#  Umbilical submanifolds of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$.

###### Abstract.

In this article we complete the classification of the umbilical submanifolds
of a Riemannian product of space forms, addressing the case of a conformally
flat product $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$, which has not been
covered in previous works on the subject. We show that there exists precisely
a $p$-parameter family of congruence classes of umbilical submanifolds of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ with substantial codimension $p$,
which we prove to be at most $\mbox{min}\,\\{k+1,n-k+2\\}$. We study more
carefully the cases of codimensions one and two and exhibit, respectively, a
one-parameter family and a two-parameter family (together with three extra
one-parameter families) that contain precisely one representative of each
congruence class of such submanifolds. In particular, this yields another
proof of the classification of all (congruence classes of) umbilical
submanifolds of $\mathbb{S}^{n}\times\mathbb{R}$, and provides a similar
classification for the case of $\mathbb{H}^{n}\times\mathbb{R}$. We determine
all possible topological types, actually, diffeomorphism types, of a complete
umbilical submanifold of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$. We also
show that umbilical submanifolds of the product model of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ can be regarded as rotational
submanifolds in a suitable sense, and explicitly describe their profile curves
when $k=n$. As a consequence of our investigations, we prove that every
conformal diffeomorphism of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ onto
itself is an isometry.

M. I. Jimenez R. Tojeiro∗ 111Corresponding author 222The first author is
supported by CAPES-PNPD Grant 88887.469213/2019-00. The second author is
partially supported by Fapesp grant 2016/23746-6 and CNPq grant 303002/2017-4.
Data availability statement: Not applicable.

_2020 Mathematics Subject Classification:_ 53 B25, 53C40.

_Key words and phrases:_ umbilical submanifolds, conformally flat products of
space forms.

##  1\. Introduction

An isometric immersion $f\colon M\to N$ of a Riemannian manifold $M$ into
another Riemannian manifold $N$ is _umbilical_ if there exists a normal vector
field $H$ along $f$ such that its second fundamental form
$\alpha_{f}\in\Gamma(T^{*}M\otimes T^{*}M\otimes N_{f}M)$ with values in the
normal bundle is given by $\alpha_{f}(X,Y)=\langle X,Y\rangle H$ for all
$X,Y\in\mathfrak{X}(M)$. Geometrically, this roughly means that $f(M)$ is
equally curved in $N$ in all tangent directions. Umbilical submanifolds are
the simplest possible submanifolds of a Riemannian manifold after the totally
geodesic ones, for which $\alpha_{f}$ vanishes identically.

It is apparent that a generic Riemannian manifold does not carry any umbilical
submanifold of dimension $n\geq 2$. For instance, it was shown by Souam and
Toubiana [8] that of the eight three-dimensional model geometries of Thurston,
only the spaces of constant curvature, the product spaces
$\mathbb{S}^{2}\times\mathbb{R}$ and $\mathbb{H}^{2}\times\mathbb{R}$, and the
Sol geometry admit umbilical surfaces, in the latter case a unique non totally
geodesic one, up to ambient isometries. Here $\mathbb{S}^{n}$ and
$\mathbb{H}^{n}$ stand for the sphere and hyperbolic space of dimension $n$,
respectively. On the other hand, by a result of Leung and Nomizu [2] (see also
Section $1.7$ of [1]), if $N$ is a Riemannian manifold that satisfies the
axiom of $r$-spheres, that is, for any point $x\in N$ and for any integer
$r\geq 2$, there always exists an extrinsic sphere of $N$ tangent at $x$ to
any prescribed $r$-dimensional subspace $V\subset T_{x}N$, then $N$ must have
constant sectional curvature. Recall that an extrinsic sphere of a Riemannian
manifold $N$ is an umbilical submanifold of $N$ whose mean curvature vector
field is parallel in the normal connection. It is also pointed out in [2] that
if the requirement on the mean curvature vector field is dropped in the axiom
of $r$-spheres, then the corresponding weaker axiom for $n>4$ and $3<r<n$
implies that M is conformally flat.

Besides the spaces of constant curvature, there are few Riemannian manifolds
for which a classification of all its umbilical submanifolds is known. For a
general symmetric space $N$, it was shown by Nikolayevsky (see Theorem $1$ of
[6]) that any umbilical submanifold of $N$ is an umbilical submanifold of a
product of space forms totally geodesically embedded in $N$. This was one of
our motivations for studying umbilical submanifolds of a product of space
forms.

For submanifolds of dimension $m\geq 3$ of a product
$\mathbb{Q}_{k_{1}}^{n_{1}}\times\mathbb{Q}_{k_{2}}^{n_{2}}$ of space forms
whose curvatures satisfy $k_{1}+k_{2}\neq 0$, the problem was reduced in [4]
to the classification of $m$-dimensional umbilical submanifolds with
codimension two of $\mathbb{S}^{n}\times\mathbb{R}$ and
$\mathbb{H}^{n}\times\mathbb{R}$. The case of $\mathbb{S}^{n}\times\mathbb{R}$
was carried out in [5], extending previous results in [8] and [9] for
hypersurfaces. The case $m=2$ was studied in [7], where it was shown that, in
addition to the examples that appear already in higher dimensions, there are
precisely two distinct two–parameter families of complete embedded flat
umbilical surfaces that lie substantially in
$\mathbb{H}_{k}^{3}\times\mathbb{R}^{2}$ and
$\mathbb{H}_{k_{1}}^{3}\times\mathbb{H}_{k_{2}}^{3}$, respectively.

In this article we address the classification of the umbilical submanifolds of
the products $\mathbb{Q}_{k_{1}}^{n_{1}}\times\mathbb{Q}_{k_{2}}^{n_{2}}$ of
space forms in the cases that have not been covered in [4], [5] and [7],
namely, the cases in which either $k_{1}+k_{2}=0$ or $k_{1}<0$ and $n_{2}=1$.
Together with the case in which $k_{1}>0$ and $n_{2}=1$, already considered in
[5], these are precisely the products of Riemannian manifolds that are
conformally flat. In fact, there exists a global conformal diffeomorphism of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ onto the complement
$\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}$ of a $(k-1)$-dimensional subspace
of $\mathbb{R}^{n+1}$ (see Section $3$). Since umbilical submanifolds are
preserved under a conformal change of the metric of the ambient manifold, one
might consider this to imply already a classification of the umbilical
submanifolds of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$: any umbilical
isometric immersion $f\colon M^{m}\to\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$
is produced by composing the standard inclusion into
$\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}$ of an open subset of either
$\mathbb{S}^{m}$ or $\mathbb{R}^{m}$ with the inverse of such diffeomorphism.

However, for a complete classification of the umbilical submanifolds of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$, one should be able to decide when
two such umbilical isometric immersions $f,\tilde{f}\colon
M^{m}\to\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ are congruent, that is, when
there exists an isometry
$\Phi\colon\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}\to\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$
such that $\tilde{f}=\Phi\circ f$. In other words, a complete classification
of the umbilical submanifolds of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$
requires a description of all congruence classes of such submanifolds as well
as the knowledge of a representative of each of them. Other natural questions
one may pose are: what is the maximal substantial codimension of an umbilical
submanifold of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$, that is, the maximal
codimension of an umbilical submanifold of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ that is not contained in a totally
geodesic submanifold? What are its geometry and topology?

We now outline the contents of the paper and the results we have obtained
while investigating these problems. In Section $2$, we recall some basic facts
on the model of Euclidean space $\mathbb{R}^{n+1}$ as a hypersurface of the
light cone in the Lorentz space $\mathbb{L}^{n+3}$, a convenient setting to
deal with such problems. In Section $3$, we introduce the conformal model
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$, and describe its isometries (Theorem
1) and totally geodesic submanifolds (Proposition 4). As a consequence of
independent interest of Theorem 1, we see that every conformal diffeomorphism
of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ is an isometry (Corollary 2),
which extends a similar result by Souam and Toubiana [8] for the particular
cases of the product spaces $\mathbb{S}^{2}\times\mathbb{R}$ and
$\mathbb{H}^{2}\times\mathbb{R}$ as well as a well-known property of
hyperbolic space. The conformal model
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ also allows us to easily determine
all possible topological types, actually, diffeomorphism types, of a complete
umbilical submanifold of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ (Proposition
6).

In Section $4$ we address the congruence problem for umbilical submanifolds of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$. First we show that any such
submanifold with codimension $p$ arises as the intersection of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}\subset\mathbb{L}^{n+3}$ with a
codimension $p$ time-like subspace of $\mathbb{L}^{n+3}$ (Proposition 8). Then
we obtain an algebraic criterion to decide when two such submanifolds are
congruent and use it to prove that there exists precisely a $p$-parameter
family of congruence classes of umbilical submanifolds of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ with substantial codimension $p$
(Theorem 15). Along the way, we show that the substantial codimension of an
umbilical submanifold of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ is at most
$\mbox{min}\,\\{k+1,n-k+2\\}$ (Corollary 14). Then we study more carefully the
cases of codimensions one and two. We give necessary and sufficient conditions
for two umbilical submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ with codimensions one and two
to be congruent in terms of their Euclidean data (Propositions 4 and 5,
respectively) and exhibit, respectively, a one-parameter family and a two-
parameter family (together with three extra one-parameter families) that
contain precisely one representative of each congruence class of such
submanifolds (Corollaries 17 and 22, respectively). In particular, we recover
the main result of [4] for the particular case of
$\mathbb{S}^{n}\times\mathbb{R}$ (Corollary 20) and obtain, as a particular
case of Corollary 22, a similar classification of all (congruence classes of)
umbilical submanifolds of $\mathbb{H}^{n}\times\mathbb{R}$.

In Section $5$, we show that umbilical submanifolds of the product model of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ can be regarded as rotational
submanifolds in a suitable sense. In particular, umbilical submanifolds of
$\mathbb{H}^{n}\times\mathbb{S}^{1}$ (and hence of its universal cover
$\mathbb{H}^{n}\times\mathbb{R}$) are rotational submanifolds with curves as
profiles, and the latter are explicitly described by means of the conformal
model of $\mathbb{H}^{n}\times\mathbb{S}^{1}$, providing an alternative
description to that in [4] of the profile curves of umbilical submanifolds of
$\mathbb{S}^{n}\times\mathbb{R}$.

##  2\. Preliminaries

In this section we summarize some basic facts on the model of Euclidean space
$\mathbb{R}^{n+1}$ as a hypersurface of the light cone in the Lorentz space
$\mathbb{L}^{n+3}$ that make it a useful tool to study Moebius geometric
problems. We refer to Chapter $9$ of [1] for details.

Let $\mathbb{L}^{n+3}$ be the $(n+3)$-dimensional Lorentzian space with the
metric induced by the inner product

$\langle v,w\rangle=-v_{0}w_{0}+v_{1}w_{1}+\cdots+v_{n+2}w_{n+2}$

for $v=(v_{0},\dots,v_{n+2})$ and $w=(w_{0},\ldots,w_{n+2})$. The set of
light-like vectors

$\mathbb{V}^{n+2}=\\{v\in\mathbb{L}^{n+3}:\langle v,v\rangle=0,\;v\neq 0\\}$

is called the _light cone_ of $\mathbb{L}^{n+3}$. For any given
$w\in\mathbb{V}^{n+2}$, the intersection

$\mathbb{E}^{n+1}=\\{v\in\mathbb{V}^{n+2}:\langle v,w\rangle=1\\}$

of $\mathbb{V}^{n+2}$ with the affine hyperplane
$\\{u\in\mathbb{L}^{n+3}:\langle u,w\rangle=1\\}$ is a model of the
$(n+1)$-dimensional Euclidean space. Namely, for a fixed
$v\in\mathbb{E}^{n+1}$ and a linear isometry
$C\colon\mathbb{R}^{n+1}\to(\mbox{span}\\{v,w\\})^{\perp}\subset\mathbb{L}^{n+3},$
the map $\Psi\colon\mathbb{R}^{n+1}\to\mathbb{L}^{n+3}$, given by

(1) $\Psi(x)=v+Cx-\frac{1}{2}\|x\|^{2}w,$

is an isometric embedding such that $\Psi(\mathbb{R}^{n+1})=\mathbb{E}^{n+1}$.
From now on we assume that a triple $(v,w,C)$ has been fixed with $w_{0}<0$,
so that $\mathbb{E}^{n+1}\subset\mathbb{V}^{n+2}_{+}$, the upper half of
$\mathbb{V}^{n+2}$.

For any hypersphere $\mathcal{S}\subset\mathbb{R}^{n+1}$ with (constant) mean
curvature $h$ with respect to a unit normal vector field $N$, the map

$x\in\mathcal{S}\mapsto\Psi_{*}(x)N(x)+h\Psi(x)\in\mathbb{L}^{n+3}$

has a constant value $z$, a unit space-like vector satisfying
$\langle\Psi(x),z\rangle=0$ for all $x\in\mathcal{S}$. It follows that
$\Psi(\mathcal{S})=\mathbb{E}^{n+1}\cap\\{z\\}^{\perp}.$ Observe that
$\mathcal{S}$ is an affine hyperplane if and only if $0=h=\langle z,w\rangle.$

If $\mathcal{S}$ is a hypersphere with (Euclidean) center $x_{0}$ and radius
$r$, oriented by its inward pointing unit normal vector field
$x\in\mathcal{S}\mapsto N(x)=\frac{1}{r}(x_{0}-x)$ with corresponding mean
curvature $h=1/r$, then its associated unit space-like vector is
$z=\frac{1}{r}\Psi(x_{0})+\frac{r}{2}w.$ On the other hand, the unit space-
like vector associated to an affine hyperplane $\mathcal{S}$ oriented by a
unit normal vector $N$ is $z=CN-cw,$ where $c\in\mathbb{R}$ is the oriented
distance from $\mathcal{S}$ to the origin in $\mathbb{R}^{n+1}$.

If $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are hyperspheres or affine
hyperplanes with corresponding unit space-like vectors $z_{1}$ and $z_{2}$,
respectively. and if $N^{1}_{x}$ and $N^{2}_{x}$ are the unit normal vectors
of $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$, respectively, at
$x\in\mathcal{S}_{1}\cap\mathcal{S}_{2}$, then $\langle
N^{1}_{x},N^{2}_{x}\rangle=\langle z_{1},z_{2}\rangle.$ In particular,
$\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ intersect orthogonally if and only if
$\langle z_{1},z_{2}\rangle=0$.

