# The Ambrose-Singer Theorem for cohomogeneity one Riemannian manifolds

J. L. Carmona Jiménez, M. Castrillón López, J. C. Díaz-Ramos

###### Abstract

We characterize isometric actions when the principal orbits are hypersurfaces
by the existence of a linear connection satisfying a set of covariant
equations. We use this results to characterize isomorphic cohomogeneity one
foliations in terms of these connections and give explicit examples of these
objects in the Euclidean space and the real hyperbolic space.

Key words. Ambrose-Singer theorem, cohomogeneity one actions, canonical
connection.

MSC: 53C05, 53C12, 53C40.

##  1 Introduction

In the early twentieth century, Élie Cartan gives a local characterization of
locally symmetric spaces: a Riemannian manifold $(M,g)$ is locally symmetric
if and only if its curvature tensor is invariant by parallelism. This result
was generalized to Riemannian homogeneous manifolds and it is the well-known
Ambrose-Singer Theorem:

######  Ambrose-Singer Theorem ([AS58]).

Let $(M,g)$ be a connected and simply-connected complete Riemannian manifold.
Then, the following statements are equivalent:

  1. 1.

The manifold $M$ is Riemannian homogeneous.

  2. 2.

The manifold $M$ admits a linear connection $\tilde{\nabla}$ satisfying:

$\tilde{\nabla}R=0,\quad\tilde{\nabla}S=0,\quad\tilde{\nabla}g=0,$

where $R$ is the curvature tensor of the Levi-Civita connection $\nabla$ and
$S=\nabla-\tilde{\nabla}$.

The explicit classification of the tensor $S$, known as _homogeneous structure
tensor_ , into $\mathrm{O}(n)$-irreducible classes, $n=\mathrm{dim}(M)$,
yields valuable geometric understanding. This classification takes the benefit
of the intricate interplay among partial differential equations, Algebra and
Geometry given by the Ambrose-Singer Theorem. This synergy was leveraged in
the program that formally started with [TV83]. It aims to understand
Riemannian homogeneous manifolds through the lens of Riemannian homogeneous
structures. For example, it characterizes a specific Riemannian homogeneous
manifold by scrutinizing all its homogeneous structures (see [CGS13, CC22a]).
Additionally, different Riemannian transitive Lie group actions give rise to
distinct Riemannian homogeneous structures. For a recent panoramic view of
this program, see [CC19].

However, this program was exclusive for pseudo-Riemannian geometry. Recently,
it was understood that these techniques can be applied for non-metric
frameworks, such as symplectic homogeneous manifolds, see [CC22]. This opens
the door to thinking about new generalizations of the Ambrose-Singer Theorem
for non-transitive actions.

A cohomogeneity one action on a Riemannian manifold is an isometric action
whose principal orbits are hypersurfaces. If a group acts on a Riemannian
manifold with cohomogeneity one, we say that such a manifold is a
cohomogeneity one manifold. These are precisely the objects of study in this
article. Techniques based on cohomogeneity one actions have successfully been
used to find examples of interesting geometric structures, such as manifolds
with positive curvature, Einstein metrics, or Ricci solitons, see [GWZ08,
GZ00, Wil06]. The idea behind these constructions is the following. Many
geometric properties of Riemannian manifolds are encoded by partial
differential equations whose solutions are usually very difficult to find.
However, if these structures turn out to be invariant by a cohomogeneity one
action, these complicated PDEs translate into ordinary differential equations
along a geodesic orthogonal to the orbits; this is a much easier problem,
whose existence and uniqueness of solutions is well understood. In general,
the use of cohomogeneity one symmetries is a technique that traces back at
least to the introduction of polar, cylindrical, or spherical coordinates in
problems with rotational symmetries, such as the motion on planets revolving
around the sun.

The aim of this article is to initiate a line of research, analogous to
Tricerri-Vanhecke’s program, adapted to cohomogeneity one Riemannian
manifolds. This objective is pursued by achieving the following goals.

Sec. 3 and Sec. 4 are devoted to proving an Ambrose-Singer Theorem for
cohomogeneity one and locally cohomogeneity one Riemannian manifolds,
respectively. We characterize connected, simply-connected, and complete
cohomogeneity one Riemannian manifolds by the existence of a geodesically
complete linear connection satisfying certain geometric covariant derivative
equations. This connection is called the cohomogeneity one AS-connection.
Analogous to the transitive framework, we introduce the difference tensor
$S=\nabla-\tilde{\nabla}$ where $\tilde{\nabla}$ is the connection under
consideration and $\nabla$ is the Levi-Civita connection. This is called the
cohomogeneity one structure. Afterwards, we relax the topological conditions
and locally characterize cohomogeneity one Riemannian manifolds, that is, we
assume that there is a Lie pseudo-group of isometries acting on the manifold
whose principal orbits are hypersurfaces. The existence of cohomogeneity one
homogeneous structures describes the nature of being a locally cohomogeneity
one manifold.

Following the program of [TV83] or [CC19], but with a non-metric perspective,
(see [CC22]), in Sec. 5, we study the $\operatorname{SO}(n)$-irreducible
submodules in the space of the tensor elements of cohomogeneity one
structures. This decomposition yields four submodules for cohomogeneity
structures. One is the homogeneous structure in the leaf; therefore, we can
scrutinize the cohomogeneity one Riemannian action by examining the projection
onto this submodule. Another is the second fundamental form of the leaves.

In Sec. 6, we give a formula for the cohomogeneity one structure constructed
in the main theorem of Sec. 3. Afterwards, we study the uniqueness of this
cohomogeneity one structure depending on the action. Finally, Sec. 7 is
dedicated to giving examples of cohomogeneity one structures in the Euclidean
space and the real hyperbolic space.

##  2 Preliminaries

Recall that a homogeneous manifold $M=G/H$ is said to be reductive if the Lie
algebra of $G$ can be decomposed as $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$
where $\mathfrak{h}$ is the Lie algebra of $H$ and $\mathfrak{m}$ is a
subspace such that $\mathrm{Ad}_{H}\mathfrak{m}\subset\mathfrak{m}$. The
following result is a generalization of the Ambrose-Singer Theorem with a non-
necessarily Riemannian geometric structure.

######  Theorem 2.1.

[CC22, Thm. 2.2] Let $M$ be a connected, simply-connected manifold equipped
with a tensor field $K$. Then there is a group $G$ of transformations
preserving $K$ and acting reductively and transitively on $M$ (that is,
$(M,K)$ is a reductive homogeneous manifold) if and only if there is a
complete linear connection $\tilde{\nabla}$ on $M$ such that

$\tilde{\nabla}\tilde{R}=0,\quad\tilde{\nabla}\tilde{T}=0,\quad\tilde{\nabla}K=0,$
(2.1)

where $\tilde{R}$ and $\tilde{T}$ are the curvature and torsion of
$\tilde{\nabla}$, respectively.

The topological conditions on $M$ as well as the completeness of the canonical
connection $\tilde{\nabla}$ are associated with the global nature of the
action. When these conditions are weakened so that a manifold satisfies (2.1)
only (that is, we have a so-called AS-manifold) we have a local version of the
result as follows.

######  Theorem 2.2.

[CC22, Thm. 3.7] Let $M$ be a differentiable manifold with a geometric
structure defined by a tensor $K$. Then the following assertions are
equivalent:

  1. 1.

The pair $(M,K)$ is a reductive locally homogeneous space, associated with a
Lie pseudo-group $\mathcal{G}$ of local transformations of $M$ preserving $K$.

  2. 2.

There exists a connection $\tilde{\nabla}$ such that:

$\tilde{\nabla}\tilde{R}=0,\quad\tilde{\nabla}\tilde{T}=0,\quad\tilde{\nabla}K=0,$

where $\tilde{R}$ and $\tilde{T}$ are the curvature and torsion of
$\tilde{\nabla}$, respectively.

In the proof of this last result, one shows that the Lie pseudo-group that is
acting on $M$ is the Lie pseudo-group of local affine transvections. Recall
that a transvection of a manifold endowed with a connection
$(M,\tilde{\nabla})$ is a diffeomorphism $f\colon M\longrightarrow M$ such
that the _natural lift_

$\displaystyle\tilde{f}\colon\mathcal{L}(M)$
$\displaystyle\longrightarrow\mathcal{L}(M)$ $\displaystyle u$
$\displaystyle\longrightarrow f_{*,\pi(u)}\circ u.$

to the frame bundle $\mathcal{L}(M)$ preserves every holonomy bundle
$P(u)\subset\mathcal{L}(M)$, i. e., $\tilde{f}(P(u))\subset P(u)$ for all
$u\in\mathcal{L}(M)$. The group $\mathrm{Tr}(M,\tilde{\nabla})$ of
transvections is a subgroup of the group of affine transformations. Therefore,
an affine transformation $f$ belongs to $\mathrm{Tr}(M,\tilde{\nabla})$ if and
only if for every point $p\in M$ there is a piecewise smooth curve $\alpha$
joining $p$ with $f(p)$ such that the tangent map $f_{*}\colon
T_{p}M\longrightarrow T_{f(p)}M$ coincides with the parallel transport along
$\alpha$.

######  Theorem 2.3.

[Kow80, Thm. I.25] Let $(M,\tilde{\nabla})$ be a connected manifold with an
affine connection. Then the following two conditions are equivalent:

  * •

$M$ can be expressed as a reductive homogeneous space $G/H$ with respect to a
reductive decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$, where $G$ is
effective on $M$, and $\tilde{\nabla}$ is the canonical connection of $G/H$.

