\qquad\text{and}\qquad
W_{j}=b_{j}X_{1}-a_{1}Y_{j}$

for any $i=2,\ldots,n$ and $j=1,\ldots,n$. Then (4.1) allows to conclude that

(4.3)
${\mathcal{H}}T_{p}S=\operatorname{span}\\{Z_{2}|_{p},\ldots,Z_{n}|_{p},W_{1}|_{p},\ldots,W_{n}|_{p}\\}.$

Step 3. Let us define

$\mathcal{Z}:=\\{z\in\pi({\mathcal{H}}T_{0}S)\,:\,Q(z,w)=0\text{ for any
}w\in\pi({\mathcal{H}}T_{0}S)\\}.$

Notice that, being $Q$ a bilinear map, then $\mathcal{Z}$ is a vector subspace
of $\pi({\mathcal{H}}T_{0}S)$. We claim that $\dim(\mathcal{Z})\leqslant
2n-2$. Indeed, assume by contradiction that $\dim(\mathcal{Z})\geqslant 2n-1$.
Then, since $\mathcal{Z}\subseteq\pi({\mathcal{H}}T_{0}S)$ and
$\dim(\pi({\mathcal{H}}T_{0}S))=2n-1$, we conclude that
$\mathcal{Z}=\pi({\mathcal{H}}T_{0}S)$. We show that this leads to a
contradiction. Assume first that $a_{2}=\ldots=a_{n}=b_{2}=\ldots=b_{n}=0$,
and set $z_{1}=(0,-1,0\ldots,0)$ and $z_{2}=(\bar{0},0,1,0,\ldots,0)$. Then
$f(z_{1})=f(z_{2})=0$ and $Q(z_{1},z_{2})=1\neq 0$, which implies that
$z_{1},z_{2}\notin\mathcal{Z}$. If it is not the case that
$a_{2}=\ldots=a_{n}=b_{2}=\ldots=b_{n}=0$, then assume without loss of
generality that $a_{2}\neq 0$. Let $z_{1}=(-a_{2},a_{1},0,\ldots,0)$ and
$z_{2}=(-b_{1},0,\ldots,0,a_{1},0,\ldots,0)$. Then $f(z_{1})=f(z_{2})=0$ and
$Q(z_{1},z_{2})=a_{1}a_{2}\neq 0$, which implies again that
$z_{1},z_{2}\notin\mathcal{Z}$. Therefore we conclude that
$\dim(\mathcal{Z})\leqslant 2n-2$, and so in particular

(4.4)
$\overline{\pi({\mathcal{H}}T_{0}S)\setminus\mathcal{K}}=\pi({\mathcal{H}}T_{0}S).$

Step 4. We claim that for any
$q=(z_{q},t_{q})=(x_{1}^{q},\ldots,x_{n}^{q},y_{1}^{q},\ldots,y_{n}^{q},t_{q})$
such that $z_{q}\in\pi({\mathcal{H}}T_{0}S)\setminus\mathcal{Z}$ there exists
$p=(z_{p},0)=(x_{1}^{p},\ldots,x_{n}^{p},y_{1}^{p},\ldots,y_{n}^{p},0)\in{\mathcal{H}}T_{0}S\cap(S\setminus
S_{0})$ and $v\in{\mathcal{H}}T_{p}S$ such that

(4.5) $q=p\cdot v.$

Indeed, let $q$ as above, and let $p\in{\mathcal{H}}T_{0}S\cap(S\setminus
S_{0})$ and $v\in{\mathcal{H}}T_{p}S$ to be chosen later. In view of (4.3), we
can express $v$ as

$v=\sum_{j=2}^{n}\alpha_{j}Z_{j}|_{p}+\sum_{j=1}^{n}\beta_{j}W_{j}|_{p}=\left(\sum_{j=2}^{n}\alpha_{j}a_{j}+\sum_{j=1}^{n}\beta_{j}b_{j}\right)X_{1}|_{p}-\sum_{j=2}^{n}\alpha_{j}a_{1}X_{j}|_{p}-\sum_{j=1}^{n}\beta_{j}a_{1}Y_{j}|_{p}.$

for some
$\alpha_{2},\ldots,\alpha_{n},\beta_{1},\ldots,\beta_{n}\in{\mathbb{R}}$.
Therefore, we infer that

$p\cdot
v=\left(x_{1}^{p}+\sum_{j=2}^{n}\alpha_{j}a_{j}+\sum_{j=1}^{n}\beta_{j}b_{j},x_{2}^{p}-\alpha_{2}a_{1},\ldots,y_{n}^{p}-\beta_{n}a_{1},Q(z_{p},v)\right).$

Let us choose

$\alpha_{i}=\frac{x_{i}^{p}-x_{i}^{q}}{a_{1}}\qquad\textit{and}\qquad\beta_{j}=\frac{y_{j}^{p}-y_{j}^{q}}{a_{1}}$

for any $i=2,\ldots,n$ and any $j=1,\ldots,n$. This choice implies that
$(p\cdot v)_{i}=x_{i}^{q}$ and $(p\cdot v)_{j}=y_{j-n}^{q}$ for any
$i=2,\ldots,n$ and any $j=n+1,\ldots,2n$. Moreover, since
$f(z_{p})=f(z_{q})=0$, it holds that

$(p\cdot
v)_{1}=x_{1}^{p}+\sum_{j=2}^{n}\alpha_{j}a_{j}+\sum_{j=1}^{n}\beta_{j}b_{j}=\frac{1}{a_{1}}\left(\sum_{j=1}^{n}(a_{j}x_{j}^{p}+b_{j}y_{j}^{p})-\sum_{j=2}^{m}a_{j}x_{j}^{q}+\sum_{j=1}^{n}b_{j}y_{j}^{q}\right)=x_{1}^{q}.$

Finally, notice that

$\begin{split}Q(z_{p},v)&=\left(\sum_{j=2}^{n}\alpha_{j}a_{j}+\sum_{j=1}^{n}\beta_{j}b_{j}\right)y_{1}^{p}-\sum_{j=2}^{n}\alpha_{j}a_{1}y_{j}^{p}+\sum_{j=1}^{n}\beta_{j}a_{1}x_{j}^{p}\\\
&=\frac{1}{a_{1}}\Big{(}\sum_{j=2}^{n}a_{j}x_{j}^{p}y_{1}^{p}-\sum_{j=2}^{n}a_{j}x_{j}^{q}y_{1}^{p}+\sum_{j=1}^{n}b_{j}y_{j}^{p}y_{1}^{p}-\sum_{j=1}^{n}b_{j}y_{j}^{q}y_{1}^{p}\\\
&-\sum_{j=2}^{n}a_{1}x_{j}^{p}y_{j}^{p}+\sum_{j=2}^{n}a_{1}x_{j}^{q}y_{j}^{p}+a_{1}x_{1}^{p}y_{1}^{p}+\sum_{j=2}^{n}a_{1}x_{j}^{p}y_{j}^{p}-\sum_{J=1}^{n}a_{1}x_{j}^{p}y_{j}^{q}\Big{)}\\\
&=\frac{1}{a_{1}}\Big{(}-\sum_{j=1}^{n}a_{j}x_{j}^{q}y_{1}^{p}-\sum_{j=1}^{n}b_{j}y_{j}^{q}y_{1}^{p}+\sum_{j=1}^{n}a_{1}x_{j}^{q}y_{j}^{p}-\sum_{j=1}^{n}a_{1}x_{j}^{p}y_{j}^{q}\Big{)}\\\
&=Q(z_{p},z_{q}),\end{split}$

where in the third equality we exploited the fact that $f(z_{p})=0$, while the
fourth equality follows from $f(z_{q})=0$. Since we assumed
$z_{q}\notin\mathcal{Z}$, then there exists uncountably many
$w\in\pi({\mathcal{H}}T_{0}S)$ such that $Q(w,z_{q})\neq 0.$ Therefore, since
$S_{0}$ is countable, it is possible to choose $w\in\pi({\mathcal{H}}T_{0}S)$
such that, setting