More generally, any sphere or affine subspace $\mathcal{S}$ with codimension
$k$ of $\mathbb{R}^{n+1}$ is given by $\Psi(\mathcal{S})=\mathbb{E}^{n+1}\cap
V^{\perp}$ for some space-like subspace $V\subset\mathbb{L}^{n+3}$ of
dimension $k$, affine subspaces being characterized by the fact that $w\in
V^{\perp}$. We refer to $V$ in the sequel as the space-like subspace of
$\mathbb{L}^{n+3}$ correspondent to $\mathcal{S}$.

If $f\colon M^{m}\to\mathbb{V}^{n+2}$ is an immersion of a differentiable
manifold and $\mu\in C^{\infty}(M)$ is any positive smooth function, then the
map $h\colon M^{m}\to\mathbb{V}^{n+2}$ given by $h=\mu f$ is also an
immersion, and the induced metrics $\langle\,,\,\rangle_{f}$ and
$\langle\,,\,\rangle_{h}$ are related by
$\langle\,,\,\rangle_{h}=\mu^{2}\langle\,,\,\rangle_{f}.$ As a consequence,
any conformal immersion $f\colon M^{m}\to\mathbb{R}^{n+1}$ with conformal
factor $\varphi\in C^{\infty}(M)$ gives rise to an isometric immersion
${\mathcal{I}}(f)\colon M^{m}\to\mathbb{V}_{+}^{n+2}$ given by
${\mathcal{I}}(f)=\frac{1}{\varphi}\Psi\circ f.$ Conversely, any isometric
immersion $F\colon M^{m}\to\mathbb{V}_{+}^{n+2}\smallsetminus\mathbb{R}w$,
where $\mathbb{R}w=\\{tw:t<0\\}$, gives rise to a conformal immersion
${\mathcal{C}}(F)\colon M^{m}\to\mathbb{R}^{n+1}$ given by
$\Psi\circ{\mathcal{C}}(F)=\Pi\circ F,$ whose conformal factor is $1/\langle
F,w\rangle$. Here
$\Pi\colon\mathbb{V}_{+}^{n+2}\smallsetminus\mathbb{R}w\to\mathbb{E}^{n+1}$
denotes the projection onto $\mathbb{E}^{n+1}$ given by $\Pi(u)=u/\langle
u,w\rangle$.

In particular, if $T\in\mathbb{O}_{1}(n+3)$, then ${\mathcal{C}}(T\circ\Psi)$
is a conformal transformation of $\mathbb{R}^{n+1}$, and every conformal
transformation of $\mathbb{R}^{n+1}$ is given in this way. For instance, if
$R\in\mathbb{O}_{1}(n+3)$ is the reflection $R(u)=u-2\langle u,z\rangle z$
with respect to the hyperplane in $\mathbb{L}^{n+3}$ orthogonal to a unit
space-like vector $z$, with $\langle z,w\rangle\neq 0$, then
${\mathcal{C}}(R\circ\Psi)=I$ is the inversion with respect to the hypersphere
such that $\Psi(\mathcal{S})=\mathbb{E}^{n+1}\cap\\{z\\}^{\perp}$. If
$G\in\mathbb{O}_{1}(n+3)$ satisfies $G(w)=\lambda w$ for some
$\lambda\in(0,+\infty)$, then ${\mathcal{C}}(G\circ\Psi)=L$ is a similarity of
ratio $\lambda$. In particular, isometries of $\mathbb{R}^{n+1}$ correspond to
the elements of $\mathbb{O}_{1}(n+3)$ that fix $w$.

##  3\. The conformal model of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$

Let $\mathbb{R}^{k-1}$ be a subspace of dimension $k-1$ of $\mathbb{R}^{n+1}$,
which we assume to be

$\mathbb{R}^{k-1}=\\{(x_{1},\ldots,x_{n+1})\in\mathbb{R}^{n+1}\,:\,x_{k}=\cdots=x_{n+1}=0\\}.$

Given $x\in\mathbb{R}^{n+1}$, we write $x=x^{T}+x^{\perp}$ according to the
orthogonal decomposition
$\mathbb{R}^{n+1}=\mathbb{R}^{k-1}\oplus\mathbb{R}^{n-k+2}$. Now let
$v,w\in\mathbb{L}^{n+3}$ be vectors in
$\mathbb{V}^{n+2}\subset\mathbb{L}^{n+3}$ with $\langle v,w\rangle=1$, let
$C\colon\mathbb{R}^{n+1}\to(\mbox{span}\\{v,w\\})^{\perp}\subset\mathbb{L}^{n+3}$
be a linear isometry, and set $V_{1}=C(\mathbb{R}^{k-1})$ and
$V_{2}=C(\mathbb{R}^{n-k+2})$. Let $\mathbb{L}^{n+3}=W_{1}\oplus W_{2}$ be the
orthogonal decomposition with $W_{1}=V_{1}\oplus\mbox{span}\\{v,w\\}$ and
$W_{2}=V_{2}$. If
$\Psi\colon\mathbb{R}^{n+1}\to\mathbb{E}^{n+1}\subset\mathbb{L}^{n+3}$ is the
isometric embedding given by (1), then the map
$\Theta\colon\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}\to\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}\subset
W_{1}\oplus W_{2}=\mathbb{L}^{k+1}\times\mathbb{R}^{n-k+2}=\mathbb{L}^{n+3}$
defined by

(2) $\displaystyle\Theta(x)$ $\displaystyle=$
$\displaystyle\|x^{\perp}\|^{-1}\Psi(x)$ $\displaystyle=$
$\displaystyle\|x^{\perp}\|^{-1}\left(v-\frac{1}{2}\|x\|^{2}w+Cx^{T}+Cx^{\perp}\right)$

is a conformal diffeomorphism with induced metric $g=\varphi^{2}g_{0}$ whose
conformal factor with respect to the Euclidean metric $g_{0}$ is given by
$\varphi(x)=\|x^{\perp}\|^{-1}=(\sum_{i=k}^{n+1}x_{i}^{2})^{-1/2}$, that is,
$\varphi(x)$ is the inverse of the distance from $x$ to $\mathbb{R}^{k-1}$. We
refer to $(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ as the _conformal
model_ of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$. The above notations will
be used throughout the paper.

###  3.1. The isometries of $(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$

The next result describes the isometries of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$.

######  Theorem 1.

The following assertions on the map
$F\colon\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}\to\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}$
are equivalent for any $1\leq k\leq n+1$:

  * (i)

$F$ is an isometry with respect to $g$;

  * (ii)

$\Theta\circ F=T\circ\Theta$, where $T\in O_{1}(n+3)$ leaves $W_{1}$ and
$W_{2}$ invariant;

  * (iii)

$F=\mathcal{C}(T\circ\Psi)$, where $T\in O_{1}(n+3)$ leaves $W_{1}$ and
$W_{2}$ invariant;

  * (iv)

$F$ is a conformal diffeomorphism with respect to the Euclidean metric
$g_{0}$;

  * (v)

$F$ is a conformal diffeomorphism with respect to $g$;

  * (vi)

$F$ is given by $F=I\circ L$, where $L$ is a composition $L=G\circ H\circ A$
of an orthogonal transformation $A\in O(n+1)$ that leaves $\mathbb{R}^{k-1}$
invariant, a homothety with respect to the origin and a translation by a
vector $v\in\mathbb{R}^{k-1}$, and $I$ is an inversion with respect to a
hypersphere centered at a point of $\mathbb{R}^{k-1}$.

Proof: $(i)\Longrightarrow(ii)$: Let $F$ be an isometry of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$. Since
$\Theta\colon(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)\to\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$
is an isometry, then $\Theta\circ F=T\circ\Theta$ for some element $T$ of the
isometry group
$\mbox{Iso}\,(\mathbb{H}^{k}\times\mathbb{S}^{n-k+1})=\mbox{Iso}\,(\mathbb{H}^{k})\times\mbox{Iso}\,(\mathbb{S}^{n-k+1})$
of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}\subset W_{1}\oplus
W_{2}=\mathbb{L}^{n+3}$, which is an orthogonal transformation $T\in
O_{1}(n+3)$ that leaves $W_{1}$ and $W_{2}$ invariant.
$(ii)\Longrightarrow(iii)$: Since $\Theta(x)=\|x^{\perp}\|^{-1}\Psi(x)$, then
$\Pi\circ\Theta=\Psi$, and hence

$\Pi\circ T\circ\Psi=\Pi\circ T\circ\Theta=\Pi\circ\Theta\circ F=\Psi\circ F.$

That $(iii)$ implies $(iv)$ is clear, as is the equivalence between $(iv)$ and
$(v)$.
$(iv)\Longrightarrow(iii)$: The assumption that
$F(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1})\subset\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}$
means that $(F(x))^{\perp}\neq 0$ whenever $x^{\perp}\neq 0$. Since
$\Psi(x)=v-\frac{1}{2}\|x\|^{2}w+Cx^{T}+Cx^{\perp}$ and $\Psi\circ F=\Pi\circ
T\circ\Psi$, this implies that $\pi_{W_{2}}(Ty)\neq 0$ whenever
$\pi_{W_{2}}(y)\neq 0$, where $\pi_{W_{2}}\colon\mathbb{L}^{n+3}\to
W_{2}=V_{2}$ is the orthogonal projection. In other words,
$\ker(\pi_{W_{2}}\circ T)\subset\ker\pi_{W_{2}}=W_{1}$, or equivalently, $y\in
W_{1}$ whenever $Ty\in W_{1}$. Thus $W_{1}$ is invariant under $T^{-1}$, and
hence both $W_{2}$ and $W_{1}$ are invariant under $T$.
$(iii)\Longrightarrow(vi)$: Define $(\bar{v},\bar{w},\bar{C})$ by
$\bar{v}=T(v),\;\;\bar{w}=T(w)\;\;\mbox{and}\;\;\bar{C}=T\circ C.$ If
$\bar{w}=\lambda w$ for some $\lambda\in(0,+\infty)$, then
${\mathcal{C}}(T\circ\Psi)=L$ is a similarity of ratio $\lambda$. Moreover,
since $T$ leaves $W_{1}$ and $W_{2}$ invariant, from $\Psi\circ L=\Pi\circ
T\circ\Psi$ it follows that $L$ leaves $\mathbb{R}^{k-1}$ and
$\mathbb{R}^{n-k+2}$ invariant. Writing $L=G\circ H\circ A$, where $A\in
O(n+1)$ is an orthogonal transformation, $H$ is a homothety and $G$ is a
translation by a vector $y\in\mathbb{R}^{n+1}$, it follows that
$y\in\mathbb{R}^{k-1}$ and that $A$ leaves $\mathbb{R}^{k-1}$ invariant.

Otherwise, consider the reflection $R(u)=u-2\langle u,z\rangle z$ determined
by the unit space-like vector ${\displaystyle
z=\frac{1}{\langle\bar{w},w\rangle}\bar{w}+\frac{1}{2}w}$, and let
$G\in\mathbb{O}_{1}(n+3)$ be given by

$G(w)=R(\bar{w})=-\frac{1}{2}\langle\bar{w},w\rangle
w,\;\;G(v)=R(\bar{v})\;\;\mbox{and}\;\;G\circ C=R\circ\bar{C}.$

Then $R\circ G$ takes $w$ to $\bar{w}$, $v$ to $\bar{v}$ and $R\circ G\circ
C=\bar{C}$, hence $R\circ G=T$. The map ${\mathcal{C}}(R\circ\Psi)=I$ is an
inversion with respect to the hypersphere of unit radius such that
$\Psi(\mathcal{S})=\mathbb{E}^{n+1}\cap\\{z\\}^{\perp}$, whereas
${\mathcal{C}}(G\circ\Psi)=L$ is a similarity of ratio
$\lambda=-(1/2)\langle\bar{w},w\rangle$. Moreover, since $T$ leaves $W_{1}$
invariant, then $\bar{w}$, and hence $z$, belongs to $W_{1}$. Therefore, the
hypersphere $\mathcal{S}$ such that
$\Psi(\mathcal{S})=\mathbb{E}^{n+1}\cap\\{z\\}^{\perp}$ is centered at a point
of $\mathbb{R}^{k-1}$. Finally, from

$\Pi\circ T\circ\Psi=\Pi\circ R\circ G\circ\Psi=\Pi\circ R\circ\Psi\circ
L=\Psi\circ I\circ L,$

it follows that $F=I\circ L$.
$(vi)\Longrightarrow(i)$: Let $x_{0}\in\mathbb{R}^{k-1}$. The inversion with
respect to $\mathbb{S}^{n}(x_{0};r)$ is given by

$I(x)=x_{0}+r^{2}\frac{x-x_{0}}{\|x-x_{0}\|^{2}},$

which is a conformal map with conformal factor
${\displaystyle\rho(x)=\frac{r^{2}}{\|x-x_{0}\|^{2}}}$. In particular,

$(I(x))^{\perp}=r^{2}\frac{x^{\perp}}{\|x-x_{0}\|^{2}}.$

Therefore

$\displaystyle g(I(x))(I_{*}(x)v,I_{*}(x)w)$ $\displaystyle=$
$\displaystyle\frac{1}{\|(I(x))^{\perp}\|^{2}}g_{0}(I(x))(I_{*}(x)v,I_{*}(x)w)$
$\displaystyle=$
$\displaystyle\frac{\|x-x_{0}\|^{4}}{r^{4}\|x^{\perp}\|^{2}}g_{0}(I(x))(I_{*}(x)v,I_{*}(x)w)$
$\displaystyle=$
$\displaystyle\frac{\|x-x_{0}\|^{4}}{r^{4}\|x^{\perp}\|^{2}}\frac{r^{4}}{\|x-x_{0}\|^{4}}g_{0}(x)(v,w)$
$\displaystyle=$ $\displaystyle g(x)(v,w).$

The verification that an orthogonal transformation $A\in O(n+1)$ that leaves
$\mathbb{R}^{k-1}$ invariant, a homothety and a translation by a vector
$y\in\mathbb{R}^{k-1}$ are also isometries with respect to $g$ is
straightforward.