  * •

The transvection group $\mathrm{Tr}(M,\tilde{\nabla})$ acts transitively on
each holonomy bundle $P(u)\subset\mathcal{L}(M)$ and in particular, it acts
transitively on $M$.

More precisely, if the first condition is satisfied, then we can write $M=G/H$
with $G=\mathrm{Tr}(M,\tilde{\nabla})$. If the second condition is satisfied,
$\mathrm{Tr}(M,\tilde{\nabla})$ is a normal subgroup of $G$.

Let $M$ be a reductive homogeneous manifold. We deduce from the previous
theorem that for any piecewise smooth curve curve $\alpha$ joining $p$ with
$q$ there exists a global transvection $f$ such that $f(p)=q$ and the tangent
map $f_{*}\colon T_{p}M\longrightarrow T_{q}M$ coincides with the parallel
transport along $\alpha$.

##  3 Main result: global version

A Riemannian manifold $(M,g)$ is said to be a regular cohomogeneity one
Riemannian manifold if there is a group of isometries
$G\subset\mathrm{Isom}(M,g)$ such that $M/G$ is connected and every orbit is
embedded, closed and principal of codimension one. If the action of a subgroup
$G\subset\mathrm{Isom}(M,g)$ is proper, and the codimension of a generic orbit
is one, then we have that the subset of principal orbits is a regular
cohomogeneity one Riemannian manifold. We refer the reader to [BCO16] for an
introduction to cohomogeneity one actions.

######  Lemma 3.1.

Let $(M,g)$ be a connected Riemannian manifold equipped with a vector field
$\xi$. Let $\tilde{\nabla}$ be an affine connection such that,

$\tilde{\nabla}\xi=0,\quad\tilde{\nabla}_{X}g=0,\quad\tilde{T}(X,Y)\in\mathcal{D},\quad\forall\,X,Y\in\mathcal{D},$

where $\mathcal{D}=\\{X:\>g(X,\xi)=0\\}$ and $\tilde{T}$ is the torsion of
$\tilde{\nabla}$. Then, the distribution $\mathcal{D}$ is integrable and the
leaves are totally geodesic submanifolds with respect to $\tilde{\nabla}$.

###### Proof.

We take two vector fields $X$, $Y$ such that $g(X,\xi)=0$ and $g(Y,\xi)=0$.
Taking derivatives of these expressions, and using $\tilde{\nabla}_{X}g=0$ and
$\tilde{\nabla}\xi=0$, we have that

$0=g(\tilde{\nabla}_{Y}X,\xi)+g(X,\tilde{\nabla}_{Y}\xi)=g(\tilde{\nabla}_{Y}X,\xi),\qquad
0=g(\tilde{\nabla}_{X}Y,\xi)+g(Y,\tilde{\nabla}_{X}\xi)=g(\tilde{\nabla}_{X}Y,\xi).$

Then $g([X,Y],\xi)=g(-\tilde{T}(X,Y),\xi)=0$ so that $\mathcal{D}$ is
integrable. From the Frobenious Theorem, we have a foliation whose leaves are
orthogonal to the vector field $\xi$. Finally, for any geodesic $\gamma(t)$ of
$\tilde{\nabla}$, the function $g(\xi,\gamma^{\prime}(t))$ is constant and the
leaves are thus totally geodesic. ∎

######  Theorem 3.2.

Let $(M,g)$ be a connected, simply-connected, orientable and complete
Riemannian manifold. Then the following two are equivalent:

  1. 1.

$(M,g)$ is a regular cohomogeneity one Riemannian manifold.

  2. 2.

There exists a complete linear connection $\tilde{\nabla}$ and a vector field
$\xi$ with $g(\xi,\xi)=1$, such that,

$\displaystyle\tilde{\nabla}\tilde{R}$
$\displaystyle=0,\quad\tilde{\nabla}\tilde{T}=0,\quad\tilde{\nabla}\xi=0,\quad$
(3.1) $\displaystyle\tilde{\nabla}_{X}g$
$\displaystyle=0,\quad\tilde{T}(X,Y)\in\mathcal{D},\quad\forall\,X,Y\in\mathcal{D},$

where $\mathcal{D}=\\{X:\>g(X,\xi)=0\\}$ is an integrable distribution whose
leaves are embedded and closed, and $\tilde{R}$ and $\tilde{T}$ are the
curvature and torsion of $\tilde{\nabla}$.

###### Proof.

The proof of (1) implies (2). Let $G$ be a Lie group of isometries of $M$
acting on $M$ such that the orbits define a cohomogeneity one foliation.

Since $M$ is orientable, any orbit admits a unit normal vector field $\xi$. We
consider a point $p\in M$. It is well known that:

######  Lemma 3.3.

For any $p\in M$, the geodesic $\mathrm{exp}(t\xi_{p})$ intersects all the
leaves of the foliation orthogonally.

Now we consider the differentiable map $\varphi\colon{\mathbb{R}}\times(G\cdot
p)\longrightarrow M$ defined by

$\varphi_{t}(g\cdot p)=\exp_{g\cdot p}(t\xi_{g\cdot p}).$

From the lemma above, this map is surjective. On the other hand, since $G$
sends geodesics to geodesics and $g\cdot\xi_{q}=\xi_{g\cdot q}$ because
$G$-orbits are principal, we have that

$g\cdot\varphi_{t}(q)=\varphi_{t}(g\cdot q),\qquad\forall q\in G\cdot p.$

In particular, since all orbits are principal (and thus, all isotropy groups
are conjugate),

$\mathrm{Isot}_{G}(p)=\mathrm{Isot}_{G}(\varphi_{t}(p)),$

where the isotropy $\mathrm{Isot}_{G}(q)$ of $q\in M$ is the Lie subgroup
$\\{h\in G:\>h\cdot q=q\\}$.

The vector field $\xi$ initially defined on $G\cdot p$ only, can now be
extended to $M$ by simply taking derivatives of $\partial\varphi/\partial t$,
$t\in\mathbb{R}$. This is well defined since if we had
$q=\varphi_{t_{1}}(p)=\varphi_{t_{2}}(g\cdot p)$ with

$\frac{\partial\varphi_{t}(p)}{\partial
t}(t_{1})=-\frac{\partial\varphi_{t}(g\cdot p)}{\partial t}(t_{2}),$ (3.2)

then

$\varphi_{\frac{t_{1}+t_{2}}{2}}(p)=\varphi_{\frac{t_{1}+t_{2}}{2}}(g\cdot
p)=g\cdot\varphi_{\frac{t_{1}+t_{2}}{2}}(p),$

that is,
$g\in\mathrm{Isot}_{G}(\varphi_{\frac{t_{1}+t_{2}}{2}}(p))=\mathrm{Isot}_{G}(p)$.
But the uniqueness of geodesics would make (3.2) impossible for $g\cdot p=p$.

The orbits of $\xi$ are all of the same type. There is thus a group
$A=\mathbb{R}$ or $\mathbb{S}^{1}$ and we have a transitive action on the left

$\displaystyle(A\times G)\times M$ $\displaystyle\longrightarrow M$
$\displaystyle((\varphi_{t},g),p)$ $\displaystyle\longmapsto\varphi_{t}(g\cdot
p)=g\cdot\varphi_{t}(p).$

Let $\mathrm{Isot}(p)=\\{(\varphi_{t},g)\in A\times G:\>\varphi_{t}(g\cdot
p)=p\\}$ denote the isotropy of $p$ by the action of $A\times G$. The subgroup
$D=\\{t\in A:\>\varphi_{t}(p)\in G\cdot p\\}\subset A$ is closed and it is
generated by the smallest $t_{0}$ such that $\varphi_{t_{0}}(p)=g\cdot p$.
Therefore,
$\mathrm{Isot}(p)=\mathcal{L}((\varphi_{t_{0}},g^{-1}))+\mathrm{Isot}_{G}(p)$
where $\mathcal{L}((\varphi_{t_{0}},g^{-1}))$ is the subgroup generated by
$(\varphi_{t_{0}},g^{-1})$ in $A\times G$ and $\mathrm{Isot}_{G}(p)$ is the
isotropy group of $p$ by the action of $G$. As
$\mathcal{L}((\varphi_{t_{0}},g^{-1}))$ is discrete, the Lie algebra
$\mathrm{Isot}(p)$ is the Lie algebra $\mathfrak{h}$ of
$\mathrm{Isot}_{G}(p)$. We have that $[\mathfrak{h},\mathfrak{a}]=0$ because
$G$ leaves $\xi$ invariant.

Since each orbit is a reductive homogeneous manifold, then we have a
connection in every leaf such that the geodesics of the connection are the
curves $\exp(tX)\cdot p$ for every $X\in\mathfrak{m}$ and $p\in M$. Moreover,
there is a decomposition
$\mathfrak{a}+\mathfrak{g}=\mathfrak{h}+(\mathfrak{a}+\mathfrak{m})$ and the
subspace $\mathfrak{a}+\mathfrak{m}$ is
$\operatorname{Ad}(\mathrm{Isot}_{G}(p))$-invariant. We have that
$\varphi_{t_{0}}\circ L_{g^{-1}}$ is a global diffeomorphism preserving $\xi$
and the horizontal distribution defined by the connection above, i. e., the
subspace $\mathfrak{m}$ is $\operatorname{Ad}(\varphi_{t_{0}}\circ
L_{g^{-1}})$-invariant. It follows that $M$ is a reductive homogeneous
manifold, and the canonical connection $\tilde{\nabla}$ associated with the
reductive decomposition above (see [KN63, Vol. 2, Ch. X]) satisfies

$\tilde{\nabla}\tilde{R}=0,\quad\tilde{\nabla}\tilde{T}=0.$

Since the Lie group $A\times G$ leaves $\xi$ invariant then
$\tilde{\nabla}\xi=0$ (see [CC19, p. 39, Prop. 1.4.15]).