$z_{p}=\frac{t_{q}}{Q(w,z_{q})}w,$

then $p\in(S\setminus S_{0})$. We conclude that
$p\in{\mathcal{H}}T_{0}S\cap(S\setminus S_{0})$ and $Q(z_{p},z_{q})=t_{q}$.
Step 5. We are now able to conclude. Indeed, thanks to (4.5) we infer that

$\pi(HT_{0}S\setminus\mathcal{K})\times\mathbb{R}\subseteq S.$

But then, being $S$ closed and recalling (4.4), we conclude that

$\pi(HT_{0}S)\times\mathbb{R}=\overline{\pi(HT_{0}S)\setminus\mathcal{K}}\times\mathbb{R}=\overline{\pi(HT_{0}S)\times\mathbb{R}}\subseteq\overline{S}=S.$

Therefore $S$ contains the vertical hyperplane $\pi(HT_{0}S)\times\mathbb{R}$.
The thesis then follows in view of the topological assumptions on $S$. ∎

###### Proof of Theorem 1.3.

Let $S$ be as in the statement. If $S$ is a vertical hyperplane, we are done.
If not, in view of Proposition 4.3, $S$ is a $t$-graph. Being $S_{0}$
countable, and recalling 3.5, up to a left translation we may assume that
$0\in S\setminus S_{0}$ and that $T|_{0}\notin T_{0}S$. Since $0\in S\setminus
S_{0}$, we infer by Proposition 4.2 that

${\mathcal{H}}T_{0}S={\mathcal{H}}_{0}\cap
S=\\{(\bar{x},\bar{y},0)\in\mathbb{H}^{n}\,:\,\langle(\bar{x},\bar{y}),(\bar{a},\bar{b})\rangle=0\\}$

for some $0\neq(\bar{a},\bar{b})\in{\mathbb{R}}^{2n}$. Again, by 3.8 we may
assume that $a_{1}\neq 0$. Being $S$ an entire $t$-graph, and since
$T|_{0}\notin T_{0}S$ and $0\in S\setminus S_{0}$, there exists
$v=(z_{v},t_{v})\in T_{0}S$ such that $f(z_{v})\neq 0$ and $t_{v}\neq 0$. Let
us set $c:=-\frac{f(z_{v})}{t_{v}}$. We claim that

$S=\\{(z,t)\in\mathbb{H}^{n}\,:\,f(z)+ct=0\\}=:S_{c}.$

Indeed, let $p=(z_{p},t_{p})\in S\setminus S_{0}.$ If $t_{p}=0$, then
$f(z_{p})=0$, and so $p\in S_{c}$. Assume then $t_{p}\neq 0$. Then, being $S$
a $t$-graph, we infer that $f(z_{p})\neq 0.$. Let $v_{1},\ldots,v_{2n-1}$ be a
basis of ${\mathcal{H}}T_{p}S$. Since $p\in S\setminus S_{0}$ and thanks to
Proposition 4.1, then $p\cdot{\mathcal{H}}T_{p}S\subseteq S$. We claim that
there exists $j=1,\ldots,2n-1$ such that $Q(z_{p},v_{j})\neq 0$. Indeed,
assume by contradiction that $Q(z_{p},v_{1})=\ldots=Q(z_{p},v_{2n-1})=0.$ In
this case, recalling that $S$ is ruled, it holds that

(4.6)
$p\cdot{\mathcal{H}}T_{p}S=\left\\{\left(z_{p}+\sum_{j=1}^{2n-1}\alpha_{j}v_{j},t_{p}\right)\,:\,\alpha_{1},\ldots,\alpha_{2n-1}\in{\mathbb{R}}\right\\}\subseteq
S.$

We claim that

(4.7)
$\operatorname{span}\\{(v_{1},0),\ldots,(v_{2n-1},0)\\}=\operatorname{span}\\{Z_{2}|_{0},\ldots,Z_{n}|_{0},W_{1}|_{0},\ldots,W_{n}|_{0}\\},$

where $Z_{2},\ldots,Z_{n},W_{1},\ldots,W_{n}$ are defined as (4.2). Indeed, if
it was not the case, then (4.6) would imply the existence of
$q=(z_{q},t_{p})\in S$ such that $z_{q}\in\pi({\mathcal{H}}T_{0}S)$. But since
$t_{p}\neq 0$ and since $(z_{q},0)\in S$, we would contradict the fact that
$S$ is a $t$-graph. Notice that (4.7) implies that

$Z_{2}|_{0},\ldots,Z_{n}|_{0},W_{1}|_{0},\ldots,W_{n}|_{0}\in H_{p}$

and so, observing that

$Z_{j}|_{0}=a_{j}\frac{\partial}{\partial x_{1}}-a_{1}\frac{\partial}{\partial
x_{j}}=a_{j}X_{1}|_{p}-a_{1}X_{j}|_{p}+(a_{1}y_{j}-a_{j}y_{1})T$

for any $j=2,\ldots,n$ and

$W_{j}|_{0}=b_{j}\frac{\partial}{\partial x_{1}}-a_{1}\frac{\partial}{\partial
y_{j}}=b_{j}X_{1}|_{p}-a_{1}Y_{j}|_{p}+(-a_{1}x_{j}-b_{j}y_{1})T$

for any $j=1,\ldots,n$, we conclude that

$z_{p}=\frac{y_{1}}{a_{1}}(-b_{1},\ldots,-b_{n},a_{1},\ldots,a_{n}),$

which implies in particular that $f(z_{p})=0$, a contradiction. In this case,
it holds that $p\cdot HT_{p}S\cap{\mathcal{H}}_{0}\cap
S=p\cdot{\mathcal{H}}T_{p}S\cap{\mathcal{H}}T_{0}S\neq 0$. Since $n\geqslant
2$, a dimensional argument as in the proof of Proposition 4.2 implies that
$\dim(p\cdot{\mathcal{H}}T_{p}S\cap{\mathcal{H}}_{0}\cap S)\geqslant 1$.
Therefore, being $S_{0}$ countable, there exists
$q=(z_{q},0)\in(p\cdot{\mathcal{H}}T_{p}S)\cap{\mathcal{H}}T_{0}S\setminus
S_{0}$. Let then $w\in{\mathcal{H}}T_{p}S$ be such that

(4.8) $(z_{q},0)=(z_{p}+w,t_{p}+Q(z_{p}w)).$

Arguing as in the proof of Proposition 4.2, recalling that $q\in S\setminus
S_{0}$ and Proposition 4.1, we see that

$P:=\\{(z_{q},0)+\alpha(z_{q},0)+\beta(w,Q(z_{q},w))\,:\,\alpha,\beta\in{\mathbb{R}}\\}\subseteq
S,$

and so we conclude as above that $(w.Q(z_{q},w))\in T_{0}S$. This means that
there exists $\tilde{w}\in\pi({\mathcal{H}}T_{0}S)$ and
$\alpha\in{\mathbb{R}}$ such that