For instance, an inversion with respect to a hypersphere of $\mathbb{R}^{n+1}$
centered on $\mathbb{R}^{k-1}$ acts on each $k$-dimensional half-space
$\mathbb{R}^{k}_{+}$ having $\mathbb{R}^{k-1}$ as boundary as an hyperbolic
isometry of $\mathbb{R}^{k}_{+}$ regarded as the half-space model of
$\mathbb{H}^{k}$ with $\mathbb{R}^{k-1}$ as asymptotic boundary. Each such
$\mathbb{H}^{k}$ is a totally geodesic submanifold of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$, which corresponds under
$\Theta$ to a slice $\mathbb{H}^{k}\times\\{x\\}$ of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ (see the next subsection for a
description of all totally geodesic submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$).

Theorem 1 has the following consequence of independent interest.

######  Corollary 2.

Any conformal diffeomorphism of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ onto
itself, $1\leq k\leq n+1$, is an isometry.

######  Remark 3.

_The assertion of Corollary 2 in the particular case of the product spaces
$\mathbb{S}^{2}\times\mathbb{R}$ and $\mathbb{H}^{2}\times\mathbb{R}$ was
proved in [8], while for $k=n+1$ it reduces to a well-known property of the
hyperbolic space $\mathbb{H}^{n}$._

###  3.2. The totally geodesic submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$

In this subsection we describe the totally geodesic submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ and their correspondent space-
like subspaces of $\mathbb{L}^{n+3}$.

######  Proposition 4.

The totally geodesic submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ are

  * (i)

Affine subspaces $x_{0}+\mathbb{R}^{l-1}\oplus\mathbb{R}^{m-l+1}$ of
$\mathbb{R}^{n+1}=\mathbb{R}^{k-1}\oplus\mathbb{R}^{n-k+2}$, with $l\leq k$
and $x_{0}\in\mathbb{R}^{k-1}$, where
$\mathbb{R}^{l-1}\subset\mathbb{R}^{k-1}$ and
$\mathbb{R}^{m-l+1}\subset\mathbb{R}^{n-k+2}$ are linear subspaces.

  * (ii)

Hyperspheres of affine subspaces as above centered at points of
$\mathbb{R}^{l-1}$.

Proof: It is well-known that if $f\colon M^{n}\to\tilde{M}^{m}$ is an
immersion and ${g}_{0},{g}_{1}$ are conformal metrics on $\tilde{M}^{m}$, then
the second fundamental forms of
$f_{j}=f\colon(M^{n},f^{*}g_{j})\to(\tilde{M}^{m},{g}_{j})$, $0\leq j\leq 1$,
are related by

(3)
$\alpha^{f_{1}}(x)(X,Y)=\alpha^{f_{0}}(x)(X,Y)-\frac{1}{\varphi(f(x))}f^{*}g_{0}(X,Y)(\mbox{grad}_{0}\varphi(f(x)))_{N_{f}M(x)}$

for all $x\in M^{n}$ and $X,Y\in T_{x}M$, where $\varphi\in C^{\infty}({M})$
is the conformal factor of $g_{1}$ with respect to $g_{0}$ and
$\mbox{grad}_{0}\varphi$ denotes the gradient of $\varphi$ with respect to
$g_{0}$. Therefore, $f_{1}$ is totally geodesic if and only if either $f_{0}$
is totally geodesic and $(\mbox{grad}_{0}\varphi(f(x)))_{N_{f}M(x)}=0$ for all
$x\in M$, or $f_{0}$ is umbilical with mean curvature vector

$H(x)=\frac{\mbox{grad}_{0}\varphi(f(x))_{N_{f}M(x)}}{\varphi(f(x))}$

for all $x\in M$.

For $\tilde{M}^{m}=\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}$, $g_{0}$ the
standard Euclidean metric and $g_{1}:=g=\varphi^{2}g_{0}$, with
$\varphi(x)=\|x^{\perp}\|^{-1}=g_{0}(x^{\perp},x^{\perp})^{-1/2}$, we have
$\mbox{grad}_{0}\varphi(f(x))=\frac{f(x)^{\perp}}{\|f(x)^{\perp}\|^{2}}$ for
all $x\in\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}$. Hence, if $f\colon
M^{n}\to\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}$ is an immersion, then
$f_{1}=f\colon(M^{m},f^{*}g)\to(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},{g})$
is totally geodesic if and only if either
$f_{0}=f\colon(M^{m},f^{*}g_{0})\to(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},{g}_{0})$
is totally geodesic and $(f(x)^{\perp})_{N_{f}M(x)}=0$ for all $x\in M^{m}$,
or $f_{0}$ is umbilical and its mean curvature vector is given by
$H(x)=-\frac{(f(x)^{\perp})_{N_{f}M}}{\|f(x)^{\perp}\|^{2}}$ for all $x\in
M^{m}$.

In the former case, $f(M)$ is an open subset of an affine subspace
$x_{0}+\mathbb{R}^{l-1}\oplus\mathbb{R}^{m-l+1}$ of
$\mathbb{R}^{n+1}=\mathbb{R}^{k-1}\oplus\mathbb{R}^{n-k+2}$, with $l\leq k$
and $x_{0}\in\mathbb{R}^{k-1}$, where
$\mathbb{R}^{l-1}\subset\mathbb{R}^{k-1}$ and
$\mathbb{R}^{m-l+1}\subset\mathbb{R}^{n-k+2}$ are linear subspaces.

In the latter case, suppose first that $m=n$, in which case $f(M)$ is an open
subset of a hypersphere centered at $x_{0}\in\mathbb{R}^{n+1}$, whose mean
curvature vector is

$H(x)=\frac{1}{r}N(x),$

where $r=\|f(x)-x_{0}\|$ is its radius and $N(x)=\frac{1}{r}(x_{0}-f(x))$ is
its inward pointing unit normal vector field. Thus,
$H(x)=-\frac{(f(x)^{\perp})_{N_{f}M}}{\|f(x)^{\perp}\|^{2}}$ if and only if

$\frac{1}{r}=-\frac{\langle
f(x)^{\perp},N\rangle}{\|f(x)^{\perp}\|^{2}}=\frac{\langle
f(x)^{\perp},f(x)-x_{0}\rangle}{r\|f(x)^{\perp}\|^{2}}.$

which is equivalent to

$\langle f(x)^{\perp},x_{0}\rangle=0$

for all $x\in M^{n}$, that is $x_{0}^{\perp}=0$. It follows that $f(M)$ is an
open subset of a hypersphere centered at $x_{0}\in\mathbb{R}^{k-1}$. For
arbitrary $m$, $f(M)$ is the intersection of such a hypersphere with an affine
subspace $x_{0}+\mathbb{R}^{l-1}\oplus\mathbb{R}^{m-l+1}$ of
$\mathbb{R}^{n+1}=\mathbb{R}^{k-1}\oplus\mathbb{R}^{n-k+2}$, with $l\leq k$
and $x_{0}\in\mathbb{R}^{k-1}$, where
$\mathbb{R}^{l-1}\subset\mathbb{R}^{k-1}$ and
$\mathbb{R}^{m-l+1}\subset\mathbb{R}^{n-k+2}$ are linear subspaces, and hence
it is an open subset of a hypersphere of such an affine subspace centered at a
point of $\mathbb{R}^{l-1}$.

For instance, hyperspheres of $\mathbb{R}^{n+1}$ centered at points of
$\mathbb{R}^{k-1}$ and affine hyperplanes
$x_{0}+\mathbb{R}^{k-2}\oplus\mathbb{R}^{n-k+2}$ correspond to totally
geodesic hypersurfaces $\mathbb{H}^{k-1}\times\mathbb{S}^{n-k+1}$.

######  Corollary 5.

The totally geodesic submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ are the (open subsets of)
spheres and affine subspaces of $\mathbb{R}^{n+1}$ correspondent to the space-
like subspaces $\tilde{V}\subset\mathbb{L}^{n+3}$ of the form
$\tilde{V}=\tilde{V}_{1}\oplus\tilde{V}_{2}$ with $\tilde{V}_{1}\subset W_{1}$
and $\tilde{V}_{2}\subset W_{2}$.

Proof: Hyperplanes $\mathbb{R}^{k-1}\oplus\mathbb{R}^{n-k+1}$ and affine
hyperplanes $x_{0}+\mathbb{R}^{k-2}\oplus\mathbb{R}^{n-k+2}$ of
$\mathbb{R}^{n+1}=\mathbb{R}^{k-1}\oplus\mathbb{R}^{n-k+2}$ correspond,
respectively, to unit space-like vectors of the type $\eta=C(N)\in
C(\mathbb{R}^{n-k+2})=W_{2}$ or $\zeta=-cw+C(N)\in\mbox{span}\\{v,w\\}\oplus
C(\mathbb{R}^{k-1})=W_{1}$, whereas hyperspheres centered at points
$x_{0}\in\mathbb{R}^{k-1}$ correspond to unit space-like vectors of the type

$\frac{1}{r}\Psi(x_{0})+\frac{r}{2}w=\frac{1}{r}v+\frac{1}{2}\left(r-\frac{\|x_{0}\|^{2}}{r}\right)w+Cx_{0}\in\mbox{span}\\{v,w\\}\oplus
C(\mathbb{R}^{k-1})=W_{1}.$

Therefore, totally geodesic submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$, as orthogonal intersections
of hyperplanes, affine hyperplanes and hyperspheres as above, correspond to
space-like subspaces that are spanned by orthogonal space-like vectors that
belong to either $W_{1}$ or $W_{2}$, and hence are space-like subspaces of the
type $\tilde{V}=\tilde{V}_{1}\oplus\tilde{V}_{2}$ with $\tilde{V}_{1}\subset
W_{1}$ and $\tilde{V}_{2}\subset W_{2}$.

###  3.3. The topology of an umbilical submanifold of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$

The topology, actually the diffeomorphism type, of a complete umbilical
submanifold of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$, is easily determined
by means of the conformal model
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$.

######  Proposition 6.

Any umbilical isometric immersion $f\colon
M^{m}\to\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$, $1\leq k\leq n$, of a
complete Riemannian manifold is an embedding, and $M^{m}$ is diffeomorphic to
either $\mathbb{S}^{m}$ or $\mathbb{R}^{m}$ if $k=1$, or else to
$\mathbb{S}^{m-d}\times\mathbb{R}^{d}$, $k-1-n+m\leq d\leq\min\\{k-1,m\\}$, if
$k\geq 2$.

Proof: We may assume that $f$ is an isometric immersion into
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$. Since umbilicity of an
isometric immersion is invariant under conformal changes of the ambient space
by (3), then there exists a standard isometric embedding
$\bar{f}\colon\bar{M}^{m}\to(\mathbb{R}^{n+1},g_{0})$, where $\bar{M}^{m}$
stands for $\mathbb{S}^{m}$ or $\mathbb{R}^{m}$, and a diffeomorphism
$\nu\colon M^{m}\to\bar{M}^{m}\setminus\bar{f}^{-1}(\mathbb{R}^{k-1})$ such
that $f=\bar{f}\circ\nu$. If
$\bar{M}^{m}\setminus\bar{f}^{-1}(\mathbb{R}^{k-1})$ is not connected, we
consider $\nu\colon M^{m}\to
U\subset\bar{M}\setminus\bar{f}^{-1}(\mathbb{R}^{k-1})$, where $U$ is a
connected component. Therefore, $f$ is an embedding and $M^{m}$ is
diffeomorphic to $\bar{M}^{m}\setminus\bar{f}^{-1}(\mathbb{R}^{k-1})$, which,
for $\bar{M}^{m}=\mathbb{S}^{m}$, is either $\mathbb{S}^{m}$ itself, if
$\bar{f}(\mathbb{S}^{m})\cap\mathbb{R}^{k-1}=\emptyset$,
$\mathbb{S}^{m}\setminus\\{p\\}$, if $\bar{f}(\mathbb{S}^{m})$ is tangent to
$\mathbb{R}^{k-1}$ at $p\in\mathbb{R}^{k-1}$, or
$\mathbb{S}^{m}\setminus\mathbb{S}^{d-1}$, if $\bar{f}(\mathbb{S}^{n})$
intersects $\mathbb{R}^{k-1}$ along a $(d-1)$-dimensional sphere
$\mathbb{S}^{d-1}$, $k-1-n+m\leq d\leq\min\\{k-1,m\\}$ (here $\mathbb{S}^{0}$
consists of two points). Clearly, in the second possibility, $M^{m}$ is
diffeomorphic to $\mathbb{R}^{m}$. In the last one, notice that
$\mathbb{S}^{m}\setminus\mathbb{S}^{d-1}$ is diffeomorphic to
$\mathbb{R}^{m}\setminus\mathbb{R}^{d-1}$ by a stereographic projection with
respect to a point of $\mathbb{S}^{d-1}$, and the latter, in turn, is
diffeomorphic to $\mathbb{H}^{d}\times\mathbb{S}^{m-d}$ by the diffeomorphism
$\Theta$ given by (2), or equivalently, to
$\mathbb{S}^{m-d}\times\mathbb{R}^{d}$, as stated. The proof when
$\bar{M}^{m}=\mathbb{R}^{m}$ is similar.

######  Remark 7.

_It is also clear from the proof of Proposition 6 that if $f\colon
M^{m}\to\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ is an umbilical isometric
immersion, then $f(M)$ is an open subset of (the image of) an umbilical
isometric embedding
$f\colon\bar{M}^{m}\to\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ of a complete
Riemannian manifold $\bar{M}^{m}$._

##  4\. The congruence problem

In this section we address the congruence problem for umbilical submanifolds
of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$. We obtain in the next subsection
an algebraic criterion to decide when two such submanifolds are congruent and
then apply it in the subsequent subsections to umbilic submanifolds with
codimension one or two of the conformal model
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$.

###  4.1. The algebraic criterion

First we show how any umbilical submanifold of the product model
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ arises.

######  Proposition 8.

The umbilical submanifolds of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ with
codimension $p$ are the intersections of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}\subset\mathbb{L}^{n+3}$ with the
codimension $p$ time-like subspaces of $\mathbb{L}^{n+3}$.