To show that $\tilde{\nabla}_{X}g=0$ for all $X\in\mathcal{D}$, we consider,
for $X\in\mathfrak{m}$, the curve $\gamma(t)=\exp(tX)\cdot p$. The parallel
transport along the curve $\gamma$ is the linear map given by the differential
$(L_{\exp(tX)})_{*}\colon T_{p}M\longrightarrow T_{\gamma(t)}M$ ([KN63, Ch. X,
Cor. 2.5]). As this map preserves the metric, we have that
$\tilde{\nabla}_{\gamma^{\prime}}g=0$. Finally, if we take $X$, $Y$ two vector
fields tangent to the leaves, since $\tilde{\nabla}_{X}g=0$, we have
$0=g(\tilde{\nabla}_{X}Y,\xi)+g(Y,\tilde{\nabla}_{X}\xi)=g(\tilde{\nabla}_{X}Y,\xi)$.
Hence the leaves are totally geodesic with respect $\tilde{\nabla}$ and
therefore $g(\tilde{T}(X,Y),\xi)=0$.

The proof of (2) implies (1).

The completeness of $\tilde{\nabla}$ together with conditions
$\tilde{\nabla}\tilde{R}=0$, $\tilde{\nabla}\tilde{T}=0$ and
$\tilde{\nabla}\xi=0$ implies (see [CC22, p. 4, Thm. 2.2]) that $M$ is a
homogeneous manifold $M=\bar{G}/\bar{H}$ ( non-necessarily Riemannian), with
reductive decomposition
$\bar{\mathfrak{g}}=\bar{\mathfrak{h}}+\bar{\mathfrak{m}}$, $\tilde{\nabla}$
is the canonical connection and the vector field $\xi$ is $\bar{G}$-invariant.
The Lie group $\bar{G}$ that acts transitively and effectively on
$M=\bar{G}/\bar{H}$ is the Lie group of transvections of $\tilde{\nabla}$ (see
Theorem 2.3). Since $\tilde{\nabla}_{X}g=0$, for $X\in\mathcal{D}$, $g$ is
invariant along parallel transport by curves orthogonal to $\xi$.

Let $\bar{\mathfrak{g}}$ be the Lie algebra of $\bar{G}$ with reductive
decomposition $\bar{\mathfrak{g}}=\bar{\mathfrak{h}}+\bar{\mathfrak{m}}$. We
can identify $\bar{\mathfrak{m}}$ with $T_{e\bar{H}}M$ via,
$\psi(X)=\frac{d}{dt}\Big{|}_{t=0}\exp(tX)\cdot e\bar{H}$. As a consequence of
this identification, we can endow $\bar{\mathfrak{m}}$ with a Riemannian
metric structure $g$ given by $g(X,Y)=g|_{e\bar{H}}(\psi X,\psi Y)$. Then, we
can consider the subspace

$\mathfrak{m}=\\{X\in\bar{\mathfrak{m}}:\>g(X,\psi^{-1}\xi)=0\\}\subset\bar{\mathfrak{m}}$

and the subalgebra

$\mathfrak{h}=\mathrm{span}\\{[X,Y]_{\bar{\mathfrak{h}}}:\>X,Y\in\mathfrak{m}\\}\subset\bar{\mathfrak{h}}.$

The algebra $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ is a Lie subalgebra of
$\bar{\mathfrak{g}}$, and $\mathfrak{h}$ is a Lie subalgebra of
$\bar{\mathfrak{h}}$. By [KN63, Vol. 2, p. 193, Cor. 2.5], the geodesics of
$\tilde{\nabla}$ starting from $e\bar{H}$ have the expression $\exp(tX)\cdot
e\bar{H}$ for all $X\in\mathfrak{m}$ and all $t\in{\mathbb{R}}$. Indeed,
parallel transport along such a geodesic is given by the differential map
$(L_{\exp(tX)})_{*}$. Since $\tilde{\nabla}_{X}g=0$, the transvection map
$L_{\exp(tX)}$ is an isometry of $M$. From
$[X,Y]_{\bar{\mathfrak{h}}}=\tilde{R}_{XY}=[\tilde{\nabla}_{X},\tilde{\nabla}_{Y}]-\tilde{\nabla}_{[X,Y]}$
and $\tilde{\nabla}_{X}g=0$, we conclude $\tilde{R}_{XY}\cdot g=0$. In
particular, the transvection map $L_{\exp(tH)}$ associated with each
$H\in\mathfrak{h}$ is an isometry. Moreover, as $\tilde{R}_{XY}\cdot\xi=0$ and
$\tilde{R}_{XY}\cdot g=0$, then $\mathfrak{m}$ is an $\mathfrak{h}$-module of
$\mathfrak{g}$ and every leaf is a reductive Riemannian homogeneous manifold.

Let $G$ be the connected Lie subgroup of $\bar{G}$ associated with
$\mathfrak{g}\subset\mathfrak{\tilde{g}}$. According to the above paragraph,
$G$ is a group of isometries of $M$.

######  Lemma 3.4.

$M$ is a cohomogeneity one foliation of principal orbits associated with the
action of $G$.

###### Proof.

The Lie group $G$ preserves every leaf since its action preserves $g$ and
$\xi$, and $G$ is connected. Since $M$ is a connected homogeneous space, any
two points $p$, $q\in M$ can be joined by a broken geodesic of
$\tilde{\nabla}$ such that each geodesic is an integral curve of $\xi$ or a
$\tilde{\nabla}$-geodesic in a leaf (we are using here connectedness of $M$).
Then, there exists $L_{g}=L_{g_{1}}\circ\varphi_{t_{1}}\circ\ \dots\ \circ
L_{g_{m}}\circ\varphi_{t_{m}}$ with $m\in\mathbb{N}$, $L_{g_{i}}\in G$ and
$\varphi$ the flow of $\xi$ such that $L_{g}p=q$. As $\xi$ is invariant for
every $L_{g_{i}}$, then $L_{g}=\varphi_{t_{1}+\ \dots\ +t_{m}}\circ
L_{g_{1}}\circ\ \dots\ \circ L_{g_{m}}$. The integral curves of $\xi$
intersect all orbits. Therefore, to prove that $G$ acts transitive on every
leaf, it is enough to prove this fact in only one leaf.

Let $N$ be the leaf of $e\bar{H}$. Since $N$ is a totally geodesic submanifold
of $M$, by Lemma. 3.1, then every point $p\in N$ is connected to $e\bar{H}$ by
a broken geodesic of $\tilde{\nabla}$. As $G$ is connected and locally
transitive on an open neighbourhood of $e\bar{H}$ (in the submanifold
topology), then $G$ acts transitively on $N$. Finally, every orbit is
principal since $\\{g\in G:\>g\cdot e\bar{H}=e\bar{H}\\}=\\{g\in
G:\>g\cdot\varphi_{t}(e\bar{H})=\varphi_{t}(e\bar{H})\\}$, for all
$t\in{\mathbb{R}}$ and $\xi$ intersects all orbits. ∎

This finishes the proof of Theorem 3.2. ∎

######  Definition 3.5.

A Riemannian manifold $(M,g)$ equipped with a unit vector field $\xi$ and a
linear connection $\tilde{\nabla}$ satisfying conditions (3.1) is called a
_cohomogeneity one Ambrose-Singer manifold_ (CO1-AS-manifold for short). In
this case, the $(1,2)$-tensor field $S=\nabla-\tilde{\nabla}$ is called
_cohomogeneity one structure_ , where $\nabla$ is the Levi-Civita connection.

######  Remark 3.6.

As it is well known, the same reductive homogeneous manifold may have
different descriptions as a quotient $G/H$ as well as different reductive
decompositions $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$. For instance,
Euclidean spheres can be regarded as homogeneous manifolds exactly in the
following ways (see [MS43] and [AHL23])

$\displaystyle\mathbb{S}^{n}$
$\displaystyle=\operatorname{SO}(n+1)/\operatorname{SO}(n),\quad\mathbb{S}^{2n+1}=\operatorname{U}(n+1)/\operatorname{U}(n),$
$\displaystyle\mathbb{S}^{2n+1}$
$\displaystyle=\operatorname{SU}(n+1)/\operatorname{SU}(n),$
$\displaystyle\mathbb{S}^{4n-1}$
$\displaystyle=\operatorname{Sp}(n)/\operatorname{Sp}(n-1),\quad\mathbb{S}^{4n-1}=\operatorname{Sp}(n+1)\operatorname{Sp}(1)/\operatorname{Sp}(n)\operatorname{Sp}(1),$
$\displaystyle\mathbb{S}^{4n-1}$
$\displaystyle=\operatorname{Sp}(n)\operatorname{U}(1)/\operatorname{Sp}(n-1)\operatorname{U}(1),$
$\displaystyle\mathbb{S}^{6}$
$\displaystyle=G_{2}/\operatorname{SU}(3),\quad\mathbb{S}^{7}=\operatorname{Spin}(7)/\operatorname{SU}(3),\quad\mathbb{S}^{15}=\operatorname{Spin}(9)/\operatorname{Spin}(7).$

Furthermore, for each of the quotient we may determine the set of all
$G$-invariant metrics. Together with the choice of the complement
$\mathfrak{m}$ to $\mathfrak{h}$, all these provide the collection of
homogeneous tensors $S$.