$(w.Q(z_{q},w))=(\tilde{w}+\alpha z_{v},\alpha t_{v}).$

Therefore, recalling (4.8), we get that

$(z_{p},t_{p})=(z_{q},0)-(w,Q(z_{p},w))=\left(z_{q}-\tilde{w}-\alpha
z_{v},-\alpha t_{v}\right).$

Therefore, since $z_{q},\tilde{w}\in\pi({\mathcal{H}}T_{0}S)$ we conclude that

$f(z_{p})+ct_{p}=-\alpha(f(z_{v})+ct_{v})=0,$

which implies that $p\in S_{c}$. Therefore we proved that $S\setminus
S_{0}\subseteq S_{c}$, and so, being $S_{0}$ countable and $S$ and $S_{c}$
closed, we conclude that $S\subseteq S_{c}$. The thesis then follows by the
topological assumptions on $S$. ∎

##  5\. Horizontal second fundamental forms on ${\mathbb{H}}^{n}$

In the current literature, different kinds of second fundamental form are
available in the sub-Riemannian setting. We recall that the _horizontal second
fundamental form of $S$ at $p\in S\setminus S_{0}$_ (cf. [32, 15, 12]) is the
map
$h_{p}:{\mathcal{H}}T_{p}S\times{\mathcal{H}}T_{p}S\longrightarrow{\mathbb{R}}$
defined by

$h_{p}(X,Y)=-\langle\nabla_{X}Y,v^{\mathbb{H}}\rangle=\langle\nabla_{X}v^{\mathbb{H}},Y\rangle$

for any $X,Y\in{\mathcal{H}}T_{p}S$, the second equality following being
$\nabla$ a metric connection. We recall that its norm is defined by

$|h_{p}|^{2}=\sum_{i,j=1}^{2n-1}h_{p}(e_{i},e_{j})^{2}$

for any $p\in S$, being $e_{1},\ldots,e_{2n-1}$ any orthonormal basis of
${\mathcal{H}}T_{p}S$. Notice that, in view of (2.1), $h$ may not be
symmetric. The _horizontal mean curvature $H_{p}$ of $S$ at $p\in S\setminus
S_{0}$_ is defined by

$H_{p}=\tau(h_{p})=\sum_{j=1}^{2n-1}h_{p}(e_{j},e_{j}),$

where here and in the following $\tau$ denote the trace operator. In analogy
with the Riemannian case, the horizontal mean curvature coincides with the
divergence of the horizontal normal (cf. [15]), meaning that

$H=\operatorname{div}_{\mathbb{H}}v^{\mathbb{H}}(p)$

for any $p\in S$. Accordingly, the following characterization of $|h|^{2}$
holds.

######  Proposition 5.1.

Let $p$ be a non-characteristic point of $S$. Let $v^{\mathbb{H}}$ be any
unitary $C^{2}$ extension of $v^{\mathbb{H}}|_{S}$. Then

$|h_{p}|^{2}=\sum_{h,k=1}^{2n}Z_{h}(v^{\mathbb{H}}_{k})Z_{k}(v^{\mathbb{H}}_{h})+4(n-1)(Td^{\mathbb{H}})^{2}.$

Moreover, if $v^{\mathbb{H}}|_{S}$ is extended as in (2.6), then

$|h_{p}|^{2}=\sum_{i,j=1}^{2n}\left(Z_{i}v^{\mathbb{H}}_{j}(p)\right)^{2}-4(Td^{\mathbb{H}})^{2}.$

###### Proof.

First, we show that the quantity
$\sum_{h,k=1}^{2n}Z_{h}(v^{\mathbb{H}}_{k})Z_{k}(v^{\mathbb{H}}_{h})$ does not
depend on the chosen unitary $C^{2}$ extension of $v^{\mathbb{H}}|_{S}$.
Indeed, in view of (2.5), we have that

$\begin{split}\sum_{h,k=1}^{2n}Z_{h}(v^{\mathbb{H}}_{k})Z_{k}(v^{\mathbb{H}}_{h})&=\sum_{h,k=1}^{2n}\delta_{h}(v^{\mathbb{H}}_{k})Z_{k}(v^{\mathbb{H}}_{h})+\sum_{h,k=1}^{2n}v^{\mathbb{H}}_{h}\langle\nabla_{\mathbb{H}}v^{\mathbb{H}}_{k},v^{\mathbb{H}}\rangle
Z_{k}(v^{\mathbb{H}}_{h})\\\
&=\sum_{h,k=1}^{2n}\delta_{h}(v^{\mathbb{H}}_{k})\delta_{k}(v^{\mathbb{H}}_{h})+\sum_{h,k=1}^{2n}\delta_{h}(v^{\mathbb{H}}_{k})v^{\mathbb{H}}_{k}\langle\nabla_{\mathbb{H}}v^{\mathbb{H}}_{h},v^{\mathbb{H}}\rangle\\\
&=\sum_{h,k=1}^{2n}\delta_{h}(v^{\mathbb{H}}_{k})\delta_{k}(v^{\mathbb{H}}_{h})+\sum_{h,k=1}^{2n}Z_{h}(v^{\mathbb{H}}_{k})v^{\mathbb{H}}_{k}\langle\nabla_{\mathbb{H}}v^{\mathbb{H}}_{h},v^{\mathbb{H}}\rangle-\left(\sum_{h,k=1}^{2n}v^{\mathbb{H}}_{h}\langle\nabla_{\mathbb{H}}v^{\mathbb{H}}_{h},v^{\mathbb{H}}\rangle\right)^{2}\\\
&=\sum_{h,k=1}^{2n}\delta_{h}(v^{\mathbb{H}}_{k})\delta_{k}(v^{\mathbb{H}}_{h}).\end{split}$

The claim then follows recalling that the horizontal tangential derivatives do
not depend on the chosen extension. Let us extend $v^{\mathbb{H}}$ as in
(2.6). Let $e_{1},\ldots,e_{2n-1}$ be an orthonormal basis of
${\mathcal{H}}T_{p}S$. For any $i=1,\ldots,2n-1$, we let
$a_{i}^{1},\ldots,a_{i}^{2n}$ be such that

$e_{i}=\sum_{j=1}^{2n}a_{i}^{j}Z_{j}.$

Then, by construction,

(5.1)
$\sum_{k=1}^{2n}a_{i}^{k}a_{j}^{k}=\delta_{ij},\qquad\sum_{k=1}^{2n}a_{i}^{k}v^{\mathbb{H}}_{k}=0\qquad\text{and}\qquad\sum_{l=1}^{2n-1}e_{k}^{l}e_{k}^{m}=\delta_{km}-v^{\mathbb{H}}_{k}v^{\mathbb{H}}_{m}$

for any $i,j=1,\ldots,2n-1$ and any $l,m=1,\ldots,2n$. Hence, recalling (2.5)
and (2.9),

$\begin{split}|h_{p}|^{2}&=\sum_{i,j=1}^{2n-1}\sum_{h,k,l,m=1}^{2n}a_{i}^{h}Z_{h}(v^{\mathbb{H}}_{k})a_{j}^{k}a_{i}^{l}Z_{l}(v^{\mathbb{H}}_{m})a_{j}^{m}\\\
&=\sum_{h,k,l,m=1}^{2n}Z_{h}(v^{\mathbb{H}}_{k})Z_{l}(v^{\mathbb{H}}_{m})(\delta_{hl}-v^{\mathbb{H}}_{h}v^{\mathbb{H}}_{l})(\delta_{km}-v^{\mathbb{H}}_{k}v^{\mathbb{H}}_{m})\\\
&=\sum_{h,k=1}^{2n}\left(Z_{h}(v^{\mathbb{H}}_{k})\right)^{2}-\sum_{k=1}^{2n}\left(\sum_{h=1}^{2n}Z_{h}(v^{\mathbb{H}}_{k})v^{\mathbb{H}}_{h}\right)^{2}\\\
&=\sum_{i,j=1}^{2n}\left(Z_{i}v^{\mathbb{H}}_{j}(p)\right)^{2}-4(Td^{\mathbb{H}})^{2}.\end{split}$