Proof: Since umbilical submanifolds are preserved under a conformal change of
the metric of the ambient manifold, the umbilical submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ are the open subsets of the
intersections $\mathcal{S}\cap(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1})$
with $(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1})$ of the spheres or affine
subspaces $\mathcal{S}\subset\mathbb{R}^{n+1}$. Therefore, the umbilical
submanifolds of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}\subset\mathbb{L}^{n+3}$ with
codimension $p$ are the open subsets of the images
$\Theta(\mathcal{S}\cap(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}))$ under
$\Theta$ of such intersections. As discussed in Section $2$, for each sphere
or affine subspace $\mathcal{S}\subset\mathbb{R}^{n+1}$ there exists a space-
like subspace $V\subset\mathbb{L}^{n+3}$ with dimension $p$ such that
$\Psi(\mathcal{S})=\mathbb{E}^{n+1}\cap V^{\perp}$. Since
$\Theta(x)=\|x^{\perp}\|^{-1}\Psi(x)$ for all
$x\in\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}$ and
$\Theta((\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1})=\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$,
it follows that
$\Theta(\mathcal{S}\cap(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}))=(\mathbb{H}^{k}\times\mathbb{S}^{n-k+1})\cap
V^{\perp}$.

The next result reduces the problem of deciding when two substantial umbilical
submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)=\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$
with the same dimension are congruent to an easier one in terms of their
corresponding space-like subspaces.

######  Proposition 9.

Let $S$ and $\tilde{S}$ be umbilical submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ with the same dimension. Let
$V$ and $\tilde{V}$ be their correspondent space-like subspaces of
$\mathbb{L}^{n+3}=W_{1}\oplus W_{2}$. Then $S$ and $\tilde{S}$ are congruent
in $(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ if and only if there exists
$T\in O_{1}(n+3)$ leaving $W_{1}$ and $W_{2}$ invariant such that
$T(V)=\tilde{V}$.

Proof: Let $S$ and $\tilde{S}$ be umbilical submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ with the same dimension, and
let $V$ and $\tilde{V}$ be their correspondent space-like subspaces of
$\mathbb{L}^{n+3}$. By Proposition 8,
$\Theta(S)=(\mathbb{H}^{k}\times\mathbb{S}^{n-k+1})\cap V^{\perp}$ and
$\Theta(\tilde{S})=(\mathbb{H}^{k}\times\mathbb{S}^{n-k+1})\cap{\tilde{V}}^{\perp}$.
Since any isometry
$F\colon(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)\to(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$
is given by $\Theta\circ F=T\circ\Theta$ for some $T\in O_{1}(n+3)$ that
leaves $W_{1}$ and $W_{2}$ invariant by Theorem 1, it follows that

$\begin{array}[]{l}F(S)=\tilde{S}\Longleftrightarrow\Theta(F(S))=\Theta(\tilde{S})\Longleftrightarrow
T(\Theta(S))=\Theta(\tilde{S})\vspace{1ex}\\\ \Longleftrightarrow
T((\mathbb{H}^{k}\times\mathbb{S}^{n-k+1})\cap
V^{\perp})=(\mathbb{H}^{k}\times\mathbb{S}^{n-k+1})\cap{\tilde{V}}^{\perp}\Longleftrightarrow
T(V^{\perp})={\tilde{V}}^{\perp}.\end{array}\hskip 5.0pt\hbox{\hskip
5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}$

######  Proposition 10.

Let $W$ be a vector space endowed with a Lorentzian inner product and let
$W=W_{1}\oplus W_{2}$ be an orthogonal direct sum decomposition, with $W_{1}$
time-like. For $w\in W$, write $w=w^{T}+w^{\perp}$ according to the
decomposition $W=W_{1}\oplus W_{2}$. Let $V$ and $\tilde{V}$ be space-like
subspaces of $W$ with the same dimension. Assume that there exists a linear
isometry $T\colon W\to W$ preserving $W_{1}$ and $W_{2}$ such that
$T(V)=\tilde{V}$. Then, for any orthonormal bases
$\\{\xi_{1},\ldots,\xi_{m}\\}$ and
$\\{\tilde{\xi}_{1},\ldots,\tilde{\xi}_{m}\\}$ of $V$ and $\tilde{V}$,
respectively, the Gram matrices
$(\langle\xi^{T}_{i},\xi^{T}_{j}\rangle)_{1\leq i,j\leq m}$ and
$(\langle\tilde{\xi}^{T}_{i},\tilde{\xi}^{T}_{j}\rangle)_{1\leq i,j\leq m}$
(or equivalently, the Gram matrices
$(\langle\xi^{\perp}_{i},\xi^{\perp}_{j}\rangle)_{1\leq i,j\leq m}$ and
$(\langle\tilde{\xi}^{\perp}_{i},\tilde{\xi}^{\perp}_{j}\rangle)_{1\leq
i,j\leq m}$), have the same eigenvalues, counted with multiplicities.

Conversely, assume that there exist orthonormal bases
$\\{\xi_{1},\ldots,\xi_{m}\\}$ and
$\\{\tilde{\xi}_{1},\ldots,\tilde{\xi}_{m}\\}$ of $V$ and $\tilde{V}$,
respectively, such that

  * (i)

the subspaces spanned by $\\{\xi^{T}_{1},\ldots,\xi^{T}_{m}\\}$ and
$\\{\tilde{\xi}^{T}_{1},\ldots,\tilde{\xi}^{T}_{m}\\}$ either are both
degenerate and have dimension $m$, or are both nondegenerate;

  * (ii)

the Gram matrices $(\langle\xi^{T}_{i},\xi^{T}_{j}\rangle)_{1\leq i,j\leq m}$
and $(\langle\tilde{\xi}^{T}_{i},\tilde{\xi}^{T}_{j}\rangle)_{1\leq i,j\leq
m}$ (or equivalently, the Gram matrices
$(\langle\xi^{\perp}_{i},\xi^{\perp}_{j}\rangle)_{1\leq i,j\leq m}$ and
$(\langle\tilde{\xi}^{\perp}_{i},\tilde{\xi}^{\perp}_{j}\rangle)_{1\leq
i,j\leq m}$) have the same eingenvalues, counted with multiplicities.

Then there is a linear isometry $T\colon W\to W$ preserving $W_{1}$ and
$W_{2}$ such that $T(V)=\tilde{V}$.

For the proof of Proposition 10 we need the following lemma.

######  Lemma 11.

Let $W$ be a vector space endowed with a Lorentzian inner product
$\langle\;,\;\rangle^{\sim}$, and let
$\\{\xi_{1},\ldots,\xi_{m}\\},\\{\tilde{\xi}_{1},\ldots,\tilde{\xi}_{m}\\}\subset
W$. Assume that the subspaces spanned by $\\{\xi_{1},\ldots,\xi_{m}\\}$ and
$\\{\tilde{\xi}_{1},\ldots,\tilde{\xi}_{m}\\}$ either are both degenerate and
have dimension $m$ or are both nondegenerate. Suppose further that
$\langle\xi_{i},\xi_{j}\rangle^{\sim}=\langle\tilde{\xi}_{i},\tilde{\xi}_{j}\rangle^{\sim}$
for all $1\leq i,j\leq m$. Then there exists a linear isometry $T\colon W\to
W$ such that $T\xi_{i}=\tilde{\xi}_{i}$, $1\leq i\leq m$.

Proof: Define $A\colon\mathbb{R}^{m}\to W$ and $B\colon\mathbb{R}^{m}\to W$ by
$Ae_{i}=\xi_{i}$ and $Be_{i}=\tilde{\xi}_{i}$, $1\leq i\leq m$, where
$\\{e_{1},\ldots,e_{m}\\}$ is the canonical basis of $\mathbb{R}^{m}$. Then
$A(\mathbb{R}^{m})=\mbox{span}\\{\xi_{1},\ldots,\xi_{m}\\}$ and
$B(\mathbb{R}^{m})=\mbox{span}\\{\tilde{\xi}_{1},\ldots,\tilde{\xi}_{m}\\}$.
Let $A^{t}\colon W\to\mathbb{R}^{m}$ and $B^{t}\colon W\to\mathbb{R}^{m}$ be
the transposes of $A$ and $B$, that is, $\langle A^{t}w,v\rangle=\langle
w,Av\rangle^{\sim}$ for all $v\in\mathbb{R}^{m}$ and $w\in W$, where
$\langle\;,\;\rangle$ is the standard inner product of $\mathbb{R}^{m}$, and
$B^{t}$ is similarly defined. Then

$\langle A^{t}Ae_{i},e_{j}\rangle=\langle
Ae_{i},Ae_{j}\rangle^{\sim}=\langle\xi_{i},\xi_{j}\rangle^{\sim}=\langle\tilde{\xi}_{i},\tilde{\xi}_{j}\rangle^{\sim}=\langle
Be_{i},Be_{j}\rangle^{\sim}=\langle B^{t}Be_{i},e_{j}\rangle$

for all $1\leq i,j\leq m$. Thus $A^{t}A=B^{t}B$. Since
$A^{t}A\colon\mathbb{R}^{m}\to\mathbb{R}^{m}$ is a symmetric operator, there
exists an orthonormal basis $v_{1},\ldots,v_{m}$ of $\mathbb{R}^{m}$ and
$\lambda_{1},\ldots,\lambda_{m}\in\mathbb{R}$ such that
$A^{t}Av_{i}=\lambda_{i}v_{i}$, $1\leq i\leq m$, that is,

$\langle Av_{i},Av_{j}\rangle=\lambda_{i}\delta_{ij}=\langle
Bv_{i},Bv_{j}\rangle,\;\;\;1\leq i,j\leq m.$

If both $A(\mathbb{R}^{m})$ and $B(\mathbb{R}^{m})$ are nondegenerate, this
implies that there exists a linear isometry $T\colon W\to W$ such that
$TAv_{i}=Bv_{i}$, $1\leq i\leq m$, and hence $TA=B$. On the other hand, if
both $A(\mathbb{R}^{m})$ and $B(\mathbb{R}^{m})$ are degenerate and have
dimension $m$, there exists exactly one $i\in\\{1,\ldots,m\\}$ such that
$\lambda_{i}=0$, in which case both $Av_{i}$ and $Bv_{i}$ are nonzero light-
like vectors. Also in this case, there exists a linear isometry $T\colon W\to
W$ such that $TAv_{i}=Bv_{i}$, $1\leq i\leq m$, and hence $TA=B$. Thus
$T\xi_{i}=TAe_{i}=Be_{i}=\tilde{\xi}_{i}$, $1\leq i\leq m$.

_Proof of Proposition 10:_ Assume first that $T\colon W\to W$ is a linear
isometry preserving $W_{1}$ and $W_{2}$ such that $T(V)=\tilde{V}$, and let
$\\{\xi_{1},\ldots,\xi_{m}\\}$ and
$\\{\tilde{\xi}_{1},\ldots,\tilde{\xi}_{m}\\}$ be orthonormal bases of $V$ and
$\tilde{V}$, respectively. Then
$T\xi_{j}=\sum_{i=1}^{m}a_{ij}\tilde{\xi}_{i}$, $1\leq j\leq m$, for some
$A=(a_{ij})\in O(m)$. Since $T$ preserves $W_{1}$ and $W_{2}$, then
$T\xi^{T}_{j}=\sum_{i=1}^{m}a_{ij}\tilde{\xi}^{T}_{i}$, and hence the Gram
matrices $G=(\langle\xi^{T}_{i},\xi^{T}_{j}\rangle)_{1\leq i,j\leq m}$ and
$\tilde{G}=(\langle\tilde{\xi}^{T}_{i},\tilde{\xi}^{T}_{j}\rangle)_{1\leq
i,j\leq m}$ are related by $G=A\tilde{G}A^{t}$. It follows that $G$ and
$\tilde{G}$ have the same eingenvalues, counted with multiplicities.

Conversely, assume that there exist orthonormal bases
$\\{\xi_{1},\ldots,\xi_{m}\\}$ and
$\\{\tilde{\xi}_{1},\ldots,\tilde{\xi}_{m}\\}$ of $V$ and $\tilde{V}$,
respectively, satisfying conditions $(i)$ and $(ii)$ in the statement.
Condition $(ii)$ implies that there exists $A=(a_{ij})\in O(m)$ such that
$G=A\tilde{G}A^{t}$. Defining
$\bar{\xi}_{j}=\sum_{i=1}^{m}a_{ij}\tilde{\xi}^{T}_{i}$, the Gram matrices
$\bar{G}=(\langle\bar{\xi}_{i},\bar{\xi}_{j}\rangle)_{1\leq i,j\leq m}$ and
$\tilde{G}$ are also related by $\bar{G}=A\tilde{G}A^{t}$, and hence
$G=\bar{G}$. By Lemma 11 and condition $(i)$, there exists a linear isometry
$T_{1}\colon W_{1}\to W_{1}$ such that $T_{1}\xi^{T}_{j}=\bar{\xi}_{j}$,
$1\leq j\leq m$.