A similar situation happens in cohomogeneity one manifolds. For example, we
might consider a warped product $M=\mathbb{R}\times_{f}\mathbb{S}^{n}$ for a
smooth function $f:\mathbb{R}\longrightarrow\mathbb{R}^{+}$ built with any the
previous metrics on the sphere. The action of the different groups $G$
obviously provides the same (trivial) foliation.

One advantage of AS or CO1-AS structures is that they can be used to
distinguish different homogeneous or cohomogeneity one descriptions. With
respect to CA1-AS structures, their classification is done up to equivalence
in the following sense.

######  Definition 3.7.

Two regular cohomogeneity one manifolds $(M,g)$ and $(M^{\prime},g^{\prime})$
with the same group $G$ are called _isomorphic_ if and only if there exists a
$G$-equivariant isometry $f\colon(M,g)\longrightarrow(M^{\prime},g^{\prime})$.

######  Definition 3.8.

Two cohomogeneity one structures $S$ in $(M,g)$ and $S^{\prime}$ in
$(M^{\prime},g^{\prime})$ are called _isomorphic_ if and only if there exists
an isometry $f\colon(M,g)\longrightarrow(M^{\prime},g^{\prime})$ that maps $S$
to $S^{\prime}$.

######  Proposition 3.9.

Let $(M,g)$ and $(M^{\prime},g^{\prime})$ be two connected, simply-connected
and complete $G$-regular cohomogeneity one manifolds with cohomogeneity one
structures $S$ and $S^{\prime}$, respectively. If $S$ and $S^{\prime}$ are
isomorphic, then $(M,g)$ and $(M^{\prime},g^{\prime})$ are isomorphic.

###### Proof.

Let $f\colon(M,g)\longrightarrow(M^{\prime},g^{\prime})$ be an isometry
sending $S$ to $S^{\prime}$. Then $f$ is an affine map from
$\tilde{\nabla}=\nabla-S$ to
$\tilde{\nabla}^{\prime}=\nabla^{\prime}-S^{\prime}$ where $\nabla$ and
$\nabla^{\prime}$ are the Levi-Civita connections of $(M,g)$ and
$(M^{\prime},g^{\prime})$, respectively. We now consider the Lie group $G$
generated by Theorem 3.2 applied for the first of the manifolds. It also acts
on $M^{\prime}$ through $f$. We know from Theorem 2.3 that this Lie group is
generated by the global transvections of the connection $\tilde{\nabla}$. For
every global transvection $F$ of $\tilde{\nabla}$, then $f\circ F\circ f^{-1}$
is a transvection map of $\tilde{\nabla}^{\prime}$. Therefore, $f$ is a
$G$-equivariant map. ∎

######  Proposition 3.10.

Let $(M,g)$ and $(M^{\prime},g^{\prime})$ be two connected, simply-connected
and complete $G$-regular cohomogeneity one manifolds. If $(M,g)$ and
$(M^{\prime},g^{\prime})$ are isomorphic by
$f\colon(M,g)\longrightarrow(M^{\prime},g^{\prime})$ and $S$ is a
cohomogeneity one structure for $(M,g)$, then $f_{*}S$ is a cohomogeneity one
structure for $(M^{\prime},g)$ that is isomorphic to $S$.

###### Proof.

This proof is direct after the observation that $f$ is an affine
diffeomorphism between the connections $\tilde{\nabla}=\nabla-S$ and
$\tilde{\nabla}^{\prime}=\nabla^{\prime}-f_{*}S$, where $\nabla$ and
$\nabla^{\prime}$ are the Levi-Civita connections of $(M,g)$ and
$(M^{\prime},g^{\prime})$, respectively. ∎

##  4 Main result: local version

Let $\mathcal{G}$ be a Lie pseudo-group of differentiable transformations on a
manifold $M$ (we refer the reader to [Spi92] or [Acc21] for an exposition on
this topic). Given a point $p_{0}\in M$, we define $\mathcal{G}(p_{0})$ as the
set of transformations for which $p_{0}$ belongs to the domain, and
$\mathcal{G}(p_{0},p_{0})\subset\mathcal{G}(p_{0})$ the set of transformations
$f$ such that $f(p_{0})=p_{0}$. The quotient
$H(p_{0})=\mathcal{G}(p_{0},p_{0})/\sim$ with respect to the relation $f\sim
f^{\prime}\iff f|_{U}=f^{\prime}|_{U}$ for some neighbourhood $U$ of $p_{0}$,
is a Lie group (cf. [Acc21, Ch. 1]). We now consider a frame
$u_{0}\in\mathcal{L}(M)$ over the point $p_{0}$. We say that the action of
$\mathcal{G}$ on $M$ is _effective_ and _closed_ if the map

$\displaystyle H(p_{0})$
$\displaystyle\longrightarrow\operatorname{GL}(n,{\mathbb{R}})$ (4.1)
$\displaystyle f$ $\displaystyle\longmapsto u_{0}^{-1}\circ f_{*}\circ u_{0}$

is a monomorphism and its image $\mathbf{H}(u_{0})$ is closed, i. e., the
image $\mathbf{H}(u_{0})$ is a Lie subgroup of
$\operatorname{GL}(n,{\mathbb{R}})$. The morphism (4.1) will be called the
isotropy representation of $\mathcal{G}$ on $M$. The effectiveness and
closedness of this representation do not depend on the choice of $u_{0}$ (see
[CC22, Prop. 3.4]). An effective and closed action of $\mathcal{G}$ on $M$
naturally induces an action of $\mathcal{G}$ on the frame bundle
$\mathcal{L}(M)$:

$(f,u)\mapsto f_{*,\pi(u)}\circ u,\quad u\in\mathcal{L}(M),$

where $\mathbf{H}(u_{0})\subset\operatorname{GL}(n,{\mathbb{R}})$ represents
the isotropy group at $u_{0}$ for the Lie groupoid defined by the germs of
differentials of $\mathcal{G}$. It is called the _linear isotropy group_.

Since two local isometries $f$, $f^{\prime}$ such that
$f(p_{0})=f^{\prime}(p_{0})$ coincide on a neighbourhood of $p_{0}$ (i. e.,
$f\sim f^{\prime}$) if and only if their differentials at $p_{0}$ are equal,
we have that the action of a pseudo-group of local isometries of a Riemannian
manifold $(M,g)$ is always effective.

If the action is both effective and transitive, given
$u_{0}\in\mathcal{L}(M)$, the bundle

$\mathcal{P}(u_{0})=\\{f_{*}(u_{0}):\>f\in\mathcal{G}\\},$

is a reduction of the frame bundle to the subgroup $\mathbf{H}(u_{0})$. Under
these conditions, for an element $\varphi\in H(p_{0})$, we define

$\displaystyle\operatorname{Ad}_{\varphi}\colon T_{u_{0}}P(u_{0})$
$\displaystyle\longrightarrow T_{u_{0}}P(u_{0})$
$\displaystyle\frac{d}{dt}\Big{|}_{t=0}(\varphi_{t})_{*}(u_{0})$
$\displaystyle\longmapsto\frac{d}{dt}\Big{|}_{t=0}(\varphi\circ\varphi_{t}\circ\varphi^{-1})_{*}(u_{0})$

where $\varphi_{t}\in\mathcal{G}$, and $t$ belongs to a certain interval
$(-\varepsilon,\varepsilon)$.

Consider a Lie pseudo-group $\mathcal{G}$ whose action is transitive,
effective and closed. In this context, we say that the action is _reductive_
(see [CC22, Def. 3.5]) if and only if the tangent space at $u_{0}$, admits a
decomposition $T_{u_{0}}P(u_{0})=\mathfrak{h}+\mathfrak{m}$, where
$\mathfrak{h}$ is the Lie algebra associated with $\mathbf{H}(u_{0})$ and
$\mathfrak{m}$ is a $\operatorname{Ad}(H(p_{0}))$-invariant subspace.

######  Theorem 4.1 ([CC19, 3.1.13]).

Let $(M,g)$ be a Riemannian manifold such that a pseudo-group
$\mathcal{G}\subset\mathrm{Isom}_{loc}(M,g)$ is acting transitively. Then, the
action of $\mathcal{G}$ is effective, closed and reductive.

######  Definition 4.2.

A Riemannian manifold $(M,g)$ is _locally cohomogeneity one_ if there exists a
pseudo-group of local isometries $\mathcal{G}$ acting on $M$ in such a way
that $M/\mathcal{G}$ is connected and every orbit is an embedded and closed
submanifold of codimension one.

A locally cohomogeneity one Riemannian manifold $(M,g)$ is said to be a
_regular_ if for every two frames $u_{0}$ and $v_{0}$ of $M$, the closed
subgroups $\mathbf{H}(u_{0})$ and $\mathbf{H}(v_{0})$ are conjugate in
$\operatorname{GL}(n,{\mathbb{R}})$. This is equivalent to the fact that all
orbits are principal.

######  Theorem 4.3.

Let $(M,\,g)$ be an orientable and connected Riemannian manifold. The
following two are equivalent:

  1. 1.

$(M,g)$ is a regular locally cohomogeneity one Riemannian manifold.

  2. 2.