To prove the second identity, notice that

$\begin{split}\sum_{h,k=1}^{2n}Z_{h}(v^{\mathbb{H}}_{k})Z_{k}(v^{\mathbb{H}}_{h})&=\sum_{h,k=1}^{2n}\left(Z_{h}(v^{\mathbb{H}}_{k})\right)^{2}+2Td^{\mathbb{H}}\sum_{h,k=1}^{n}X_{h}(v^{\mathbb{H}}_{n+k})-2Td^{\mathbb{H}}\sum_{h,k=1}^{n}Y_{h}(v^{\mathbb{H}}_{k})\\\
&=\sum_{h,k=1}^{2n}\left(Z_{h}(v^{\mathbb{H}}_{k})\right)^{2}+2Td^{\mathbb{H}}\sum_{k=1}^{n}[X_{k},Y_{k}]d^{\mathbb{H}}\\\
&=\sum_{h,k=1}^{2n}\left(Z_{h}(v^{\mathbb{H}}_{k})\right)^{2}-4n(Td^{\mathbb{H}})^{2}\end{split}$

∎

In the first Heisenberg group ${\mathbb{H}}^{1}$, ${\mathcal{H}}TS$ is a one
dimensional distribution generated by $J(v^{\mathbb{H}})$. In particular (cf.
[44]), $h$ completely determines the behavior of
$\nabla_{J(v^{\mathbb{H}})}J(v^{\mathbb{H}})$, meaning that

$\nabla_{J(v^{\mathbb{H}})}J(v^{\mathbb{H}})=-h(J(v^{\mathbb{H}}),J(v^{\mathbb{H}}))v^{\mathbb{H}}.$

This consideration is a first crucial step in the study of minimal surfaces,
since it allows to infer that, when $H=0$, then $S$ is ruled by horizontal
lines. A horizontal line is a horizontal curve
$\Gamma:I\longrightarrow\mathbb{H}^{n}$ such that

$\langle\ddot{\Gamma}(s),Z_{j}|_{\Gamma(s)}\rangle=0$

for any $s\in I$ and any $j=1,\ldots,2n$, where here and in the following $I$
is a domain of ${\mathbb{R}}$ containing $0$. Indeed the following simple
characterization holds.

######  Proposition 5.2.

Let $\Gamma:I\longrightarrow{\mathbb{H}}^{n}$ be a horizontal curve. The
following are equivalent.

  * $(i)$

$\nabla_{\dot{\Gamma}}\dot{\Gamma}=0$ along $\Gamma$.

  * $(ii)$

$\Gamma$ is a horizontal line.

###### Proof.

Let $A=\sum_{j=1}^{2n}A_{j}Z_{j}$ be any $C^{2}$ extension of $\dot{\Gamma}$.
$\Gamma$ is a horizontal line if and only if $t\mapsto A_{j}(\Gamma(t))$ is
constant on $I$ for any $j=1,\ldots,2n$. Notice that

$\nabla_{A}A\big{|}_{\Gamma(s)}=\sum_{k=1}^{2n}A(A_{k})|_{\Gamma(s)}Z_{k}|_{\Gamma(s)}=\sum_{k=1}^{2n}\dot{\Gamma}(A_{k})|_{\Gamma(s)}Z_{k}|_{\Gamma(s)}=\sum_{k=1}^{2n}\frac{d(A_{k}(\Gamma(t))}{dt}\Big{|}_{s}Z_{k}|_{\Gamma(s)}$

for any $s\in I$. The thesis then follows. ∎

The higher dimensional case is typically more involved, since it is not always
true that

$\nabla_{X}X=-h(X,X)v^{\mathbb{H}}.$

Nevertheless, there is a particular situation in which the second fundamental
form provides global information.

######  Definition 5.3.

Let $S$ be a hypersurface of class $C^{2}$. We say that $S$ is _horizontally
totally geodesic_ when

(5.2) $h(X,X)=0$

for any $X\in C^{1}(S,{\mathcal{H}}TS)$, that is when $h$ is an _alternating_
bilinear form.

Notice that (5.2) is equivalent to require that the symmetric part of $h$ is
identically vanishing. Let us denote the latter by $\tilde{h}$, that is

$\tilde{h}(X,Y)=\frac{h(X,Y)+h(Y,X)}{2}$

For any $X,Y\in C^{1}(S,{\mathcal{H}}TS)$. This kind of symmetric second
fundamental form has already been considered, although through different but
equivalent definitions, by several authors (cf. e.g. [42, 15, 43]). It is
clear that

$h_{p}=0\implies\tilde{h}_{p}=0$

for any non-characteristic point $p\in S$, while the converse implication may
be false in general. More precisely, $|h|$ and $|\tilde{h}|$ can be related in
the following way.

######  Proposition 5.4.

Let $p$ be a non-characteristic point of $S$. Let $v^{\mathbb{H}}$ be any
unitary $C^{2}$ extension of $v^{\mathbb{H}}|_{S}$. Then

$|\tilde{h}_{p}|^{2}=\sum_{h,k=1}^{2n}Z_{h}(v^{\mathbb{H}}_{k})Z_{k}(v^{\mathbb{H}}_{h})+2(n-1)\left(Td^{\mathbb{H}}(p)\right)^{2}.$

Moreover,

$|h_{p}|^{2}=|\tilde{h}_{p}|^{2}+2(n-1)\left(Td^{\mathbb{H}}(p)\right)^{2}.$

Finally, if $v^{\mathbb{H}}$ is extended as in (2.6), then

$|\tilde{h}_{p}|^{2}=\sum_{h,k=1}^{2n}\left(Z_{h}v^{\mathbb{H}}_{k}(p)\right)^{2}-2(n+1)\left(Td^{\mathbb{H}}(p)\right)^{2}$

###### Proof.

Let $e_{1},\ldots,e_{2n-1}$ be as in the proof of Proposition 5.1. Notice that

$|\tilde{h}_{p}|^{2}=\tau\left(\tilde{h}_{p}\cdot\tilde{h}_{p}^{T}\right)=\tau\left(\frac{(h_{p}+h_{p}^{T})^{2}}{4}\right)=\frac{1}{2}|h_{p}|^{2}+\frac{1}{2}\tau\left(h_{p}^{2}\right).$

Arguing as in the proof of Proposition 5.1,

$\begin{split}\tau\left(h_{p}^{2}\right)&=\sum_{i,j=1}^{2n-1}\sum_{h,k,l,m=1}^{2n}a_{i}^{h}Z_{h}(v^{\mathbb{H}}_{k})a_{j}^{k}a_{j}^{l}Z_{l}(v^{\mathbb{H}}_{m})a_{i}^{m}\\\
&=\sum_{h,k,l,m=1}^{2n}Z_{h}(v^{\mathbb{H}}_{k})Z_{l}(v^{\mathbb{H}}_{m})\sum_{i=1}^{2n-1}a_{i}^{h}a_{i}^{m}\sum_{j=1}^{2n-1}a_{j}^{k}a_{j}^{l}\\\
&=\sum_{h,k,l,m=1}^{2n}Z_{h}(v^{\mathbb{H}}_{k})Z_{l}(v^{\mathbb{H}}_{m})(\delta_{hm}-v^{\mathbb{H}}_{h}v^{\mathbb{H}}_{m})(\delta_{kl}-v^{\mathbb{H}}_{k}v^{\mathbb{H}}_{l})\\\
&=\sum_{h,k=1}^{2n}Z_{h}(v^{\mathbb{H}}_{k})Z_{k}(v^{\mathbb{H}}_{h}).\end{split}$