Now define $\hat{\xi}_{j}=\sum_{i=1}^{m}a_{ij}\tilde{\xi}^{\perp}_{i}$. We
have

$\displaystyle\langle\hat{\xi}_{j},\hat{\xi}_{k}\rangle$ $\displaystyle=$
$\displaystyle\langle\sum_{i=1}^{m}a_{ij}\tilde{\xi}^{\perp}_{i},\sum_{\ell=1}^{m}a_{\ell
k}\tilde{\xi}^{\perp}_{\ell}\rangle\vspace{1ex}$ $\displaystyle=$
$\displaystyle\sum_{i,\ell=1}^{m}a_{ij}a_{\ell
k}\langle\tilde{\xi}^{\perp}_{i},\tilde{\xi}^{\perp}_{\ell}\rangle\vspace{1ex}$
$\displaystyle=$ $\displaystyle\sum_{i,\ell=1}^{m}a_{ij}a_{\ell
k}(\delta_{i\ell}-\langle\tilde{\xi}^{T}_{i},\tilde{\xi}^{T}_{\ell}\rangle)\vspace{1ex}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{m}a_{ij}a_{ik}-\sum_{i,\ell=1}^{m}a_{ij}a_{\ell
k}\langle\tilde{\xi}^{T}_{i},\tilde{\xi}^{T}_{\ell}\rangle\vspace{1ex}$
$\displaystyle=$
$\displaystyle\delta_{jk}-\langle{\xi}^{T}_{j},{\xi}^{T}_{k}\rangle\vspace{1ex}$
$\displaystyle=$
$\displaystyle\langle{\xi}^{\perp}_{j},{\xi}^{\perp}_{k}\rangle.$

Since $W_{2}$ is space-like, there exists a linear isometry $T_{2}\colon
W_{2}\to W_{2}$ such that $T_{2}\xi^{\perp}_{j}=\hat{\xi}_{j}$, $1\leq j\leq
m$. Defining $T\colon W\to W$ by $T|_{W_{1}}=T_{1}$ and $T|_{W_{2}}=T_{2}$, it
follows that $T$ is a linear isometry preserving $W_{1}$ and $W_{2}$ such that

$T\xi_{j}=T_{1}\xi^{T}_{j}+T_{2}\xi^{\perp}_{j}=\sum_{i=1}^{m}a_{ij}\tilde{\xi}^{T}_{i}+\sum_{i=1}^{m}a_{ij}\tilde{\xi}^{\perp}_{i}=\sum_{i=1}^{m}a_{ij}\tilde{\xi}_{i},$

and hence $T(V)=\tilde{V}$.

######  Remark 12.

_In Lemma 11, if the subsets $\\{\xi_{1},\ldots,\xi_{m}\\}$ and
$\\{\tilde{\xi}_{1},\ldots,\tilde{\xi}_{m}\\}$ are assumed to be linearly
independent, then the hypothesis that
$\langle\xi_{i},\xi_{j}\rangle^{\sim}=\langle\tilde{\xi}_{i},\tilde{\xi}_{j}\rangle^{\sim}$
for all $1\leq i,j\leq m$ already implies that the subspaces spanned by
$\\{\xi_{1},\ldots,\xi_{m}\\}$ and
$\\{\tilde{\xi}_{1},\ldots,\tilde{\xi}_{m}\\}$ either are both degenerate or
both nondegenerate, for the degeneracy of the subspace spanned by a linearly
independent subset $\\{\xi_{1},\ldots,\xi_{m}\\}$ is characterized by the
vanishing of the determinant of the Gram matrix
$(\langle\xi_{i},\xi_{j}\rangle)_{1\leq i,j\leq m}$. This follows from the
proof of Lemma 11, which shows that if $A(\mathbb{R}^{m})$ has dimension $m$
then it is degenerate if and only if one of the eigenvalues
$\lambda_{1},\ldots,\lambda_{m}$ of the operator $A^{t}A$ is zero.
Accordingly, if in the converse statement of Proposition 10 the subsets
$\\{\xi^{T}_{1},\ldots,\xi^{T}_{m}\\}$ and
$\\{\tilde{\xi}^{T}_{1},\ldots,\tilde{\xi}^{T}_{m}\\}$ are assumed to be
linearly independent, then condition $(i)$ is a consequence of condition
$(ii)$._

The next result gives an algebraic criterion to decide whether an umbilical
submanifold of $(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ is substantial.

######  Proposition 13.

An umbilical submanifold $S$ of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ with corresponding space-like
subspace $V\subset\mathbb{L}^{n+3}$ is substantial if and only if $V\cap
W_{1}$ and $V\cap W_{2}$ are trivial.

Proof: Since $\Psi(S)=\mathbb{E}^{n+1}\cap V^{\perp}$, then $S$ is contained
in a totally geodesic submanifold $\tilde{S}$, with
$\Psi(\tilde{S})=\mathbb{E}^{n+1}\cap{\tilde{V}}^{\perp}$ for a space-like
subspace $\tilde{V}\subset\mathbb{L}^{n+3}$, if $V\supset\tilde{V}$. By
Corollary 5, $\tilde{V}=\tilde{V}_{1}\oplus\tilde{V}_{2}$ with
$\tilde{V}_{1}\subset W_{1}$ and $\tilde{V}_{2}\subset W_{2}$. Therefore, $S$
is contained in some totally geodesic submanifold $\tilde{S}$ if and only if
either $V\cap W_{1}$ or $V\cap W_{2}$ is non trivial.

######  Corollary 14.

The substantial codimension of an umbilical submanifold of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ is at most
$\mbox{min}\,\\{k+1,n-k+2\\}$.

We are now able to determine how many congruence classes of umbilical
submanifolds of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ with a given
substantial codimension are there.

######  Theorem 15.

There exists precisely a $p$-parameter family of congruence classes of
umbilical submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)=\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$
with substantial codimension $p$.

Proof: By Proposition 9, two umbilical submanifolds $S$ and $\tilde{S}$ of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ are congruent in
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ if and only if there exists
$T\in O_{1}(n+3)$ leaving $W_{1}$ and $W_{2}$ invariant such that
$T(V)=\tilde{V}$, where $V$ and $\tilde{V}$ are the space-like subspaces
correspondent to $S$ and $\tilde{S}$, respectively.

Let $\\{\xi_{1},\ldots,\xi_{p}\\}$ and
$\\{\tilde{\xi}_{1},\ldots,\tilde{\xi}_{p}\\}$ be orthonormal bases of $V$ and
$\tilde{V}$, respectively. Since $V$ and $\tilde{V}$ are substantial, by
Proposition 13 the subspaces spanned by $\\{\xi^{T}_{1},\ldots,\xi^{T}_{m}\\}$
and $\\{\tilde{\xi}^{T}_{1},\ldots,\tilde{\xi}^{T}_{m}\\}$ have dimension $p$.
It follows from Proposition 10 that there exists $T\in O_{1}(n+3)$ leaving
$W_{1}$ and $W_{2}$ invariant such that $T(V)=\tilde{V}$ if and only if the
Gram matrices $(\langle\xi^{T}_{i},\xi^{T}_{j}\rangle)_{1\leq i,j\leq m}$ and
$(\langle\tilde{\xi}^{T}_{i},\tilde{\xi}^{T}_{j}\rangle)_{1\leq i,j\leq m}$
have the same eingenvalues, counted with multiplicities. Therefore, the
congruence classes of umbilical submanifolds of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ with substantial codimension
$p$ are in one-to-one correspondence with the sets of eigenvalues of such Gram
matrices.

###  4.2. The hypersurface case

.

The next result gives necessary and sufficient conditions for two umbilical
hypersurfaces of $(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ to be
congruent in terms of their Euclidean data.

######  Proposition 16.

The following assertions hold:

  * (i)

The hyperspheres $\mathbb{S}^{n}(x_{0};r)$ and
$\mathbb{S}^{n}(\tilde{x}_{0};\tilde{r})$ are congruent in
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ if and only if
${\displaystyle\frac{\|{x}_{0}^{\perp}\|}{r}=\frac{\|\tilde{x}_{0}^{\perp}\|}{\tilde{r}}}$.

  * (ii)

The hypersphere $\mathbb{S}^{n}(x_{0};r)$ and the affine hyperplane
$\mathcal{H}(N;c)$ are congruent in
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ if and only if
${\displaystyle\frac{\|{x}_{0}^{\perp}\|}{r}=\|N^{\perp}\|}$ and $c\neq 0$ if
$N^{T}=0$.

  * (iii)

Two affine hyperplanes $\mathcal{H}(N;c)$ and
$\mathcal{H}(\tilde{N};\tilde{c})$ are congruent in
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ if and only if
$\|N^{\perp}\|=\|\tilde{N}^{\perp}\|$ and either $c\neq 0\neq\tilde{c}$ or
$c=0=\tilde{c}$ if $N^{T}$ (and hence $\tilde{N}^{T}$) vanishes.

Proof: $(i)$ Let ${\displaystyle\zeta=\frac{1}{r}\Psi(x_{0})+\frac{r}{2}w}$
and
${\displaystyle\tilde{\zeta}=\frac{1}{\tilde{r}}\Psi(\tilde{x}_{0})+\frac{\tilde{r}}{2}w}$
be the unit space-like vectors associated with $\mathbb{S}^{n}(x_{0};r)$ and
$\mathbb{S}^{n}(\tilde{x}_{0};\tilde{r})$, respectively. Write
$x_{0}=x_{0}^{T}+x_{0}^{\perp}$, with $x_{0}^{T}\in\mathbb{R}^{k-1}$ and
$x_{0}^{\perp}\in\mathbb{R}^{n-k+2}=(\mathbb{R}^{k-1})^{\perp}$, and decompose
$\tilde{x}_{0}$ accordingly. Then $\zeta=\zeta^{T}+\zeta^{\perp}$, with

$\zeta^{T}=\frac{1}{r}v+\frac{1}{2}\left(r-\frac{\|x_{0}\|^{2}}{r}\right)w+\frac{1}{r}Cx_{0}^{T}\,\,\,\mbox{and}\,\,\,\zeta^{\perp}=\frac{1}{r}Cx_{0}^{\perp},$

is the decomposition of $\zeta$ with respect to the orthogonal decomposition
$\mathbb{L}^{n+3}=W_{1}\oplus W_{2}$, with
$W_{1}=V_{1}\oplus\mbox{span}\\{v,w\\}$ and $W_{2}=V_{2}$. A similar
decomposition holds for $\tilde{\zeta}$.

By Proposition 9, $\mathbb{S}^{n}(x_{0};r)$ and
$\mathbb{S}^{n}(\tilde{x}_{0};\tilde{r})$ are congruent in
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ if and only if there exists
$T\in O_{1}(n+3)$ leaving $W_{1}$ and $W_{2}$ invariant such that
$T(\zeta)=\tilde{\zeta}$. It follows from Proposition 10 that this is the case
if and only if either $\zeta^{T}$ and $\tilde{\zeta}^{T}$ are nonzero light-
like vectors or if neither of them is light-like and
$\langle\zeta^{T},\zeta^{T}\rangle=\langle\tilde{\zeta}^{T},\tilde{\zeta}^{T}\rangle$
(or equivalently,
$\langle\zeta^{\perp},\zeta^{\perp}\rangle=\langle\tilde{\zeta}^{\perp},\tilde{\zeta}^{\perp}\rangle$),
that is, if and only if
${\displaystyle\frac{\|{x}_{0}^{\perp}\|}{r}=\frac{\|\tilde{x}_{0}^{\perp}\|}{\tilde{r}}}$.

$(ii)$ Let $\eta=CN-cw$ be the unit space-like vector representing
$\mathcal{H}(N;c)$. By Proposition 9, $\mathbb{S}^{n}(x_{0};r)$ and
$\mathcal{H}(N;c)$ are congruent in
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ if and only if there exists
$T\in O_{1}(n+3)$ leaving $W_{1}$ and $W_{2}$ invariant such that
$T(\zeta)=\eta$. It follows from Proposition 10 that this is the case if and
only if either $\zeta^{T}$ and $\eta^{T}$ are nonzero light-like vectors or
neither of them is light-like, and
$\langle\zeta^{\perp},\zeta^{\perp}\rangle=\langle\eta^{\perp},\eta^{\perp}\rangle$.
This is the case if and only if
${\displaystyle\frac{\|{x}_{0}^{\perp}\|}{r}=\|N^{\perp}\|}$ and $c\neq 0$ if
$N^{T}=0$.

$(iii)$ Follows in a similar way as in $(i)$ and $(ii)$.

######  Corollary 17.

There exists precisely a one-parameter family of umbilical non totally
geodesic hypersurfaces of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)=\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$.
More precisely, for a fixed $x_{0}\in\mathbb{R}^{n+1}$ with $x_{0}^{\perp}\neq
0$, the set $\\{\mathbb{S}^{n}(x_{0};r),r>0\\}$ contains exactly one
representative of each congruence class of such hypersurfaces.

Proof: By part $(i)$ of Proposition 16, $\mathbb{S}^{n}(x_{0};r)$ is not
congruent to $\mathbb{S}^{n}({x}_{0};\tilde{r})$ if $r\neq\tilde{r}$. On the
other hand, any $\mathbb{S}^{n}(\tilde{x}_{0};\tilde{r})$ with
$\tilde{x}_{0}^{\perp}\neq 0$ is congruent to $\mathbb{S}^{n}(x_{0};r)$ for
${\displaystyle
r=\frac{\tilde{r}\|x_{0}^{\perp}\|}{\|\tilde{x}_{0}^{\perp}\|}}$, and any
$\mathcal{H}(N;c)$ with $N^{\perp}\neq 0$ and with $c\neq 0$ if $N^{T}=0$ is
congruent to $\mathbb{S}^{n}(x_{0};r)$ for ${\displaystyle
r=\frac{\|x_{0}^{\perp}\|}{\|N^{\perp}\|}}$.

###  4.3. The codimension two case

In this section we solve the congruence problem for umbilical submanifolds
with codimension two of $(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$. In
particular, this implies a classification of (the congruence classes of) all
umbilical submanifolds of $\mathbb{S}^{n}\times\mathbb{R}$ and
$\mathbb{H}^{n}\times\mathbb{R}$.

Notice that any sphere of $\mathbb{R}^{n+1}$ with codimension two can be
produced (in a unique way) as the orthogonal intersection of a hypersphere
$\mathbb{S}^{n}(x_{0};r)$ and an affine hyperplane $\mathcal{H}(N;c)$. Also,
any affine subspace of $\mathbb{R}^{n+1}$ with codimension two whose distance
to the origin is $c>0$ can be produced (in a unique way) as the orthogonal
intersection of an affine hyperplane $\mathcal{H}(N_{1};c)$ and a hyperplane
$\mathcal{H}(N_{2};0)$. For a linear subspace with codimension $2$, however,
the representation $\mathcal{H}(N_{1},0)\cap\mathcal{H}(N_{2},0)$ is not
unique. Notice also that, if in the latter representation the vectors
$N_{1}^{T}$ and $N_{2}^{T}$ are linearly dependent, then one can replace it by
another such representation for which $N_{2}^{T}=0$.