There exists a connection $\tilde{\nabla}$ and a unit vector field
$\xi\in\mathfrak{X}(M)$ such that

$\displaystyle\tilde{\nabla}\tilde{R}$
$\displaystyle=0,\quad\tilde{\nabla}\tilde{T}=0,\quad\tilde{\nabla}\xi=0,\quad$
(4.2) $\displaystyle\tilde{\nabla}_{X}g$
$\displaystyle=0,\quad\tilde{T}(X,Y)\in\mathcal{D},\quad$
$\displaystyle\forall\,X,Y\in\mathcal{D},$

where $\mathcal{D}=\\{X\in\mathfrak{X}(M):\>g(X,\xi)=0\\}$, and $\tilde{R}$,
$\tilde{T}$ are the curvature and torsion of $\tilde{\nabla}$, respectively.
Furthermore, the maximal integral leaves of the distribution $\mathcal{D}$
(which is integrable according to Lemma 3.1) are embedded and closed.

###### Proof.

The main ideas of the proof are summarized in Theorem 3.2.

Proof of (2) implies (1). From [CC22, Thm. 3.7], we know that there exists
reductive Lie pseudo-group $\bar{\mathcal{G}}$ that acts transitively on
$(M,g)$ and leaves $\xi$ invariant. Moreover, this pseudo-group
$\bar{\mathcal{G}}$ can be identified with the Lie pseudo-group of local
transvections of $\tilde{\nabla}$. We thus consider the subset $\mathcal{G}$
of $\bar{\mathcal{G}}$ consisting of local transvections associated with
parallel transports by curves in the leaves. Since $\tilde{\nabla}\xi=0$ and
$\tilde{\nabla}_{X}g=0$ for all $X\in\mathcal{D}$, these local transvections
preserve the vector field $\xi$, the Riemannian tensor $g$ and thus the
codimension one distribution $\mathcal{D}$.

######  Lemma 4.4.

The subset $\mathcal{G}$ is a Lie pseudo-group.

###### Proof.

This follows directly from the definition of local transvection, i. e., it is
a local diffeomorphism that preserves every holonomy bundle. ∎

######  Lemma 4.5.

For any $f\in\mathcal{G}$ and any point $q\in\mathrm{dom}(f)$, both $q$ and
$f(q)$ belong to the same leaf. Furthermore, the action of $\mathcal{G}$ on
every leaf is transitive.

###### Proof.

The local transvection $f\in\mathcal{G}$ is associated to a
$\tilde{\nabla}$-parallel transport along a broken geodesic in one leaf $O$,
and that leaf is thus invariant. But the point $q\in\mathrm{dom}(f)$ needs not
belong to that leaf. In that case, there exists $t_{0}\in{\mathbb{R}}$ such
that $\varphi_{t_{0}}(q)\in O$, where $\varphi_{t}$ is the flow of $\xi$. The
vector field is geodesic for $\tilde{\nabla}$ and the differential of the flow
is the $\tilde{\nabla}$-parallel transport along $\xi$ and send leaves to
leaves. As $f$ preserves $\xi$, the points $q$ and $f(q)$ belong to the same
leaf $\varphi_{-t_{0}}(O)$.

The transitiveness comes from the existence of (broken) geodesics connecting
any two point of any leaf. We consider the element of $\mathcal{G}$ defined by
that geodesic. ∎

Proof of (1) implies (2). The procedure is to find a connection
$\tilde{\nabla}$ satisfying the first row of (4.2). Again, from [CC22, Thm.
3.7], it is sufficient to show that there exists a Lie pseudo-group
$\bar{\mathcal{G}}$ whose action is transitive (Lemma 4.6), effective (Lemma
4.7), closed (Lemma 4.8) and reductive (Lemma 4.9) on the leaves. Afterwards,
we prove the second row of (4.2).

Since $M$ is orientable, any orbit admits a unit normal vector field $\xi$. As
$M/G$ is connected, in analogy with Theorem 3.2, the vector field $\xi$ is
globally defined on $M$. That is, for every orbit $\mathcal{G}\cdot p_{o}$
there exists an $\varepsilon>0$ such that the flow
$\varphi\colon(-\varepsilon,\varepsilon)\times U\longrightarrow M$, where $U$
is an open set containing $\mathcal{G}\cdot p_{0}$, is defined. Let
$\mathcal{G}$ be the Riemannian pseudo-group acting with cohomogeneity one on
$M$. We define the pseudo-group $\bar{\mathcal{G}}$ generated by $\mathcal{G}$
and $\mathcal{A}$, the latter being the pseudo-group generated by flows of
unit local vector fields orthogonal to the leaves.

######  Lemma 4.6.

The Lie pseudo-group $\bar{\mathcal{G}}$ acts transitively on $M$.

###### Proof.

Let $p$ and $q$ be two points. If $p$ and $q$ belong to the same leaf, then
there exists a local transformation $f\in\mathcal{G}$ such that $f(p)=q$. In
general, it is easy to see that $A=\\{x\in
N:\>f(p)=x,\,f\in\bar{\mathcal{G}}\\}$ is open and closed. Since $M$ is
connected, we get the result. ∎

As $\xi$ is invariant by the isometries of $\mathcal{G}$, then every geodesic
flow orthogonal to the leaves commutes with the local isometries of
$\mathcal{G}$ by Lemma 3.3. Actually, every transformation
$\bar{f}\in\bar{\mathcal{G}}$ can be written as $\bar{f}=\varphi\circ f$,
where $\varphi\in\mathcal{A}$ and $f\in\mathcal{G}$.

######  Lemma 4.7.

The Lie pseudo-group $\bar{\mathcal{G}}$ is effective.

###### Proof.

According to the first paragraph of Section 4, let $H_{\mathcal{G}}(p_{0})$ be
the quotient space of elements $f\in\mathcal{G}$ fixing a point $p_{0}$. Thus,
analogously, the space of elements $\bar{f}\in\bar{\mathcal{G}}$ fixing a
point $p_{0}$ is given by

$H(p_{0})=H_{\mathcal{G}}(p_{0})+\mathcal{L}\left(\varphi_{t_{0}}\circ
f^{-1}\right),$

where $\mathcal{L}(\varphi_{t_{0}}\circ f^{-1})$ is the discrete Lie group
generated by the element $\varphi_{t_{0}}\circ f^{-1}$ and $t_{0}$ is the
smallest $t_{0}>0$ such that $\varphi_{t_{0}}(p_{0})=f(p)$ for some
$f\in\mathcal{G}$. Note that, the set
$D=\\{\varphi_{k\,t_{0}}(p_{0}):\>k\in\mathbb{Z}\\}$ is the intersection of
$\\{\varphi_{t}(p_{0}):\>t\in{\mathbb{R}}\\}$ and $\mathcal{G}\cdot p_{0}$,
and both subsets are closed in $N$.

Note that $H_{\mathcal{G}}(p_{0})$ is effective because it arises from a
Riemannian Lie pseudo-group. The question is, if $(\varphi_{t_{0}}\circ
f^{-1})_{*}(p_{0})=Id_{T_{p_{0}}M}$, then $\varphi_{t_{0}}\circ f^{-1}$ is the
identity map in an open neighbourhood $U$ of $M$. There is an open
neighbourhood $U$ such that both domain and image of $\varphi_{t_{0}}\circ
f^{-1}$ are contained in $U$. This map is $\mathcal{G}(U)$-equivariant by
local isometries $h\in\mathcal{G}$ with domain $U$, i. e., for every
$h\in\mathcal{G}(U)$ there exits $h^{\prime}\in\mathcal{G}(U)$ such that
$\varphi_{t_{0}}\circ f^{-1}\circ h=h^{\prime}\circ\varphi_{t_{0}}\circ
f^{-1}$. For every local Killing vector field $X^{*}$, its flow comes from a
family $F_{t}$ of isometries with $t\in I$, a closed interval, and
$F_{t}(p_{0})$ is an integral curve of $X^{*}$. Due to the equivariance, there
exists a local Killing vector field $Y^{*}$ and a family $H_{t}$ of isometries
such that,

$\varphi_{t_{0}}\circ f^{-1}\circ F_{t}=H_{t}\circ\varphi_{t_{0}}\circ
f^{-1}.$

When applied to $p_{0}$, we get

$\varphi_{t_{0}}\circ f^{-1}\circ F_{t}(p_{0})=H_{t}(p_{0}),$

i. e., $(\varphi_{t_{0}}\circ f^{-1})_{*}(X^{*}(p_{0}))=Y^{*}(p_{0})$.
Necessarily, $X^{*}=Y^{*}$; therefore $H_{t}=F_{t}$. Consequently, by applying
this argument for every local Killing vector field, we conclude that
$(\varphi_{t_{0}}\circ f^{-1})$ fixes integral curves of Killing vector fields
with the initial point $p_{0}$. In these conditions $\varphi_{t_{0}}\circ
f^{-1}=Id_{U}$. ∎

######  Lemma 4.8.

The action of the Lie pseudo-group $\bar{\mathcal{G}}$ is closed.

###### Proof.

The image of $H(p_{0})$ by the isotropy map (4.1) is

$\mathbf{H}(u_{0})=\mathbf{H}_{\mathcal{G}}(u_{0})+\mathcal{L}(u_{0}{}^{-1}\circ(\varphi_{t_{0}}\circ
f^{-1})_{*}\circ u_{0})$

where $\mathbf{H}_{\mathcal{G}}(u_{0})$ is the image of
$H_{\mathcal{G}}(p_{0})$. On the one hand, as every leaf $\mathcal{G}\cdot p$
is a locally homogeneous Riemannian manifold and its action is reductive (see
[Tri92]), then $\mathbf{H}_{\mathcal{G}}(u_{0})$ is equal to the holonomy
group of one AS-connection. Consequently, it is closed in
$\operatorname{GL}(n,{\mathbb{R}})$.