Exploiting Proposition 5.1, the thesis follows. ∎

In view of Proposition 5.1 and Proposition 5.4, non-characteristic
hypersurfaces of class $C^{2}$ with $h\equiv 0$ are trivially vertical
hyperplanes, provided that $n\geqslant 2$. Indeed, if $S$ is such a
hypersurface, $N$ is its Euclidean unit normal and $v^{\mathbb{H}}$ its
horizontal unit normal, then Proposition 5.4 and (2.10) imply that
$\tilde{h}\equiv 0$, $N_{2n+1}\equiv 0$, $N=N(\bar{x},\bar{y})$ and
$v^{\mathbb{H}}=(N_{1},\ldots,N_{2n})$. Hence

$0=|\tilde{h}|^{2}=\sum_{i,j=1}^{2n}Z_{i}v^{\mathbb{H}}_{j}Z_{j}v^{\mathbb{H}}_{i}=\sum_{i,j=1}^{2n+1}\frac{\partial
N_{j}}{\partial z_{i}}\frac{\partial N_{i}}{\partial z_{j}},$

where the last term coincides with the squared norm of the Euclidean second
fundamental form of $S$. Hence $S$ is a hyperplane, which is vertical since
$N_{2n+1}\equiv 0$. As already mentioned, when $n\geqslant 2$ it is not in
general true that $\tilde{h}=0$ implies $h=0$.

######  Example 5.5.

As an instance, consider in ${\mathbb{H}}^{2}$ the non-vertical hyperplane

$S:=\\{(\bar{x},\bar{y},t)\in{\mathbb{H}}^{2}\,:\,a_{1}x_{1}+a_{2}x_{2}+b_{1}y_{1}+b_{2}y_{2}+t+d=0\\}$

for some $a_{1},a_{2},b_{1},b_{2},d\in{\mathbb{R}}$. An easy computation shows
that

$N(p)=\frac{(a_{1},a_{2},b_{1},b_{2},1)}{\sqrt{1+a_{1}^{2}+a_{2}^{2}+b_{1}^{2}+b_{2}^{2}}}\qquad\text{and}\qquad
N^{\mathbb{H}}(p)=\frac{(a_{1}+y_{1},a_{2}+y_{2},b_{1}-x_{1},b_{2}-x_{2})}{\sqrt{1+a_{1}^{2}+a_{2}^{2}+b_{1}^{2}+b_{2}^{2}}}$

for any $p\in S$. Therefore, $S$ has a unique characteristic point
$p_{0}=(b_{1},b_{2},-a_{1},-a_{2},-d)$. Far from $p_{0}$, $v^{\mathbb{H}}$ can
be expressed by

$v^{\mathbb{H}}(p)=\frac{(a_{1}+y_{1},a_{2}+y_{2},b_{1}-x_{1},b_{2}-x_{2})}{\sqrt{(a_{1}+y_{1})^{2}+(a_{2}+y_{2})^{2}+(b_{1}-x_{1})^{2}+(b_{2}-x_{2})^{2}}}$

for any $p\in S\setminus S_{0}$.

Recalling (2.10), a tedious but simple computations shows that

$\sum_{h,k=1}^{4}Z_{h}(v^{\mathbb{H}}_{k})Z_{k}(v^{\mathbb{H}}_{h})=-\frac{2}{(a_{1}+y_{1})^{2}+(a_{2}+y_{2})^{2}+(b_{1}-x_{1})^{2}+(b_{2}-x_{2})^{2}}=-2(Td^{\mathbb{H}})^{2}.$

Hence, Proposition 5.4 implies that $\tilde{h}\equiv 0$ on $S\setminus S_{0}$.
Nevertheless, in view of the previous computation and Proposition 5.1, we
conclude that

$|h_{p}|^{2}=\frac{2}{(a_{1}+y_{1})^{2}+(a_{2}+y_{2})^{2}+(b_{1}-x_{1})^{2}+(b_{2}-x_{2})^{2}}$

for any $p\in S\setminus S_{0}$.

##  6\. Local existence of geodesics on hypersurfaces

Let $S$ be a hypersurface of class $C^{2}$. Let $p\in S\setminus S_{0}$ and
$w\in{\mathcal{H}}T_{p}S$. We wish to find a curve $\Gamma\in C^{2}(I,S)$
solving the differential problem

(6.1) $\left\\{\begin{aligned} &\Gamma\text{ is horizontal}\\\
&\nabla^{S}_{\dot{\Gamma}}\dot{\Gamma}=0&&\text{on }I\\\ &\Gamma(0)=p\\\
&\dot{\Gamma}(0)=w\end{aligned}\right.$

Arguing for instance as in [33], it is not difficult to show that solutions to
(6.1) are geodesics in the Carnot-Carathéodory space associated with the sub-
Riemannian structure $(S,\langle\cdot,\cdot\rangle_{S})$. First, notice that,
by means of [22, Theorem 6.5] and [4, Theorem 1.2] and without loss of
generality, there exists $\Omega\subseteq{\mathbb{R}}^{2n}$ and $\varphi\in
C(\Omega)$ such that $\nabla^{\varphi}\varphi\in
C(\Omega,{\mathbb{R}}^{2n-1})$, $U=i(\Omega)\cdot j({\mathbb{R}})$ is an open
neighborhood of $p$ and

$S\cap U=\operatorname{graph}_{Y_{1}}(\varphi,\Omega)\cap U.$

We need the following lemma.

######  Proposition 6.1.

Let $\varphi\in C(\Omega)$ be such that $\nabla^{\varphi}\varphi\in
C(\Omega,{\mathbb{R}}^{2n-1})$. Assume that
$\operatorname{graph}_{Y_{1}}(\varphi,\Omega)$ is a non-characteristic
hypersurface of class $C^{2}$. Then $\varphi\in C^{2}(\Omega)$.

###### Proof.

Let us consider the map $g:{\mathbb{H}}^{n}\longrightarrow{\mathbb{H}}^{n}$
defined by

$g(\bar{x},\bar{y},t)=(\bar{x},\bar{y},t-x_{1}y_{1})$

for any $(\bar{x},\bar{y},t)\in\mathbb{H}^{n}$. Notice that $g$ is smooth,
bijective and and $\det(Dg)\equiv 1$. Hence $g$ is a smooth diffeomorphism.
Let us set $\hat{S}:=g(S)$. Notice that $\hat{S}$ is of class $C^{2}$. It is
easy to check that

$\hat{S}\cap
g(U)=\\{(\bar{\xi},\varphi(\bar{\xi},\tilde{\eta},\tau),\tilde{\eta},\tau)\,:\,(\tilde{\xi},\bar{\eta},\tau)\in\Omega\\}.$

Therefore the thesis follows provided that $(\hat{N}(\hat{p}))_{n+1}\neq 0$
for any $\hat{p}\in\hat{S}\cap g(U)$, being $\hat{N}(\hat{p})$ the Euclidean
normal to $\hat{S}$ at $\hat{p}$. Assume by contradiction that there exists
$\hat{p}\in\hat{S}\cap g(U)$ such that $(\hat{N}(\hat{p}))_{n+1}=0$. This
implies that $(\bar{0},1,\tilde{0},0)\in T_{\hat{p}}\hat{S}$. Let $p\in S$ be
such that $g(p)=\hat{p}$. Noticing that

$(dg)|_{p}(Y_{1}|_{p})=(\bar{0},1,\tilde{0},0)\in T_{\hat{p}}\hat{S},$

we infer that $Y_{1}|_{p}\in T_{p}S$. Since $S$ is non-characteristic, (2.4)
implies that $(v^{\mathbb{H}}(p))_{n+1}=0$. On the other hand, we know from
[4, Theorem 1.2] that $(v^{\mathbb{H}}(p))_{n+1}\neq 0$, a contradiction.