######  Proposition 18.

$(a)$ The spheres $\mathbb{S}^{n}(x_{0};r)\cap\mathcal{H}(N;c)$ and
$\mathbb{S}^{n}(\tilde{x}_{0};\tilde{r})\cap\mathcal{H}(\tilde{N};\tilde{c})$
are congruent in $(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ if and only
if one of the following holds:

  * (i)

$(N^{T},c)=(0,0)=(\tilde{N}^{T},\tilde{c})$ and
$r^{-1}\|x_{0}^{\perp}\|=\tilde{r}^{-1}\|\tilde{x}_{0}^{\perp}\|$; or

  * (ii)

$(N^{T},c)\neq(0,0)\neq(\tilde{N}^{T},\tilde{c})$ and

(4)
$\left\\{\begin{array}[]{l}{\displaystyle\frac{\|x_{0}^{\perp}\|^{2}}{r^{2}}+\|N^{\perp}\|^{2}=\frac{\|\tilde{x}_{0}^{\perp}\|^{2}}{\tilde{r}^{2}}+\|\tilde{N}^{\perp}\|^{2}\vspace{1ex}}\\\
{\displaystyle\frac{1}{r^{2}}(\|x_{0}^{\perp}\|^{2}\|N^{\perp}\|^{2}-\langle
x_{0}^{\perp},N^{\perp}\rangle^{2})=\frac{1}{\tilde{r}^{2}}(\|\tilde{x}_{0}^{\perp}\|^{2}\|\tilde{N}^{\perp}\|^{2}-\langle\tilde{x}_{0}^{\perp},\tilde{N}^{\perp}\rangle^{2}}).\end{array}\right.$

$(b)$ The sphere $\mathbb{S}^{n}(x_{0};r)\cap\mathcal{H}(N;c)$ and the affine
subspace $\mathcal{H}(N_{1};c_{1})\cap\mathcal{H}(N_{2};0)$ are congruent in
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ if and only if one of the
following holds:

  * (i)

$(N^{T},c)=(0,0)$, $N_{2}^{T}=0$, $r^{-1}\|x_{0}^{\perp}\|=\|N_{1}^{\perp}\|$
and $c_{1}\neq 0$ if $N_{1}^{T}=0$; or

  * (ii)

$(N^{T},c)\neq(0,0)$, $N_{2}^{T}\neq 0$, with $\\{N_{1}^{T},N_{2}^{T}\\}$
linearly independent if $c_{1}=0$, and

(5)
$\left\\{\begin{array}[]{l}{\displaystyle\frac{\|x_{0}^{\perp}\|^{2}}{r^{2}}+\|N^{\perp}\|^{2}=\|N^{\perp}_{1}\|^{2}+\|N^{\perp}_{2}\|^{2}\vspace{1ex}}\\\
{\displaystyle\|x_{0}^{\perp}\|^{2}\|N^{\perp}\|^{2}-\langle
x_{0}^{\perp},N^{\perp}\rangle^{2}=r^{2}(\|N^{\perp}_{1}\|^{2}\|N^{\perp}_{2}\|^{2}-\langle
N^{\perp}_{1},N^{\perp}_{2}\rangle^{2}).}\end{array}\right.$

$(c)$ The affine subspaces $\mathcal{H}(N_{1};c)\cap\mathcal{H}(N_{2};0)$ and
$\mathcal{H}(\tilde{N}_{1};\tilde{c})\cap\mathcal{H}(\tilde{N}_{2};0)$ are
congruent in $(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ if and only if
one of the following holds:

  * (i)

$N_{2}^{T}=0=\tilde{N}_{2}^{T}$, $\|N_{1}^{\perp}\|=\|\tilde{N}_{1}^{\perp}\|$
and either $c\neq 0\neq\tilde{c}$ or $c=0=\tilde{c}$ if $N_{1}^{T}$ (and hence
$\tilde{N}_{1}^{T}$) vanishes; or

  * (ii)

$N_{2}^{T}\neq 0\neq\tilde{N}_{2}^{T}$, with $\\{N_{1}^{T},N_{2}^{T}\\}$
linearly independent if $c=0$ and $\\{\tilde{N}_{1}^{T},\tilde{N}_{2}^{T}\\}$
linearly independent if $\tilde{c}=0$, and

(6)
$\left\\{\begin{array}[]{l}{\displaystyle\|N^{\perp}_{1}\|^{2}+\|N^{\perp}_{2}\|^{2}=\|\tilde{N}^{\perp}_{1}\|^{2}+\|\tilde{N}^{\perp}_{2}\|^{2}\vspace{1ex}}\\\
{\displaystyle\|N^{\perp}_{1}\|^{2}\|N^{\perp}_{2}\|^{2}-\langle
N^{\perp}_{1},N^{\perp}_{2}\rangle^{2}=\|\tilde{N}^{\perp}_{1}\|^{2}\|\tilde{N}^{\perp}_{2}\|^{2}-\langle\tilde{N}^{\perp}_{1},\tilde{N}^{\perp}_{2}\rangle^{2}.}\end{array}\right.$

Proof: The spheres $\mathbb{S}^{n}(x_{0};r)\cap\mathcal{H}(N;c)$ and
$\mathbb{S}^{n}(\tilde{x}_{0};\tilde{r})\cap\mathcal{H}(\tilde{N};\tilde{c})$
are represented by the two-dimensional space-like subspaces
$V=\mbox{span}\\{\xi_{1},\xi_{2}\\}$ and
$\tilde{V}=\mbox{span}\\{\tilde{\xi}_{1},\tilde{\xi}_{2}\\}$, where

$\xi_{1}=\frac{1}{r}\Psi(x_{0})+\frac{r}{2}w=\frac{1}{r}v+\frac{1}{2}\left(r-\frac{\|x_{0}\|^{2}}{r}\right)w+\frac{1}{r}Cx_{0},\;\;\;\;\xi_{2}=-cw+CN,$

and $\tilde{\xi}_{1}$, $\tilde{\xi}_{2}$ are given in a similar way. Thus

$\xi_{1}^{T}=\frac{1}{r}v+\frac{1}{2}\left(r-\frac{\|x_{0}\|^{2}}{r}\right)w+\frac{1}{r}Cx_{0}^{T},\,\,\,\,\,\,\,\,\xi_{1}^{\perp}=\frac{1}{r}Cx_{0}^{\perp}$

and

$\xi_{2}^{T}=-cw+CN^{T},\,\,\,\,\,\,\,\,\,\xi_{2}^{\perp}=CN^{\perp}$

are the decompositions of $\xi_{1}$ and $\xi_{2}$, respectively, with respect
to the orthogonal decomposition $\mathbb{L}^{n+3}=W_{1}\oplus W_{2}$, with
similar decompositions for $\tilde{\xi}_{1}$ and $\tilde{\xi}_{2}$. The
orthogonality condition between $\mathbb{S}^{n}(x_{0};r)$ and
$\mathcal{H}(N;c)$ (respectively, $\mathbb{S}^{n}(\tilde{x}_{0};\tilde{r})$
and $\mathcal{H}(\tilde{N};\tilde{c})$) is that $\langle N,x_{0}\rangle=c$
(respectively, $\langle\tilde{N},\tilde{x}_{0}\rangle=\tilde{c}$).

By Proposition 9, the spheres $\mathbb{S}^{n}(x_{0};r)\cap\mathcal{H}(N;c)$
and
$\mathbb{S}^{n}(\tilde{x}_{0};\tilde{r})\cap\mathcal{H}(\tilde{N};\tilde{c})$
are congruent in $(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ if and only
if there exists $T\in O_{1}(n+3)$ leaving $W_{1}$ and $W_{2}$ invariant such
that $T(V)=\tilde{V}$. By Proposition 10, if
$\dim\mbox{span}\\{\xi_{1}^{T},\xi_{2}^{T}\\}=\mbox{span}\\{\tilde{\xi}_{1}^{T},\tilde{\xi}_{2}^{T}\\}=2$,
this is the case if and only if the Gram matrices
$(\langle\xi^{T}_{i},\xi^{T}_{j}\rangle)_{1\leq i,j\leq 2}$ and
$(\langle\tilde{\xi}^{T}_{i},\tilde{\xi}^{T}_{j}\rangle)_{1\leq i,j\leq 2}$
(or equivalently, the Gram matrices
$(\langle\xi^{\perp}_{i},\xi^{\perp}_{j}\rangle)_{1\leq i,j\leq 2}$ and
$(\langle\tilde{\xi}^{\perp}_{i},\tilde{\xi}^{\perp}_{j}\rangle)_{1\leq
i,j\leq 2})$ have the same eigenvalues, counted with multiplicities. Since
these Gram matrices have order two, they have the same eigenvalues, counted
with multiplicities, if and only if they have equal traces and determinants.
The latter conditions are equivalent to (4). Notice that
$\dim\mbox{span}\\{\xi_{1}^{T},\xi_{2}^{T}\\}=1$ if and only if
$(N^{T},c)=(0,0)$. If this is the case, then there exists $T\in O_{1}(n+3)$
leaving $W_{1}$ and $W_{2}$ invariant such that $T(V)=\tilde{V}$ if and only
if also $(\tilde{N}^{T},\tilde{c})=(0,0)$ and there exists $T_{1}\in
O_{1}(k+1)$ such that $T_{1}\xi_{1}^{T}=\tilde{\xi}_{1}^{T}$. The latter holds
if and only if
$\langle\xi_{1}^{T},\xi_{1}^{T}\rangle=\langle\tilde{\xi}_{1}^{T},\tilde{\xi}_{1}^{T}\rangle$,
that is, if and only if
$\displaystyle{\frac{x_{0}^{\perp}}{r}=\frac{\tilde{x}_{0}^{\perp}}{\tilde{r}}}$
The proofs of $(ii)$ and $(iii)$ are similar.

######  Corollary 19.

$a)$ The spheres $\mathbb{S}^{n}(x_{0};r)\cap\mathcal{H}(N;c)$ and
$\mathbb{S}^{n}(\tilde{x}_{0};\tilde{r})\cap\mathcal{H}(\tilde{N};\tilde{c})$
are congruent in $(\mathbb{R}^{n+1}\setminus\\{0\\},g)$ if and only if
${\displaystyle\frac{\|\tilde{x}_{0}\|}{\tilde{r}}=\frac{\|{x}_{0}\|}{r}\;\;\;\mbox{and}\;\;\;\frac{\tilde{c}^{2}}{\tilde{r}^{2}}=\frac{c^{2}}{r^{2}}.}$
$b)$ The sphere $\mathbb{S}^{n}(x_{0};r)\cap\mathcal{H}(N;c)$ and the affine
subspace $\mathcal{H}(N_{1};c_{1})\cap\mathcal{H}(N_{2};0)$ are congruent in
$(\mathbb{R}^{n+1}\setminus\\{0\\},g)$ if and only if $c=0$, $\|x_{0}\|=r$ and
$c_{1}\neq 0$.
$c)$ Two affine subspaces $\mathcal{H}(N_{1};c)\cap\mathcal{H}(N_{2};0)$ and
$\mathcal{H}(\tilde{N}_{1};\tilde{c})\cap\mathcal{H}(\tilde{N}_{2};0)$ are
congruent in $(\mathbb{R}^{n+1}\setminus\\{0\\},g)$ if and only if either
$c=0=\tilde{c}$ or $c\neq 0\neq\tilde{c}$.

The preceding corollary allows us to recover the classification in [4] of the
umbilical submanifolds of $\mathbb{S}^{n}\times\mathbb{R}$.

######  Corollary 20.

For any $x_{0}\in\mathbb{R}^{n+1}\setminus\\{0\\}$ and any unit vector
$N\in\mathbb{R}^{n+1}$ with $\langle x_{0},N\rangle=0$, the two-parameter
family $\\{\mathbb{S}^{n}(x_{c};r)\cap\mathcal{H}(N;c),\;x_{c}=x_{0}+cN,c\geq
0,r>0\\}$, together with the one-parameter family
$\\{\mathbb{S}^{n}(x_{0};r)\cap\mathcal{H}(x_{0}/\|x_{0}\|;\|x_{0}\|),\;r>0\\}$,
contain precisely one representative of each congruence class of umbilical non
totally geodesic submanifolds of codimension two of
$(\mathbb{R}^{n+1}\setminus\\{0\\},g)=\mathbb{S}^{n}\times\mathbb{R}$. The
sphere $\mathbb{S}^{n}(x_{c};r)\cap\mathcal{H}(N;c)$ has substantial
codimension two if and only if $c>0$.

Proof: By Proposition 19, if $\mathbb{S}^{n}(x_{c};r)\cap\mathcal{H}(N;c)$ and
$\mathbb{S}^{n}(x_{\tilde{c}};\tilde{r})\cap\mathcal{H}(\tilde{N};\tilde{c})$,
with $c,\tilde{c}>0$ and $r,\tilde{r}>0$, are congruent, then
${\displaystyle\frac{\|{x}_{\tilde{c}}\|}{\|{x}_{c}\|}=\frac{\tilde{r}}{r}=\frac{\tilde{c}}{c}}$.
Thus
${\displaystyle\frac{\tilde{c}^{2}+\|x_{0}\|^{2}}{{c}^{2}+\|x_{0}\|^{2}}=\frac{\|{x}_{\tilde{c}}\|^{2}}{\|{x}_{c}\|^{2}}=\frac{\tilde{c}^{2}}{c^{2}}}$,
which implies that $\tilde{c}=c$, and hence $\tilde{r}=r$.

Similarly, Proposition 19 also implies that
$\mathbb{S}^{n}(x_{0};r)\cap\mathcal{H}(x_{0}/\|x_{0}\|;\|x_{0}\|)$ and
$\mathbb{S}^{n}(x_{0};\tilde{r})\cap\mathcal{H}(x_{0}/\|x_{0}\|;\|x_{0}\|)$
are congruent if and only if $r=\tilde{r}$.