On the other hand, let $A=u_{0}{}^{-1}\circ(\varphi_{t_{0}}\circ
f^{-1})_{*}\circ u_{0}\in\operatorname{GL}(n,{\mathbb{R}})$. Then, the space
$\mathcal{L}(A)$ consists of the powers $A^{k}$ for $k\in\mathbb{Z}$. The
matrix
$A\in\operatorname{GL}(n,{\mathbb{R}})\subset\operatorname{GL}(n,{\mathbb{C}})$
diagonalizes as $A=P\cdot D\cdot P^{-1}$. We consider a convergent sequence
$\\{A^{k_{n}}\\}$ in $\operatorname{GL}(n,{\mathbb{C}})$. Its eigenvalues
cannot have a norm different from 1; otherwise the limit would have a singular
eigenvalue. If the eigenvalues are of the form $e^{iv}$ with $v$ irrational,
$\\{A^{k_{n}}\\}$ would not converge. Then the exponents of the eigenvalues
are rational and the group $\mathcal{L}(A)$ is cyclic, finite and closed.
Finally, since $\mathbf{H}_{\mathcal{G}}(u_{0})$ and $\mathcal{L}(A)$ are
closed, it follows that $\mathbf{H}(u_{0})$ is closed. ∎

As $\mathcal{L}((\varphi_{t_{0}}\circ g^{-1}))$ is discrete, the Lie algebra
of $H(p_{0})$ is equal to the Lie algebra $\mathfrak{h}$ of
$\mathrm{H}_{\mathcal{G}}(p_{0})$.

######  Lemma 4.9.

The Lie pseudo-group $\bar{\mathcal{G}}$ acts reductively.

###### Proof.

Given a point $p_{0}\in M$ and a frame $u_{0}$ on it, we consider

$\mathcal{P}(u_{0})=\left\\{f_{*}\circ
u_{0}\in\mathcal{L}(M):\>f\in\bar{\mathcal{G}}\right\\}$

and $\mathcal{P}(u_{0},\mathcal{G}\cdot p_{0})=\\{f_{*}\circ
u_{0}\in\mathcal{L}(\mathcal{G}\cdot p_{0}):\>f\in\mathcal{G}\\}$, the
reduction of the frame bundle of $M$ and the orbit $\mathcal{G}\cdot p_{0}$,
respectively. Since every locally homogeneous Riemannian manifold is reductive
(see Theorem 4.1 above), we have the reductive decomposition in the leaves:
$T_{u_{0}}\mathcal{P}(u_{0},\mathcal{G}\cdot
p_{0})=\mathfrak{h}+\mathfrak{m}$, where $\mathfrak{m}$ is an
$\operatorname{Ad}(H_{\mathcal{G}}(p_{0}))$-invariant subspace.

We consider the decomposition
$T_{u_{0}}P(u_{0})=\mathfrak{h}+(\mathfrak{a}+\mathfrak{m})$, where

$\mathfrak{a}=\left\\{\frac{d}{dt}\Big{|}_{t=0}(\varphi)_{*}\circ
u_{0}:\>\varphi\in\mathcal{A}\right\\}.$

As every unit vector field $\xi$ orthogonal to the leaves is invariant under
the local transformations of $\mathcal{G}$, the flow of $\xi$ commutes with
every $f\in\mathcal{G}$. Consequently, $\mathfrak{a}$ is
$\operatorname{Ad}(\mathrm{H}_{\mathcal{G}}(p_{0}))$-invariant and
$\operatorname{Ad}(\varphi_{t_{0}}\circ L_{g^{-1}})$-invariant. Finally, as
$\varphi_{t_{0}}\circ L_{g^{-1}}$ preserves $\xi$ and the horizontal
distribution defined by the connection above, the subspace $\mathfrak{m}$ is
$\operatorname{Ad}(\varphi_{t_{0}}\circ L_{g^{-1}})$-invariant. ∎

Therefore, from [CC22, Thm. 3.7], there exists a global connection
$\tilde{\nabla}$ such that

$\tilde{\nabla}\tilde{R}=0,\quad\tilde{\nabla}\tilde{T}=0.$

Finally, the proof of

$\tilde{\nabla}\xi=0,\quad\tilde{\nabla}_{X}g=0,\quad\tilde{T}(X,Y)\in\mathcal{D}$

is analogous to the corresponding proof in Theorem 3.2. ∎

We also have the following properties of CO1-AS-manifolds.

######  Proposition 4.10.

Let $(M,g,\tilde{\nabla})$ be a CO1-AS-manifold and let
$S=\nabla-\tilde{\nabla}$ be the corresponding cohomogeneity one structure
tensor.

  1. 1.

For every $X\in\mathcal{D}$, we have $\tilde{\nabla}_{X}S=0$.

  2. 2.

For every $X\in\mathcal{D}$, $S_{X}\cdot g=0$.

  3. 3.

For every $X$, $Y\in\mathcal{D}$ then
$g(S_{X}Y,\xi)=g(\nabla_{X}Y,\xi)=g(\mathrm{II}(X,Y),\xi)$, where
$\mathrm{II}$ is the second fundamental form of the leaves.

  4. 4.

We have that $S_{\xi}\xi=0$.

###### Proof.

1. Since $\tilde{\nabla}_{X}g=0$, every parallel transport defined by $\tilde{\nabla}$ along curves belonging to one leaf preserves the Levi-Civita connection and the connection $\nabla$. Therefore, they also preserve the difference tensor $S=\nabla-\tilde{\nabla}$, that is, $\tilde{\nabla}_{X}S=0$.

2. From $\tilde{\nabla}_{X}g=0$ and $\nabla g=0$, we have $(\nabla-\tilde{\nabla})_{X}g=S_{X}g=0$.

3. First, $g(S_{X}Y,\xi)=g((\nabla-\tilde{\nabla})_{X}Y,\xi)=g(\nabla_{X}Y,\xi)-g(\tilde{\nabla}_{X}Y,\xi)$. Finally, from $\tilde{\nabla}_{X}g=0$, this is equal to $g(\nabla_{X}Y,\xi)+g(Y,\tilde{\nabla}_{X}\xi)=g((\nabla_{X}Y)^{\perp},\xi)=g(\mathrm{II}(X,Y),\xi)$.

4. It is a consequence of the fact that $\xi$ is a geodesic vector field and $\tilde{\nabla}\xi=0$. ∎

##  5 Decomposition of cohomogeneity one Structures

We now explore the infinitesimal models associated with CO1-AS manifolds as
well as their classification.

Let $(V,g)$ be a vector space of dimension $n+1$, $n\in\mathbb{N}$, endowed
with a positive definite inner product $g$ and a unit vector $\xi$. Let

$\tilde{R}\colon V\wedge V\longrightarrow\mathrm{End}(V),\quad\tilde{T}\colon
V\longrightarrow\mathrm{End}(V),$

be two linear homomorphisms. We say that $(\tilde{R},\tilde{T})$ is an
infinitesimal cohomogeneity one model if it satisfies

$\displaystyle\tilde{T}_{X}Y+\tilde{T}_{Y}X=0,$
$\displaystyle\tilde{R}_{XY}Z+\tilde{R}_{YX}Z=0,$
$\displaystyle\tilde{R}_{XY}\cdot\tilde{T}=\tilde{R}_{XY}\cdot\tilde{R}=0,$
$\displaystyle\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}\tilde{R}_{XY}Z+\tilde{T}_{\tilde{T}_{X}Y}Z=0,$
$\displaystyle\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}\tilde{R}_{\tilde{T}_{X}YZ}=0,$
$\displaystyle\tilde{R}_{XY}\cdot\xi=0,\quad$

and, for all $X$, $Y\in D=\mathcal{L}(\xi)^{\perp}$,

$\displaystyle\tilde{R}_{XY}\cdot g=0,\quad\forall X,Y\in D,$
$\displaystyle\tilde{R}_{XY}\cdot S=0,\quad\forall X,Y\in D,$
$\displaystyle\tilde{T}_{X}Y\in D,\quad\forall X,Y\in D,$

where $\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}$ is the
cyclic sum, and the dot stands for the action of $\tilde{R}_{XY}$ on $V$ as a
derivation.

Two CO1-AS infinitesimal models $(V,\tilde{T},\tilde{R},\xi,g,S)$,
$(V,\tilde{T}^{\prime},\tilde{R}^{\prime},\xi^{\prime},g^{\prime},S^{\prime})$
are said to be isomorphic if there is an isometry

$f\colon(V,g)\longrightarrow(V^{\prime},g^{\prime})$

such that,

$f\,\tilde{R}=\tilde{R}^{\prime},\quad f\,\tilde{T}=\tilde{T}^{\prime},\quad
f(\xi)=\xi,\quad f\,S=S^{\prime}.$

######  Remark 5.1.

In particular, at any point $p$ of a CO1-AS manifold $M$ there is an
infinitesimal cohomogeneity one model by taking $V=T_{p}M$, $g_{p}$,
$\xi_{p}$, $\tilde{R}_{p}$ and $\tilde{T}_{p}$. According to Theorem 4.3, two
points $p$, $p^{\prime}$ in the same leaf of the foliation of $M$ define
isomorphic infinitesimal cohomogeneity one models.