∎

Therefore we reduce (6.1) to a differential problem for curves in $\Omega$. To
this aim, fix $q\in\Omega$ such that $\Psi(q)=p$, and let
$\gamma(s)=(\bar{\xi}(s),\tilde{\eta}(s),\tau(s)):I\longrightarrow\Omega$. If
we lift $\gamma$ to a curve $\Gamma:I\longrightarrow\mathbb{H}^{n}$ by letting

$\Gamma(s)=\Psi(\gamma(s))=(\bar{\xi}(s),\varphi(\gamma(s)),\tilde{\eta}(s)),\tau(s)-\xi_{1}(s)\varphi(\gamma(s)))$

for any $s\in I$, then by construction $\Gamma(I)\subseteq S$. From now on, we
fix the notation $\alpha(s):=\varphi(\gamma(s))$. To give a meaning to (6.1)
we need that $\dot{\Gamma}$ is horizontal. Notice that

$\begin{split}\dot{\Gamma}&=(\dot{\xi_{1}},\ldots,\dot{\xi_{n}},\dot{\alpha},\dot{\eta_{2}},\ldots,\dot{\eta_{n}},\dot{\tau}-\dot{\xi_{1}}\alpha-\xi_{1}\dot{\alpha})\\\
&=\sum_{j=1}^{n}\dot{\xi_{j}}X_{j}+\dot{\alpha}Y_{1}+\sum_{2=1}^{n}\dot{\eta_{j}}Y_{j}+\left(\dot{\tau}-2\alpha\dot{\xi_{1}}-\sum_{j=2}^{n}\eta_{j}\dot{\xi_{j}}+\sum_{j=2}^{n}\xi_{j}\dot{\eta_{j}}\right)T.\end{split}$

Therefore $\dot{\Gamma}$ admits a $C^{1}$ extension to the whole
${\mathcal{H}}TS$ if and only if

(6.2)
$\dot{\tau}=2\alpha\dot{\xi_{1}}+\sum_{j=2}^{n}\eta_{j}\dot{\xi_{j}}-\sum_{j=2}^{n}\xi_{j}\dot{\eta_{j}},$

that is if and only if $\gamma$ is horizontal in $(\Omega,d_{\varphi})$. Let
us denote such an extension by $A=\sum_{j=1}^{2n}\psi_{j}Z_{j}$. This means
that $A\in C^{1}(S,{\mathcal{H}}TS)$ and

$\psi_{j}(\Gamma(s))=\dot{\Gamma}_{j}(s)$

for any $s\in I$ and any $j=1,\ldots,2n$. Thanks to the aforementioned
properties of $\nabla^{S}$ and recalling (2.2), then

$\begin{split}\nabla^{S}_{\dot{\Gamma}}\dot{\Gamma}\big{|}_{\Gamma(s)}&=\nabla_{\dot{\Gamma}}\dot{\Gamma}\big{|}_{\Gamma(s)}-\left\langle\nabla_{\dot{\Gamma}}\dot{\Gamma}\big{|}_{\Gamma(s)},v^{\mathbb{H}}\big{|}_{\Gamma(s)}\right\rangle
v^{\mathbb{H}}\big{|}_{\Gamma(s)}\\\
&=\sum_{j=1}^{2n}\langle\dot{\Gamma}(s),\nabla_{H}\psi_{j}(\Gamma(s))\rangle
Z_{j}\big{|}_{\Gamma(s)}-\left(\sum_{k=1}^{2n}\langle\dot{\Gamma}(s),\nabla_{H}\psi_{j}(\Gamma(s))\rangle
v^{\mathbb{H}}_{k}\big{|}_{\Gamma(s)}\right)v^{\mathbb{H}}\big{|}_{\Gamma(s)}\\\
&=\sum_{j=1}^{2n}\ddot{\Gamma}_{j}(s)Z_{j}\big{|}_{\Gamma(s)}-\left(\sum_{k=1}^{2n}\ddot{\Gamma}_{k}(s)v^{\mathbb{H}}_{k}\big{|}_{\Gamma(s)}\right)v^{\mathbb{H}}\big{|}_{\Gamma(s)}\end{split}$

for any $s\in I$. Hence $\nabla^{S}_{\dot{\Gamma}}\dot{\Gamma}=0$ if and only
if

(6.3)
$\ddot{\Gamma}_{j}-v^{\mathbb{H}}_{j}\langle\ddot{\Gamma},v^{\mathbb{H}}\rangle=0$

for any $j=1,\ldots,2n$. We need to traduce (6.3) in terms of $\gamma$. To
this aim, recalling (6.2), notice that

$\ddot{\Gamma}=\sum_{j=1}^{n}\ddot{\xi_{j}}X_{j}+\ddot{\alpha}Y_{1}+\sum_{2=1}^{n}\ddot{\eta_{j}}Y_{j}.$

######  Lemma 6.2.

It holds that

$\langle\ddot{\Gamma},v^{\mathbb{H}}\rangle=-W^{-\frac{1}{2}}\left(2\tilde{T}\varphi\dot{\alpha}\dot{\xi}_{1}+\langle
D^{2}\varphi\dot{\gamma},\dot{\gamma}\rangle\right).$

###### Proof.

Notice that

(6.4)
$\dot{\alpha}(s)=\langle\dot{\gamma},D\varphi(\gamma(s))\rangle\qquad\text{and}\qquad\ddot{\alpha}(s)=\langle\ddot{\gamma}(s),D\varphi(\gamma(s))\rangle+\langle
D^{2}\varphi(\gamma(s))\dot{\gamma}(s),\dot{\gamma}(s)\rangle$

for any $s\in I$. Moreover, taking derivatives in (6.2), we see that

(6.5)
$\ddot{\tau}=2\dot{\alpha}\dot{\xi}_{1}+2\alpha\ddot{\xi}_{1}+\sum_{j=2}^{n}\eta_{j}\ddot{\xi}_{j}-\sum_{j=2}^{n}\xi_{j}\ddot{\eta}_{j}.$