Now, given any
$\mathbb{S}^{n}(\tilde{x}_{0};\tilde{r})\cap\mathcal{H}(\tilde{N};\tilde{c})$
with $\|\tilde{x}_{0}\|\neq 0$, set
${\displaystyle\lambda^{2}=\|\tilde{x}_{0}\|^{2}}/{\tilde{c}^{2}}$. Notice
that $\lambda\geq 1$ by the Cauchy-Schwarz inequality, for
$\langle\tilde{N},\tilde{x}_{0}\rangle=\tilde{c}$. If $\lambda^{2}=1$, that
is, $\tilde{c}^{2}=\|\tilde{x}_{0}\|^{2}$, then
$\mathbb{S}^{n}(\tilde{x}_{0};\tilde{r})\cap\mathcal{H}(\tilde{N};\tilde{c})$
is congruent to
$\mathbb{S}^{n}(x_{0};r)\cap\mathcal{H}(x_{0}/\|x_{0}\|;\|x_{0}\|)$ for
${\displaystyle r=\frac{\tilde{r}\|x_{0}\|}{\|\tilde{x}_{0}\|}}$.

If $\lambda^{2}>1$, choose ${\displaystyle
c=\sqrt{\frac{\|x_{0}\|^{2}}{\lambda^{2}-1}}}$ and ${\displaystyle
r=\frac{c\tilde{r}}{\tilde{c}}}$. Then
${\displaystyle\frac{\|\tilde{x}_{0}\|}{\|x_{c}\|}=\frac{\tilde{r}}{r}=\frac{\tilde{c}}{c},}$
hence
$\mathbb{S}^{n}(\tilde{x}_{0};\tilde{r})\cap\mathcal{H}(\tilde{N};\tilde{c})$
is congruent to $\mathbb{S}^{n}(x_{c};r)\cap\mathcal{H}(N;c)$ by Proposition
19.

Finally, any $\mathcal{H}(N_{1};c_{1})\cap\mathcal{H}(N_{2};0)$ with
$c_{1}\neq 0$ is congruent to $\mathbb{S}^{n}(x_{c};r)\cap\mathcal{H}(N;c)$
with $c=0$ and $r=\|x_{0}\|$.

In the remaining of this section we prove a version of Corollary 20 for the
conformal model $(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$, $2\leq k\leq n$. As a particular
case, it provides a classification of the congruence classes of all umbilical
submanifolds of $\mathbb{H}^{n}\times\mathbb{S}^{1}$, and hence of
$\mathbb{H}^{n}\times\mathbb{R}$. First we make the following observation.

######  Lemma 21.

Let $v\in\mathbb{R}^{m}$ be such that $\|v\|\leq 1$. Then, for any
$w\in\mathbb{R}^{m}$, the smallest eigenvalue of the Gram matrix $G$ of
$\\{v,w\\}$ satisfies $0\leq\bar{\lambda}\leq 1$, and $\bar{\lambda}=1$ if and
only if $\langle v,w\rangle=0$, $\|v\|=1$ and $\|w\|\geq 1$.

Proof:  Given any $w\in\mathbb{R}^{m}$, let
$A\colon\mathbb{R}^{2}\to\mathbb{R}^{m}$ be the linear map such that
$Ae_{1}=v$ and $Ae_{2}=w$, where $\\{e_{1},e_{2}\\}$ is the canonical basis of
$\mathbb{R}^{2}$. Then the eigenvalues of $G$ are the eigenvalues of
$A^{t}A\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$. Thus its smallest eigenvalue
$\bar{\lambda}$ satisfies

$1\geq\|v\|^{2}=\langle
A^{t}Ae_{1},e_{1}\rangle\geq\mbox{min}\left\\{\frac{\langle
A^{t}Av,v\rangle}{\langle v,v\rangle}\;:\>v\neq 0\right\\}=\bar{\lambda}\geq
0.$

Moreover, if $\bar{\lambda}=1$, then $\|v\|=1$ and $e_{1}$ is an eigenvector
of $A^{t}A$ associated to $\bar{\lambda}$. Thus $e_{2}$ is also an eigenvector
of $A^{t}A$, and hence $\langle v,w\rangle=\langle
A^{t}Ae_{1},e_{2}\rangle=0$. In particular, the eigenvalues of $A^{t}A$ are
$1=\|v\|^{2}$ and $\|w\|^{2}$, and hence $\|w\|\geq 1$. The “if” part is
trivial.

######  Corollary 22.

Let $p_{0}\in\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1}$, $2\leq k\leq n$, be
such that $p_{0}=p_{0}^{\perp}$. Let $e\in(\mathbb{R}^{k-1})^{\perp}$ and
$\eta\in\mathbb{R}^{k-1}$ be such that $\|e\|=\|\eta\|=1$ and $\langle
p_{0},e\rangle=0$. Set

$x_{c}=p_{0}+\sqrt{1+c^{2}}\,\eta\;\;\;\mbox{and}\;\;\;N_{c}=\frac{1}{\sqrt{1+c^{2}}}e+\frac{c}{\sqrt{1+c^{2}}}\eta,\;c\geq
0.$

Then the two-parameter family

(7) $\\{\mathbb{S}^{n}(x_{c},r)\cap\mathcal{H}(N_{c},c)\colon
c>0,\;\;0<r<\sqrt{1+c^{2}}\|p_{0}\|\\}$

and the one-parameter families

$\displaystyle\\{\mathbb{S}^{n}(x_{c},\sqrt{1+c^{2}}\|p_{0}\|)$
$\displaystyle\cap\mathcal{H}(N_{c},c)\colon
c>0\\},\;\;\;\\{\mathbb{S}^{n}(p_{0},r)\cap\mathcal{H}(e,0)\colon r>0\\}$ (8)
$\displaystyle\mbox{and}\;\;\;\\{\mathbb{S}^{n}(p_{0},r)\cap\mathcal{H}(\eta,0)\colon
r>0\\}$

contain, together, precisely one representative of each congruence class of
umbilical non-totally geodesic submanifolds of codimension two in
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$.

Proof:  First notice that in the family (7) we have $\langle
x_{c}^{\perp},N_{c}^{\perp}\rangle=0$. Then the Gram matrix of
$\\{r^{-1}x_{c}^{\perp},N_{c}^{\perp}\\}$ is always diagonal with distinct
eigenvalues $\|p_{0}\|^{2}/r^{2}$ and $(1+c^{2})^{-1}$. Moreover, since we are
taking $r<\sqrt{1+c^{2}}\|p_{0}\|$, then $(1+c^{2})^{-1}$ is the smallest one.

Assume that $\mathbb{S}^{n}(x_{c},r)\cap\mathcal{H}(N_{c},c)$ and
$\mathbb{S}^{n}(x_{\tilde{c}},\tilde{r})\cap\mathcal{H}(N_{\tilde{c}},\tilde{c})$
belong to the family (7) and are congruent. Then the eigenvalues of the Gram
matrices of $\\{r^{-1}x_{c}^{\perp},N_{c}^{\perp}\\}$ and
$\\{\tilde{r}^{-1}x_{\tilde{c}}^{\perp},N_{\tilde{c}}^{\perp}\\}$, in
particular their smallest ones, coincide, and hence $c=\tilde{c}$ and
$r=\tilde{r}$. Similarly, one can check that distinct elements in any of the
one-parameter families in (22) lie in distinct congruence classes, as well as
elements of distinct families.

Let
$\mathbb{S}^{n}(\tilde{x}_{0},\tilde{r})\cap\mathcal{H}(\tilde{N},\tilde{c})$
be an umbilical non-totally geodesic submanifold with codimension two of
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{k-1},g)$ and let $\tilde{G}$ be the
Gram matrix of $\\{\tilde{r}^{-1}\tilde{x}_{0}^{\perp},\tilde{N}^{\perp}\\}$.
Since $\|\tilde{N}^{\perp}\|\leq 1$, we see from Lemma 21 that the smallest
eigenvalue $\tilde{\lambda}$ of $\tilde{G}$ satisfies $\tilde{\lambda}\leq 1$.
If $(\tilde{N}^{T},\tilde{c})=(0,0)$, then $\|\tilde{N}^{\perp}\|=1$ and
$\langle\tilde{x}_{0}^{\perp},\tilde{N}\rangle=0$. Therefore, in this case
Proposition 18 implies that
$\mathbb{S}^{n}(\tilde{x}_{0},\tilde{r})\cap\mathcal{H}(\tilde{N},\tilde{c})$
is congruent to $\mathbb{S}^{n}(p_{0},r)\cap\mathcal{H}(e,0)$ with
$r=\tilde{r}\|p_{0}\|/\|\tilde{x}_{0}^{\perp}\|$. Notice that, if
$\tilde{\lambda}=1$, then
$\langle\tilde{x}_{0}^{\perp},\tilde{N}^{\perp}\rangle=0$ and
$\|\tilde{N}^{\perp}\|=1$ by the second assertion in Lemma 21, implying that
$(\tilde{N}^{T},\tilde{c})=(0,0)$.

From now on we assume that $(\tilde{N}^{T},\tilde{c})\neq(0,0)$, and thus
$\tilde{\lambda}<1$. If $\tilde{\lambda}=0$, then
$\mathbb{S}^{n}(\tilde{x}_{0},\tilde{r})\cap\mathcal{H}(\tilde{N},\tilde{c})$
is congruent to $\mathbb{S}^{n}(p_{0},r)\cap\mathcal{H}(\eta,0)$ with
$r^{2}=\|p_{0}\|^{2}(\tilde{r}^{-2}\|\tilde{x}_{0}^{\perp}\|^{2}+\|\tilde{N}^{\perp}\|^{2})^{-1}$.
Now assume that $0<\tilde{\lambda}<1$. If $\tilde{\lambda}$ has multiplicity
$2$, then
$\mathbb{S}^{n}(\tilde{x}_{0},\tilde{r})\cap\mathcal{H}(\tilde{N},\tilde{c})$
is congruent to
$\mathbb{S}^{n}(x_{c},\sqrt{1+c^{2}}\|p_{0}\|)\cap\mathcal{H}(N_{c},c)$ with
$c=\sqrt{\tilde{\lambda}^{-1}-1}$. Finally, if $\tilde{G}$ has two distinct
eigenvalues $\tilde{\lambda}<\tilde{\lambda}^{\prime}$, then
$\mathbb{S}^{n}(\tilde{x}_{0},\tilde{r})\cap\mathcal{H}(\tilde{N},\tilde{c})$
is congruent to $\mathbb{S}^{n}(x_{c},r)\cap\mathcal{H}(N_{c},c)$ with
$c=\sqrt{\tilde{\lambda}^{-1}-1}$ and
$r^{2}=\|p_{0}\|^{2}/\tilde{\lambda}^{\prime}$.

As for an umbilical non-totally geodesic
$\mathcal{H}(N_{1};c_{1})\cap\mathcal{H}(N_{2};0)$, first notice that, since
it is not totally geodesic, it can not contain $\mathbb{R}^{k-1}$. Then, its
image by an inversion with respect to a hypersphere centered at some point
$y\in\mathbb{R}^{k-1}\setminus\mathcal{H}(N_{1};c_{1})\cap\mathcal{H}(N_{2};0)$
is a non-totally geodesic
$\mathbb{S}^{n}(\tilde{x}_{0},\tilde{r})\cap\mathcal{H}(\tilde{N},\tilde{c})$,
which is congruent to some element of the family (7) or one of the families in
(22) by the part that has already been proved.

######  Remark 23.

_In Corollary 22, the elements of the families (7) and (22) that are
substantial are those in the family (7) and those of the first family in (22).
_

##  5\. Umbilical submanifolds of $\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ as
rotational submanifolds.

The next result shows that any umbilical submanifold of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ with substantial codimension $p$
(which is necessarily less than or equal to $\mbox{min}\,\\{k+1,n-k+2\\}$ by
Corollary 14) is a rotational submanifold in a suitable sense, under mild
further restrictions on $p$ and $k$.

Given a space-like subspace $V\subset\mathbb{L}^{n+3}=W_{1}\oplus
W_{2}\supset\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$, for $1\leq i\leq 2$ set
$U_{i}=V^{\perp}\cap W_{i}$, and let $U_{i}^{\perp}$ be the orthogonal
complement of $U_{i}$ in $W_{i}$. Denote by $O(U^{\perp}_{1})$ (respectively,
$O(U^{\perp}_{2})$) the subgroup of all elements of $O_{1}(k+1)$
(respectively, $O(n-k+2)$) that fix $U^{\perp}_{1}$ (respectively,
$U^{\perp}_{2}$) pointwise.

######  Proposition 24.

Let $S=(\mathbb{H}^{k}\times\mathbb{S}^{n-k+1})\cap V^{\perp}$, where
$V\subset\mathbb{L}^{n+3}$ is a space-like subspace of dimension $p$, be a
substantial umbilical submanifold with codimension $p$ of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}\subset\mathbb{L}^{n+3}$. Then the
following assertions hold:

  * •

If $p\leq k-1$ then the subgroup $\\{I\\}\times O(U^{\perp}_{2})$ of
$O_{1}(k+1)\times O(n-k+2)$ leaves $S$ invariant and acts on $S$ with
cohomogeneity $n-k+1$, the orbit through (a regular point) $(x,y)\in S$ being
the $(k-p)$-dimensional umbilical submanifold
$V^{\perp}\cap(\mathbb{H}^{k}\times\\{y\\})$;

  * •

If $p\leq n-k$ then the subgroup $\\{I\\}\times O(U^{\perp}_{2})$ of
$O_{1}(k+1)\times O(n-k+2)$ leaves $S$ invariant and acts on $S$ with
cohomogeneity $k$, the orbit through (a regular point) $(x,y)\in S$ being the
$(n-k-p+1)$-dimensional sphere
$V^{\perp}\cap(\\{x\\}\times\mathbb{S}^{n-k+1})$.

Proof: Since $S=(\mathbb{H}^{k}\times\mathbb{S}^{n-k+1})\cap V^{\perp}$ for
some space-like subspace $V\subset\mathbb{L}^{n+3}$ with dimension $p$, any
isometry $T\in O_{1}(k+1)\times O(n-k+2)$ of
$\mathbb{H}^{k}\times\mathbb{S}^{n-k+1}$ such that $T(V^{\perp})=V^{\perp}$
leaves $S$ invariant. Since $S$ is in substantial codimension $p$, then $V\cap
W_{i}=\\{0\\}$ for $1\leq i\leq 2$ by Proposition 13, hence
$U_{i}=V^{\perp}\cap W_{i}$ has codimension $p$ in $W_{i}$ for $1\leq i\leq
2$. Therefore, if $p\leq k-1$ (respectively, if $p\leq n-k$), then $\dim
U_{1}\geq 2$ (respectively, $\dim U_{2}\geq 2$), hence
$O(U^{\perp}_{1})\times\\{I\\}$ (respectively, $\\{I\\}\times
O(U^{\perp}_{2})$ leaves $S$ invariant, the orbit of (a regular point)
$(x,y)\in S$ being the $(k-p)$-dimensional (respectively,
$(n-k-p+1)$-dimensional) umbilical submanifold
$V^{\perp}\cap(\mathbb{H}^{k}\times\\{y\\})$ (respectively, sphere
$V^{\perp}\cap(\\{x\\}\times\mathbb{S}^{n-k+1})$).