Given an infinitesimal cohomogeneity one model, we define

$\mathcal{S}(V)=\\{S\in V^{*}\otimes\mathfrak{gl}(V):\>S_{X}\cdot
g=0,S_{\xi}\xi=0,\,\forall\,X\in D\\}$

as the _space of infinitesimal cohomogeneity one structures_. From the
decomposition $V=U\oplus D$ as irreducible $\operatorname{SO}(n)$-submodules,
$U=\mathcal{L}(\xi)$, $D=U^{\perp}$, we have

$\mathcal{S}(V)=\mathcal{S}_{D}(V)+\mathcal{S}_{U}(V)$

with

$\displaystyle\mathcal{S}_{D}(V)$ $\displaystyle{}=\\{S\in
D^{*}\otimes\mathfrak{gl}(V):\>S_{X}\cdot g=0,\forall\,X\in D\\}\simeq
D^{*}\otimes\mathfrak{so}(V)=D^{*}\otimes V\wedge V,$
$\displaystyle\mathcal{S}_{U}(V)$ $\displaystyle{}=\\{S\in
U^{*}\otimes\mathfrak{gl}(V):\>S_{\xi}\xi=0\\}.$

We now explore each of these $\operatorname{SO}(n)$-submodules. For that, as
usual, we can identify spaces and their duals through the metric $g$.

We begin with $\mathcal{S}_{D}(V)$. It can be decomposed in
$\operatorname{SO}(n)$-submodules as

$\displaystyle\mathcal{S}_{D}(V)$ $\displaystyle{}=D^{*}\otimes V^{*}\wedge
V^{*}$ $\displaystyle{}=D^{*}\otimes D^{*}\wedge D^{*}+D^{*}\otimes
U^{*}\otimes D^{*},$

decomposition that is denoted by

$\displaystyle\mathcal{T}(V)$ $\displaystyle{}=D^{*}\otimes D^{*}\wedge
D^{*},$ $\displaystyle\mathcal{II}(V)$ $\displaystyle{}=D^{*}\otimes
U^{*}\otimes D^{*}.$

As we will see when we understand these structures coming from a CO1-AS
manifold, this notation is motivated by the fact that $\mathcal{T}(V)$ is the
space of all possible infinitesimal homogeneous structures tensors of the
leaves and $\mathcal{II}(V)$ is the space of all possible second fundamental
forms of the leaves at each point. The decomposition of these modules into
irreducible submodules can be derived from the results in the literature. They
are as follows.

######  Proposition 5.2.

The submodule $\mathcal{T}(V)$ decomposes as
$\operatorname{SO}(n)$-irreducible submodules as,

$\mathcal{T}(V)=\mathcal{T}_{1}(V)+\mathcal{T}_{2}(V)+\mathcal{T}_{3}(V)$

with explicit expressions,

$\displaystyle\mathcal{T}_{1}(V)$
$\displaystyle=\Bigl{\\{}S\in\mathcal{T}(V):\>S_{XYZ}=g(X,Y)\theta(Z)-g(X,Z)\theta(Y),\,\theta\in
D^{*},\,X,Y,Z\in D\Bigr{\\}},$ $\displaystyle\mathcal{T}_{2}(V)$
$\displaystyle=\Bigl{\\{}S\in\mathcal{T}(V):\>\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}S_{XYZ}=0,\,c_{12}(S)=0,\,X,Y,Z\in
D\Bigr{\\}},$ $\displaystyle\mathcal{T}_{3}(V)$
$\displaystyle=\Bigl{\\{}S\in\mathcal{T}(V):\>S_{XYZ}+S_{YXZ}=0,\,X,Y,Z\in
D\Bigr{\\}},$

where $c_{12}(S)(Z)=\sum_{i=1}^{n}S_{e_{i}e_{i}Z}$ for any orthonormal basis
$\\{e_{1},\ \ldots\ ,e_{n}\\}$ of $D$.

###### Proof.

The proof follows directly from [TV83, Thm. 3.1] applied to our notation. ∎

######  Proposition 5.3.

The submodule $\mathcal{II}(V)$ decomposes as
$\operatorname{SO}(n)$-irreducible submodules as,

$\mathcal{II}(V)=\mathcal{II}_{1}(V)+\mathcal{II}_{2}(V)+\mathcal{II}_{3}(V)$

with explicit expressions,

$\displaystyle\mathcal{II}_{1}(V)$
$\displaystyle=\Bigl{\\{}S\in\mathcal{II}(V):\>S_{XYZ}=\lambda
g(X,Z)g(Y,\xi),\,\lambda\in{\mathbb{R}},\,X,Y,Z\in D\Bigr{\\}},$
$\displaystyle\mathcal{II}_{2}(V)$
$\displaystyle=\Bigl{\\{}S\in\mathcal{II}(V):\>S_{XYZ}=g(Y,\xi)S_{X\xi
Z},\,S_{X\xi Z}=S_{Z\xi X},\,c_{13}(S)=0,\,X,Y,Z\in D\Bigr{\\}},$
$\displaystyle\mathcal{II}_{3}(V)$
$\displaystyle=\Bigl{\\{}S\in\mathcal{II}(V):\>S_{XYZ}=g(Y,\xi)S_{X\xi
Z},\,S_{X\xi Z}=-S_{Z\xi X},\,X,Y,Z\in D\Bigr{\\}},$

where $c_{13}(S)(Y)=\sum_{i=1}^{n}S_{e_{i}Ye_{i}}$ for any orthonormal basis
$\\{e_{1},\ \ldots\ ,e_{n}\\}$ of $D$.

###### Proof.

Recall that $\mathcal{II}(V)=D^{*}\otimes U^{*}\otimes D^{*}$. Indeed, for any
$S\in D^{*}\otimes U^{*}\otimes D^{*}$, we have $S_{XYZ}=g(Y,\xi)S_{X\xi Z}$,
and $S_{\cdot\xi\cdot}\in D^{*}\otimes D^{*}$. This space decomposes into
irreducible $\mathrm{SL}(D)$ submodules as

$\mathcal{II}(V)=D^{*}\wedge D^{*}+S^{2}D^{*},$

in symmetric and skew-symmetric endomorphisms, respectively. To get the
$\operatorname{SO}(D)\subset\mathrm{SL}(D)$ irreducible submodules we have to
take traces in the endomorphisms, i. e.,

$\mathcal{II}(V)=D^{*}\wedge D^{*}+\\{S_{X\xi Z}\in
S^{2}D^{*}:\>\mathrm{c}_{12}(S)(\xi)=0\\}+\\{S_{X\xi Z}\in
S^{2}D^{*}:\>\mathrm{c}_{12}(S)(\xi)\neq 0\\}$

and this decomposition into irreducible submodules yields the three
expressions $\mathcal{II}_{3}(V)$, $\mathcal{II}_{2}(V)$ and
$\mathcal{II}_{1}(V)$, respectively. ∎

We now analyse

$\mathcal{S}_{U}(V)=\\{S\in
U^{*}\otimes\mathfrak{gl}(V):\>S_{\xi}\xi=0\\}=U^{*}\otimes V^{*}\otimes V.$

From $V=U\oplus D$ we get

$U^{*}\otimes V^{*}\otimes V^{*}=U^{*}\otimes U^{*}\otimes V^{*}+U^{*}\otimes
D^{*}\otimes D^{*}+U^{*}\otimes D^{*}\otimes U^{*}$

and since $S_{\xi}\xi=0$, the space $\mathcal{S}_{U}(V)$ is exactly

$\mathcal{S}_{U}(V)=U^{*}\otimes D^{*}\otimes D^{*}+U^{*}\otimes D^{*}\otimes
U^{*},$

decomposition that is denoted by

$\displaystyle\mathcal{Z}(V)$ $\displaystyle=U^{*}\otimes D^{*}\otimes D^{*},$
$\displaystyle S_{U}^{1}(V)$ $\displaystyle=U^{*}\otimes D^{*}\otimes U^{*}.$

######  Proposition 5.4.

The submodule $\mathcal{Z}(V)$ decomposes as
$\operatorname{SO}(n)$-irreducible submodules as,

$\mathcal{Z}(V)=\mathcal{Z}_{1}(V)+\mathcal{Z}_{2}(V)+\mathcal{Z}_{3}(V)$

with explicit expressions,

$\displaystyle\mathcal{Z}_{1}(V)$
$\displaystyle=\Bigl{\\{}S\in\mathcal{Z}(V):\>S_{XYZ}=\lambda
g(Y,Z)g(X,\xi),\,\lambda\in{\mathbb{R}},\,X,Y,Z\in D\Bigr{\\}},$
$\displaystyle\mathcal{Z}_{2}(V)$
$\displaystyle=\Bigl{\\{}S\in\mathcal{Z}(V):\>S_{XYZ}=g(X,\xi)S_{\xi
YZ},\,S_{\xi YZ}=S_{\xi YX},\,c_{23}(S)=0,\,X,Y,Z\in D\Bigr{\\}},$
$\displaystyle\mathcal{Z}_{3}(V)$
$\displaystyle=\Bigl{\\{}S\in\mathcal{Z}(V):\>S_{XYZ}=g(X,\xi)S_{\xi
YZ},\,S_{\xi YZ}=-S_{\xi ZY},\,X,Y,Z\in D\Bigr{\\}},$

where $c_{23}(S)(X)=\sum_{i=1}^{n}S_{Xe_{i}e_{i}}$ for any orthonormal basis
$\\{e_{1},\ \ldots\ ,e_{n}\\}$ of $D$.