Exploiting (2.11), (6.4) and (6.5), we see that

$\begin{split}W^{\frac{1}{2}}\langle\ddot{\Gamma},v^{\mathbb{H}}\rangle&=W^{\varphi}\varphi\ddot{\xi}_{1}+\sum_{j=2}^{n}\tilde{X}_{j}\varphi\ddot{\xi}_{j}+\sum_{j=2}^{n}\tilde{Y}_{j}\varphi\ddot{\eta}_{j}-\ddot{\alpha}\\\
&=\ddot{\xi}_{1}\varphi_{\xi_{1}}+2\ddot{\xi}_{1}\alpha\varphi_{\tau}+\sum_{j=2}^{n}\ddot{\xi}_{j}\varphi_{\xi_{j}}+\sum_{j=2}^{n}\eta_{j}\ddot{\xi}_{j}\varphi_{\tau}+\sum_{j=2}^{n}\ddot{\eta}_{j}\varphi_{\eta_{j}}-\sum_{j=2}^{n}\xi_{j}\ddot{\eta}_{j}\varphi_{\tau}\\\
&\quad-\ddot{\xi}_{1}\varphi_{\xi_{1}}-\sum_{j=2}^{n}\ddot{\xi}_{j}\varphi_{\xi_{j}}-\sum_{j=2}^{n}\ddot{\eta}_{j}\varphi_{\eta_{j}}-\ddot{\tau}\varphi_{\tau}-\langle
D^{2}\varphi\dot{\gamma},\dot{\gamma}\rangle\\\
&=\tilde{T}\varphi\left(2\alpha\ddot{\xi}_{1}+\sum_{j=2}^{n}\eta_{j}\ddot{\xi}_{j}-\sum_{j=2}^{n}\ddot{\eta}_{j}\xi_{j}-\ddot{\tau}\right)-\langle
D^{2}\varphi\dot{\gamma},\dot{\gamma}\rangle\\\
&=-2\tilde{T}\varphi\dot{\alpha}\dot{\xi}_{1}-\langle
D^{2}\varphi\dot{\gamma},\dot{\gamma}\rangle.\end{split}$

∎

In the following, we let $M=2\tilde{T}\varphi\dot{\alpha}\dot{\xi}_{1}+\langle
D^{2}\varphi\dot{\gamma},\dot{\gamma}\rangle$. Notice that, by Lemma 6.2, the
term $\langle\ddot{\Gamma},v^{\mathbb{H}}\rangle$ does not involve second
derivatives of $\gamma$. Therefore (6.1) is equivalent to the following
differential problem.

(6.6) $\left\\{\begin{aligned} &\ddot{\xi}_{1}+W^{-1}W^{\varphi}\varphi
M=0&&\text{on }I,\qquad\xi_{1}(0)=\xi_{1}^{0},\qquad\dot{\xi}_{1}(0)=w_{1}\\\
&\ddot{\xi}_{j}+W^{-1}\tilde{X}_{j}\varphi M=0&&\text{on
}I,\qquad\xi_{j}(0)=\xi_{j}^{0},\qquad\dot{\xi}_{j}(0)=w_{j}\qquad\text{$j=2,\ldots,n$}\\\
&\ddot{\alpha}-W^{-1}M=0&&\text{on
}I,\qquad\alpha(0)=y_{1},\qquad\dot{\alpha}(0)=w_{n+1}\\\
&\ddot{\eta}_{j}+W^{-1}\tilde{Y}_{j}\varphi M=0&&\text{on
}I,\qquad\eta_{j}(0)=\eta_{j}^{0},\qquad\dot{\eta}_{j}(0)=w_{n+j}\qquad\text{$j=2,\ldots,n$}\\\
&\dot{\tau}=2\alpha\dot{\xi_{1}}+\sum_{j=2}^{n}\eta_{j}\dot{\xi_{j}}-\sum_{j=2}^{n}\xi_{j}\dot{\eta_{j}}&&\text{on
}I,\qquad\tau(0)=t+\xi_{1}^{0}\varphi(q)\end{aligned}\right.$

A key step consist in showing that the third line of (6.6) is redundant.

######  Lemma 6.3.

A curve $\gamma\in C^{2}(I,\Omega)$ solves (6.6) if and only if it solves the
following differential system.

(6.7) $\left\\{\begin{aligned} &\ddot{\xi}_{1}+W^{-1}W^{\varphi}\varphi
M=0&&\text{on }I,\qquad\xi_{1}(0)=\xi_{1}^{0},\qquad\dot{\xi}_{1}(0)=w_{1}\\\
&\ddot{\xi}_{j}+W^{-1}\tilde{X}_{j}\varphi M=0&&\text{on
}I,\qquad\xi_{j}(0)=\xi_{j}^{0},\qquad\dot{\xi}_{j}(0)=w_{j}\qquad\text{$j=2,\ldots,n$}\\\
&\ddot{\eta}_{j}+W^{-1}\tilde{Y}_{j}\varphi M=0&&\text{on
}I,\qquad\eta_{j}(0)=\eta_{j}^{0},\qquad\dot{\eta}_{j}(0)=w_{n+j}\qquad\text{$j=2,\ldots,n$}\\\
&\dot{\tau}=2\alpha\dot{\xi_{1}}+\sum_{j=2}^{n}\eta_{j}\dot{\xi_{j}}-\sum_{j=2}^{n}\xi_{j}\dot{\eta_{j}}&&\text{on
}I,\qquad\tau(0)=t+\xi_{1}^{0}\varphi(q)\end{aligned}\right.$

###### Proof.

If $\gamma\in C^{2}(I,\Omega)$ solves (6.6), then clearly solves (6.7).
Conversely, assume that $\gamma\in C^{2}(I,\Omega)$ solves (6.7). Since
$y_{1}=\varphi(q)$, then $\alpha(0)=\varphi(\gamma(0))=\varphi(q)=y_{1}$.
Moreover, notice that

$\begin{split}\dot{\alpha}(0)&=\langle\dot{\gamma}(0),D\varphi(q)\rangle\\\
&=\dot{\xi}_{1}(0)\varphi_{\xi_{1}}(q)+\sum_{j=2}^{n}\dot{\xi}_{j}(0)\varphi_{\xi_{j}}(q)+\sum_{j=2}^{n}\dot{\eta}_{j}(0)\varphi_{\eta_{j}}(q)+\dot{\tau}(0)\varphi_{\tau}(q)\\\
&=w_{1}W^{\varphi}\varphi(q)+\sum_{j=2}^{n}w_{j}\tilde{X}_{j}\varphi(q)+\sum_{j=2}^{n}w_{n+j}\tilde{Y}_{j}\varphi(q)\\\
&=w_{n+1},\end{split}$

where the last equality follows from (2.11) and the fact that
$w\in{\mathcal{H}}T_{p}S$. Observe that, recalling (6.5) and exploiting all
the second-order equations in (6.7),

$\begin{split}\langle\ddot{\gamma},D\varphi\rangle&=\ddot{\xi}_{1}\varphi_{\xi_{1}}+\sum_{j=2}^{n}\ddot{\xi}_{j}\varphi_{\xi_{j}}+\sum_{j=2}^{n}\ddot{\eta}_{j}\varphi_{\eta_{j}}+\ddot{\tau}\tilde{T}\varphi\\\
&=\ddot{\xi}_{1}W^{\varphi}\varphi+\sum_{j=2}^{n}\ddot{\xi}_{j}\tilde{X}_{j}\varphi+\sum_{j=2}^{n}\ddot{\eta}_{j}\tilde{Y}_{j}\varphi+2\tilde{T}\varphi\dot{\alpha}\dot{\xi}_{1}\\\
&=-W^{-1}M|\nabla^{\varphi}\varphi|^{2}+2\tilde{T}\varphi\dot{\alpha}\dot{\xi}_{1}.\end{split}$