######  Corollary 25.

Let $S=(\mathbb{H}^{k}\times\mathbb{S}^{n-k+1})\cap V^{\perp}$ be as in
Proposition 24.

  * (i)

If $p\leq\mbox{min}\,\\{k-1,n-k\\}$, then the subgroup $O(U^{\perp}_{1})\times
O(U^{\perp}_{2})$ of $O_{1}(k+1)\times O(n-k+2)$ leaves $S$ invariant and acts
on $S$ with cohomogeneity $p$, the orbit through (a regular point) $(x,y)\in
S$ being the product
$(V^{\perp}\cap(\mathbb{H}^{k}\times\\{y\\}))\times(V^{\perp}\cap(\\{x\\}\times\mathbb{S}^{n-k+1}))$;

  * (ii)

If $k=1=p$, then the subgroup $\\{I\\}\times O(U^{\perp}_{2})$ of
$O_{1}(2)\times O(n+1)$ leaves $S$ invariant and acts on $S$ with
cohomogeneity $1$, the orbit through (a regular point) $(x,y)\in S$ being the
$(n-1)$-dimensional sphere $V^{\perp}\cap(\\{x\\}\times\mathbb{S}^{n})$;

  * (iii)

If $k=1$, $p=2$ and $n\geq 3$, then the subgroup $\\{I\\}\times
O(U^{\perp}_{2})$ of $O_{1}(2)\times O(n+1)$ leaves $S$ invariant and acts on
$S$ with cohomogeneity $1$, the orbit through (a regular point) $(x,y)\in S$
being the $(n-2)$-dimensional sphere
$V^{\perp}\cap(\\{x\\}\times\mathbb{S}^{n})$;

  * (iv)

If $k=n$ and $p=1$, then the subgroup $O(U^{\perp}_{1})\times\\{I\\}$ of
$O_{1}(n+1)\times O(2)$ leaves $S$ invariant and acts on $S$ with
cohomogeneity $1$, the orbit through (a regular point) $(x,y)\in S$ being the
$(n-1)$-dimensional umbilical submanifold
$V^{\perp}\cap(\mathbb{H}^{n}\times\\{y\\})$;

  * (v)

If $k=n\geq 3$ and $p=2$, then the subgroup $O(U^{\perp}_{1})\times\\{I\\}$ of
$O_{1}(n+1)\times O(2)$ leaves $S$ invariant and acts on $S$ with
cohomogeneity $1$, the orbit through (a regular point) $(x,y)\in S$ being the
$(n-2)$-dimensional umbilical submanifold
$V^{\perp}\cap(\mathbb{H}^{n}\times\\{y\\})$.

It follows from parts $(iv)$ and $(v)$ of Corollary 25 that umbilical
submanifolds of $\mathbb{H}^{n}\times\mathbb{S}^{1}$ (or equivalently,
$\mathbb{H}^{n}\times\mathbb{R}$) with substantial codimension one
(respectively, substantial codimension two), are rotational submanifolds with
curves in $\mathbb{H}^{1}\times\mathbb{S}^{1}$ (respectively,
$\mathbb{H}^{2}\times\mathbb{S}^{1}$) as profiles. In the next section these
curves are explicitly described.

###  5.1. The profile curves of umbilical submanifolds of
$\mathbb{H}^{n}\times\mathbb{S}^{1}$

We first consider the codimension $1$ case, that is, umbilical non-totally
geodesic hypersurfaces of $\mathbb{H}^{n}\times\mathbb{S}^{1}$. Let
$(\mathbb{R}^{n+1}\setminus\mathbb{R}^{n-1},g)$ be the conformal model of
$\mathbb{H}^{n}\times\mathbb{S}^{1}$ and let $\\{e_{1},\dots,e_{n+1}\\}$ be an
orthonormal basis of $\mathbb{R}^{n+1}$ such that $\\{e_{1},\dots,e_{n-1}\\}$
spans $\mathbb{R}^{n-1}$. By Corollary 17, it is enough to consider the one-
parameter family of hyperspheres $\\{\mathbb{S}^{n}(x_{0},r),r>0\\}$ centered
at a point $x_{0}$ with $x_{0}^{\perp}\neq 0$. For simplicity, we take
$x_{0}=e_{n+1}$. The hypersphere $\mathbb{S}^{n}(e_{n+1},r)$ is represented by
the unit space-like vector
$z=\frac{1}{r}\Psi(e_{n+1})+\frac{r}{2}w\in\mathbb{L}^{n+3}$, whose
$W_{1}$-component is

$z^{T}=\frac{1}{r}v+\frac{1}{2}\left(r-\frac{1}{r}\right)w.$

Thus $z^{T}$ is space-like, light-like or time-like, according to whether
$r>1$, $r=1$ or $r<1$, respectively. Hence, with the notations as in
Proposition 25, the subspace $U_{1}$ is, accordingly, time-like, degenerate or
space-like, respectively. We call these cases _hyperbolic_ , _parabolic_ or
_spherical_ , respectively.

Let $\mathbb{R}^{2}=(\mathbb{R}^{n-1})^{\perp}=\mbox{span}\\{e_{n},e_{n+1}\\}$
and consider the curve $c\colon I\to\mathbb{R}^{2}$ given by

$c(\theta)=e_{n+1}+r\cos(\theta)e_{n+1}+r\sin(\theta)e_{n},$

where $I=(-\pi,\pi]$, $I=(-\pi,\pi)$ or $I=[-\pi+\arccos(r),\pi-\arccos(r)]$,
depending on whether we are in the hyperbolic, parabolic or spherical case,
respectively. Notice that the image $\Theta(c(I))$ under the diffeomorphism
$\Theta\colon\mathbb{R}^{n+1}\setminus\mathbb{R}^{n-1}\to\mathbb{H}^{n}\times\mathbb{S}^{1}$
given by (2) is contained in a totally geodesic
$\mathbb{H}^{1}\times\mathbb{S}^{1}\subset\mathbb{H}^{n}\times\mathbb{S}^{1}$
and that, for each $\theta\in I$, the point $c(\theta)$ lies in a different
slice $\mathbb{H}^{n}\times\\{q\\}$, $q\in\mathbb{S}^{1}$. Since the the
subgroup of isometries $O(U_{1}^{\perp})\times\\{I\\}$ acts on each slice, and
this action is transitive on the intersection
$\Theta(\mathbb{S}^{n}(e_{n+1},r))\cap\mathbb{H}^{n}\times\\{q\\}$ of the
umbilical hypersurface $\Theta(\mathbb{S}^{n}(e_{n+1},r))$ of
$\mathbb{H}^{n}\times\mathbb{S}^{1}$ with each slice
$\mathbb{H}^{n}\times\\{q\\}$, it follows that
$\Theta(\mathbb{S}^{n}(e_{n+1},r))$ is generated by the action of
$O(U_{1}^{\perp})\times\\{I\\}$ on the curve $\Theta(c(I))$. Notice also that
if $\Theta(c(\theta))$ lies in the slice $\mathbb{H}^{n}\times\\{q\\}$, then
the orbit $\Theta(\mathbb{S}^{n}(e_{n+1},r))\cap\mathbb{H}^{n}\times\\{q\\}$
of $\Theta(c(\theta))$ under $O(U_{1}^{\perp})\times\\{I\\}$ is an equidistant
hypersurface, horosphere or sphere in that slice (which can degenerate into a
point), depending on whether we are in the hyperbolic, parabolic or spherical
case, respectively.

Let us now consider the case of umbilical submanifolds with codimension $2$.
By Corollary 22, it is enough to consider the elements of the two families of
umbilical submanifolds given by (7) and (22) that have substantial codimension
$2$. Following the notations in that proposition, set $p_{0}=e_{n+1}$,
$e=e_{n}$, $\eta=e_{1}$ and let $x_{c}$ and $N_{c}$ be given by

$x_{c}=e_{n+1}+\sqrt{1+c^{2}}\,e_{1}\;\;\mbox{and}\;\;N_{c}=\frac{1}{\sqrt{1+c^{2}}}e_{n}+\frac{c}{\sqrt{1+c^{2}}}e_{1},\,\,\,c>0.$

Let $z_{1}$ and $z_{2}$ be the unit space-like vectors associated to
$\mathbb{S}^{n}(x_{c},r)$ and $\mathcal{H}(N_{c},c)$, respectively. Then
$\\{z_{1},z_{2}\\}$ is an orthonormal basis of the space-like subspace
$V\subset\mathbb{L}^{n+3}$ associated to the umbilical submanifold
$\mathbb{S}^{n}(x_{c},r)\cap\mathcal{H}(N_{c},c)$. Notice that the $W_{1}$
components $z_{1}^{T}$ and $z_{2}^{T}$ are orthogonal, and that $z_{2}^{T}$ is
always space-like. Moreover, $\langle
z_{1}^{T},z_{1}^{T}\rangle=1-\frac{1}{r^{2}}$. Thus the subspace $U_{1}\subset
W_{1}$ is time-like, degenerate or space-like depending on whether $r>1$,
$r=1$ or $r<1$, respectively. We call these cases accordingly _hyperbolic_ ,
_parabolic_ or _spherical_.

Consider the curve $c\colon I\to\mathcal{H}(N_{c},c)$ given by

$c(\theta)=x_{c}+r\cos(\theta)e_{n+1}+r\sin(\theta)\eta_{c},$

where

(9) $\eta_{c}=-\frac{c}{\sqrt{1+c^{2}}}e_{n}+\frac{1}{\sqrt{1+c^{2}}}e_{1}.$

Here $I=(-\pi,\pi]$, $I=(-\pi,\pi)$ or $I=[-\pi+\arccos(r),\pi-\arccos(r)]$,
depending on whether we are in the hyperbolic, parabolic or spherical case,
respectively. Notice that $\Theta(c(I))$ is contained in a totally geodesic
$\mathbb{H}^{2}\times\mathbb{S}^{1}\subset\mathbb{H}^{n}\times\mathbb{S}^{1}$
and that, for each $\theta\in I$, the point $\Theta(c(\theta))$ lies in a
different slice $\mathbb{H}^{n}\times\\{q\\}$, $q\in\mathbb{S}^{1}$. Since the
the subgroup of isometries $O(U_{1}^{\perp})\times\\{I\\}$ acts on each slice,
and this action is transitive on the intersection
$\Theta(\mathbb{S}^{n}(x_{c},r)\cap\mathcal{H}(N_{c},c))\cap\mathbb{H}^{n}\times\\{q\\}$
of the umbilical submanifold
$\Theta(\mathbb{S}^{n}(x_{c},r)\cap\mathcal{H}(N_{c},c))$ with each slice
$\mathbb{H}^{n}\times\\{q\\}$, it follows as before that
$\Theta(\mathbb{S}^{n}(x_{c},r)\cap\mathcal{H}(N_{c},c))$ is generated by the
action of $O(U_{1}^{\perp})\times\\{I\\}$ on the curve $\Theta(c(I))$.

As for the umbilical submanifolds in the one-parameter family
$\\{\mathbb{S}^{n}(x_{c},\sqrt{1+c^{2}})\cap\mathcal{H}(N_{c},c)$, $c>0$, they
are always hyperbolic. Here we define
$c\colon(-\pi,\pi]\to\mathcal{H}(N_{c},c)$ by

$c(\theta)=x_{c}+\sqrt{1+c^{2}}\cos(\theta)e_{n+1}+\sqrt{1+c^{2}}\sin(\theta)\eta_{c},$

where $\eta_{c}$ is given by (9). Then we see as in the previous case that the
submanifold
$\Theta(\\{\mathbb{S}^{n}(x_{c},\sqrt{1+c^{2}})\cap\mathcal{H}(N_{c},c))$ is
generated by the action of $O(U_{1}^{\perp})\times\\{I\\}$ on the curve
$\Theta(c(I))$.

######  Remark 26.

_A similar description can be given as above of the profile curves of the
umbilical submanifolds of $\mathbb{S}^{n}\times\mathbb{R}$, which is much
simpler than those given in [4], [8] and [9] and, in particular, avoids
solving any ODE._

_Acknowledgement._ After a preliminary version of this paper was completed, we
learned that a classification of umbilical _hypersurfaces_ of
$\mathbb{H}^{n}\times\mathbb{R}$ using a different method was provided in [3].

## References

  * [1] Dajczer, M. and Tojeiro, R., “Submanifold Theory beyond an introduction”, Springer, New York, 2019, Universitext Series.
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  * [3] de Lima, R. F. and dos Santos, J. P., _Totally umbilical hypersurfaces of product spaces_. Preprint. (available at arXiv : 2010.00634).
  * [4] Mendonça, B. and Tojeiro, R., _Umbilical submanifolds of $\mathbb{S}^{n}\times\mathbb{R}$_, Canadian J. Math. 66 (2104), 400–428.
  * [5] Mendonça, B. and Tojeiro, R., _Submanifolds of products of space forms_ , Indiana Univ. Math. J. 62 (4) (2013), 1283–1314.
  * [6] Nikolayevsky, Y., _Totally umbilical submanifolds of symmetric spaces_ , Mat. Fiz. Anal. Geom. 1 (1994), 314–357.
  * [7] Orjuela, J. and Tojeiro, R., _Umbilical surfaces of products of space forms_ , Tôhoku Math. J. 68 (2016), 471–486.
  * [8] Souam, R. and Toubiana, E., Totally umbilic surfaces in homogeneous $3$-manifolds. Comment. Math. Helv. 84 (3) (2009), 673-704.
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