Therefore, we conclude that

$\begin{split}\ddot{\alpha}-W^{-1}M&=W^{-1}(W\langle\ddot{\gamma},D\varphi\rangle+W\langle
D^{2}\varphi\dot{\gamma},\dot{\gamma}\rangle-2\tilde{T}\varphi\dot{\alpha}\dot{\xi}_{1}-\langle
D^{2}\varphi\dot{\gamma},\dot{\gamma}\rangle)\\\
&=W^{-1}(\langle\ddot{\gamma},D\varphi\rangle+|\nabla^{\varphi}\varphi|^{2}\langle\ddot{\gamma},D\varphi\rangle+|\nabla^{\varphi}\varphi|^{2}\langle
D^{2}\varphi\dot{\gamma},\dot{\gamma}\rangle-2\tilde{T}\varphi\dot{\alpha}\dot{\xi}_{1})\\\
&=W^{-1}(-W^{-1}M|\nabla^{\varphi}\varphi|^{2}+|\nabla^{\varphi}\varphi|^{2}\langle\ddot{\gamma},D\varphi\rangle+|\nabla^{\varphi}\varphi|^{2}\langle
D^{2}\varphi\dot{\gamma},\dot{\gamma}\rangle)\\\
&=\frac{|\nabla^{\varphi}\varphi|^{2}}{1+|\nabla^{\varphi}\varphi|^{2}}(\ddot{\alpha}-W^{-1}M),\end{split}$

which is equivalent to say that

$W^{-1}(\ddot{\alpha}-W^{-1}M)=0.$

Being $W^{-1}\neq 0$, the thesis follows. ∎

We can summarize the previous achievements in the following statement.

######  Proposition 6.4.

The following properties hold.

  * $(i)$

If $\Gamma\in C^{2}(I,S)$ solves (6.1), then $\gamma:I\longrightarrow\Omega$
defined by

$\gamma(s):=\Pi(\Gamma(s))$

for any $s\in I$ solves (6.7).

  * $(ii)$

If $\gamma\in C^{2}(\Omega)$ solves (6.7), then
$\Gamma:I\longrightarrow\Omega$ defined by

$\Gamma(s):=\Psi(\gamma(s))$

for any $s\in I$ solves (6.1).

###### Proof.

$(ii)$ follows thanks to Lemma 6.3. To prove $(i)$, notice that, if $\Gamma$
is as in the statement, then $\Gamma=\Psi(\sigma)$, where
$\sigma=\Pi(\Gamma)$, and so $(i)$ easily follows. ∎

######  Theorem 6.5.

The initial value problem (6.1) admits a unique local solution $\Gamma\in
C^{2}(I,S)$.

###### Proof.

In view of Proposition 6.4, it suffices to show that the initial value problem
(6.7) admits locally a unique solution. Notice that (6.7) can be seen as a
fist-order initial value problem by means of a standard doubling variable
argument. More precisely, let us introduce the equations

(6.8)
$\dot{\xi}_{1}=\Xi_{1},\qquad\dot{\xi}_{j}=\Xi_{j}\qquad\text{and}\qquad\dot{\eta}_{j}=H_{j}$

for any $j=2,\ldots,n$, let us define the curve
$\tilde{\Gamma}:I\longrightarrow{\mathbb{R}}^{4n-1}$ by

$\tilde{\Gamma}=(\xi_{1},\ldots,\xi_{n},\eta_{2},\ldots,\eta_{n},\tau,\Xi_{1},\ldots,\Xi_{n},H_{2},\ldots,H_{n}),$

and let
$\tilde{q}=(x_{1},\ldots,x_{n},y_{2},\ldots,y_{n},t-x_{1}y_{1},w_{1},\ldots,w_{n},w_{n+2},\ldots,w_{2n})$.
Then (6.7) is equivalent to the first-order initial value problem

(6.9) $\left\\{\begin{aligned}
&\dot{\tilde{\Gamma}}(s)=F(s,\Gamma(s))\quad\text{on }I\\\
&\tilde{\Gamma}(0)=\tilde{q}\end{aligned}\right.$

where $F:I\times{\mathbb{R}}^{4n-1}\longrightarrow{\mathbb{R}}^{4n-1}$ is
defined in the obvious way taking into account (6.7) and (6.8). Thanks to
Proposition 6.1, $F$ is of class $C^{1}$ in a neighborhood of $(0,\tilde{q})$.
Hence the thesis follows by means of the classical Picard-Lindelöf Theorem
(cf. e.g. [31]). ∎

###### Proof of Theorem 1.4.

Fix $p=(\bar{x},\bar{y},t)\in S\setminus S_{0}$. Assume first that there
exists an open neighborhood $U$ of $p$ such that $\tilde{h}\equiv 0$ on $U$.
Fix $w\in{\mathcal{H}}T_{p}S$. As before, recalling also Proposition 6.1, we
can assume that there exists $\Omega\subseteq{\mathbb{R}}^{2n}$ and
$\varphi\in C^{2}(\Omega)$ such that

$S\cap V=\operatorname{graph}_{Y_{1}}(\varphi,\Omega)\cap V,$

where $V=i(\Omega)\cdot j({\mathbb{R}})$. In view of Theorem 6.5, there exists
a small domain $I\subseteq{\mathbb{R}}$ such that $0\in I$ and a curve
$\Gamma\in C^{2}(I,S)$ solving (6.1) with initial data $\Gamma(0)=p$ and
$\dot{\Gamma}(0)=w$. Since $\tilde{h}\equiv 0$, and recalling Proposition 5.2,
we conclude that $\Gamma(s):=p\cdot(sw,0)$. Hence $S$ is locally ruled at $p$.
Conversely, assume that $S$ is locally ruled in a suitable neighborhood $U$ of
$p$. Assume also that $U\cap S_{0}=\emptyset$. Fix $\bar{p}\in U$ and
$w\in{\mathcal{H}}T_{\bar{p}}S$. Since $\Gamma(s):=\bar{p}\cdot(sw,0)$ lies
locally in $S$, then $h_{\bar{p}}(w,w)=0$, and so $\tilde{h}_{\bar{p}}=0$. ∎

###### Proof of Theorem 1.1.

The first equivalence follows from Theorem 1.3. If in addition $S$ is
topologically closed and $n\geqslant 2$, arguing as in [8, Proposition 4.1] it
is easy to see that the fact that $S$ is horizontally totally geodesic implies
that $S_{0}$ is constituted by isolated points, and so it is countable. The
thesis then follows by Theorem 1.4. ∎

###### Proof of Theorem 1.5.

$S$ is clearly a smooth hypersurface. Let $p\in S\setminus S_{0}$. It is well-
known that

$N(p)=\frac{1}{\sqrt{1+|Du(z)|_{\mathbb{R}^{2n}}^{2}}}(Du(z),-1)=\frac{1}{\sqrt{1+x_{1}^{2}+y_{1}^{2}}}(x_{1},0,\ldots,0,-y_{1},0,\ldots,0,-1),$

and so

$v^{\mathbb{H}}(p)=v^{\mathbb{H}}(z)=\frac{1}{\sqrt{2(x_{1}-y_{1})^{2}+\sum_{j=2}^{n}(x_{j}^{2}+y_{j}^{2})}}(x_{1}-y_{1},-y_{2},\ldots,-y_{n},x_{1}-y_{1},x_{2},\ldots,x_{n}).$

Since in this case $v^{\mathbb{H}}$ does not depend on $t$, an easy
computation shows that

(6.10)
$\operatorname{div}_{\mathbb{H}}v^{\mathbb{H}}(p)=\operatorname{div}_{\mathbb{R}^{2n}}v^{\mathbb{H}}(z)=0$

for any $p\in S\setminus S_{0}$. Since $n\geqslant 2$, (6.10) allows us to
apply [11, Corollary F] and [5, Theorem 2.3], which, together with [46,
Example 5.29], imply that $S$ is minimal. We conclude noticing that, in view
of Theorem 1.1, $S$ is not horizontally totally geodesic. ∎

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