###### Proof.

As $\mathcal{Z}(V)\simeq\mathcal{II}(V)$, this decomposition is analogous to
Theorem 5.3 doing a permutation in the indices. ∎

######  Proposition 5.5.

The submodule $S_{U}^{1}(V)$ is $\operatorname{SO}(D)$-irreducible and its
explicit expression is,

$\mathcal{S}_{U}^{1}(V)=\Bigl{\\{}S_{XYZ}=g(Y,\eta)g(X,\xi)g(Z,\xi),\,\eta\in
W\Bigr{\\}}.$

###### Proof.

It is direct from the fact $S_{U}^{1}(V)=U^{*}\otimes D^{*}\otimes U^{*}\simeq
D^{*}$ and it is irreducible. Its explicit expression is direct. ∎

Therefore, we can write the space of cohomogeneity one structures as

$\mathcal{S}(V)=\mathcal{T}(V)+\mathcal{II}(V)+\mathcal{Z}(V)+S_{U}^{1}(V).$

From the previous four propositions we have that, in fact, $\mathcal{S}(V)$
decomposes into the sum of ten irreducible $\mathrm{SO}(n)$-submodules. The
local isometries in CO1-AS manifolds in Theorem 4.3 give the following result.

######  Proposition 5.6.

If the cohomogeneity one structure $S_{p}$ of a CO1-AS manifold $(M,g)$ at a
point $p\in M$ belongs to a certain irreducible submodule or sum of
irreducible submodules, then $S_{p^{\prime}}$ will belong to the same
submodule or sum of submodules for any other point $p^{\prime}$ of the leaf of
$p$.

We now give a few simple geometric results stemming from the classification
above. It is easy to see that the projections to each submodule are

$\displaystyle\Pi_{\mathcal{T}(V)}(S)_{XYZ}$ $\displaystyle=\sum_{i,j=0}^{n}\
g(X,e_{i})g(Y,e_{j})S_{e_{i}e_{j}Z},$
$\displaystyle\Pi_{\mathcal{II}(V)}(S)_{XYZ}$ $\displaystyle=\sum_{i=0}^{n}\
g(X,e_{i})g(Y,\xi)S_{e_{i}\xi Z},$
$\displaystyle\Pi_{\mathcal{Z}(V)}(S)_{XYZ}$ $\displaystyle=\sum_{i=0}^{n}\
g(Y,e_{i})g(X,\xi)S_{e_{i}\xi Z},$ $\displaystyle\Pi_{S_{U}^{1}(V)}(S)_{XYZ}$
$\displaystyle=g(X,\xi)g(Z,\xi)S_{\xi Y\xi}.$

######  Proposition 5.7.

Let $(M,g,\xi)$ be a CO1-AS manifold and $p\in M$.

If $\Pi_{\mathcal{T}(V)}(S_{p})_{XYZ}=0$, the leaf of $p$ is a locally
symmetric Riemannian manifold.

If $\Pi_{\mathcal{T}(V)}(S_{p})_{XYZ}\in\mathcal{T}_{1}(V)$, then the leaf of
$p$ is locally isometric to the real hyperbolic space.

If $\Pi_{\mathcal{II}(V)}(S_{p})_{XYZ}=0$, then the leaf of $p$ is a totally
geodesic submanifolds.

##  6 The canonical cohomogeneity one structure

The proof (1) implies (2) of the classical Ambrose-Singer Theorem, provides
the canonical connection associated with the given reductive decomposition.
Recall that this canonical connection is characterized algebraically (see
[Nom54, p. 43, Thm. 8.1]) and geometrically (see [KN69, p. 192, Prop. 2.4]).
The aim of this section is the analysis of the connection built in the proof
(1) implies (2) of Theorem 3.2 along a geodesic that intersects all the orbits
of a cohomogeneity one action without singular orbits. This can be understood
as a canonical connection for cohomogeneity one manifolds.

We consider a reductive homogeneous space $M=G/H$ (not necessarily
Riemannian), with reductive decomposition
$\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$. For any $X\in\mathfrak{g}$ we denote
by $X^{*}$ the fundamental vector field associated with $X$. The canonical
connection $\tilde{\nabla}$ of $M$ with respect to the above decomposition is
determined by (cf. [KN69, p. 188-192])

$\bigl{(}\tilde{\nabla}_{X^{*}}Y^{*}\bigr{)}_{o}=-\tilde{T}_{o}(Y_{o}^{*},X_{o}^{*})=\tilde{T}_{o}(X_{o}^{*},Y_{o}^{*}),$

where $\tilde{T}$ is the torsion of $\tilde{\nabla}$, for any $X$,
$Y\in\mathfrak{m}$, $o=eH$. Since ([KN69, Ch. X, Thm. 2.6]), we also have
$\tilde{T}_{o}(X_{o}^{*},Y_{o}^{*})=-[X,Y]_{\mathfrak{m}}$ and, therefore,

$\bigl{(}\tilde{\nabla}_{X^{*}}Y^{*}\bigr{)}_{o}=-[X,Y]_{\mathfrak{m}},$

for $X$, $Y\in\mathfrak{m}$.

######  Lemma 6.1.

If $X\in\mathfrak{m}$ and $Y\in\mathfrak{g}$ then
$\tilde{\nabla}_{X}Y^{*}=-[X,Y]_{\mathfrak{m}}$.

###### Proof.

Using the formulas for $\tilde{\nabla}$ and the torsion $\tilde{T}_{o}$, when
the corresponding vectors are in $\mathfrak{m}$, we have

$\displaystyle\tilde{\nabla}_{X}Y^{*}$
$\displaystyle{}=\tilde{\nabla}_{X^{*}_{o}}Y^{*}=\bigl{(}\tilde{\nabla}_{X^{*}}Y^{*}\bigr{)}_{o}$
$\displaystyle{}=\bigl{(}\tilde{\nabla}_{Y^{*}}X^{*}\bigr{)}_{o}+[X^{*},Y^{*}]_{o}+\tilde{T}_{o}(X^{*},Y^{*})$
$\displaystyle{}=\tilde{\nabla}_{Y^{*}_{o}}X^{*}-[X,Y]^{*}_{o}+\tilde{T}_{o}(X^{*}_{o},Y^{*}_{o})$
$\displaystyle{}=\tilde{\nabla}_{Y_{\mathfrak{m}}}X^{*}-[X,Y]_{\mathfrak{m}}+\tilde{T}_{o}(X,Y_{\mathfrak{m}})$
$\displaystyle{}=-[Y_{\mathfrak{m}},X]_{\mathfrak{m}}-[X,Y]_{\mathfrak{m}}-[X,Y_{\mathfrak{m}}]_{\mathfrak{m}}=-[X,Y]_{\mathfrak{m}},$

as we wanted to show. ∎

Now we calculate $\tilde{\nabla}$ at any other point.

######  Lemma 6.2.

If $X$, $Y\in\mathfrak{m}$ and $f\in G$, we have

$\bigl{(}\tilde{\nabla}_{X^{*}}Y^{*}\bigr{)}_{f(o)}=-f_{*o}\bigl{[}\bigl{(}\operatorname{Ad}(f^{-1})X\bigr{)}_{\mathfrak{m}},\operatorname{Ad}(f^{-1})Y\bigr{]}_{\mathfrak{m}}.$

###### Proof.

Let $X\in\mathfrak{g}$. We consider $\psi_{t}=\exp(tX)$ the $1$-parameter
group generated by $X$. We denote by $I_{f^{-1}}$ the conjugation by $f^{-1}$,
and by $\operatorname{Ad}(f^{-1})=I_{f^{-1}*}$ the corresponding adjoint
representation of $\mathfrak{g}$. We have

$X^{*}_{f(o)}=\frac{d}{dt}\Big{|}_{t=0}\psi_{t}(f(o))=\frac{d}{dt}\Big{|}_{t=0}fI_{f^{-1}}(\psi_{t})(o)=f_{*o}\bigl{(}\operatorname{Ad}(f^{-1})X\bigr{)}_{o}.$

Hence,
$\bigl{(}f_{*}^{-1}X^{*}\bigr{)}_{o}=\bigl{(}\operatorname{Ad}(f^{-1})X\bigr{)}_{o}$.
Since $\tilde{\nabla}$ is $G$-invariant, Lemma 6.1 yields

$\displaystyle\bigl{(}\tilde{\nabla}_{X^{*}}Y^{*}\bigr{)}_{f(o)}$
$\displaystyle{}=f_{*o}\bigl{(}\tilde{\nabla}_{f_{*}^{-1}X^{*}}f_{*}^{-1}Y^{*}\bigr{)}_{o}=f_{*o}\tilde{\nabla}_{\bigl{(}\operatorname{Ad}(f^{-1})X\bigr{)}_{\mathfrak{m}}}\bigl{(}\operatorname{Ad}(f^{-1})Y\bigr{)}^{*}$
$\displaystyle{}=f_{*o}\bigl{[}\bigl{(}\operatorname{Ad}(f^{-1})X\bigr{)}_{\mathfrak{m}},\operatorname{Ad}(f^{-1})Y\bigr{]}_{\mathfrak{m}},$

as claimed. ∎

Now we come back to our original problem. We consider an oriented Riemannian
manifold $M$ that is being acted upon by a group of isometries $G$ of
cohomogeneity one, all whose orbits are principal. As we have seen, we can
define a global unit vector field $\xi$ that is orthogonal to all the orbits
of $G$. We denote by $\varphi_{t}$ the flow of $\xi$. Then $\varphi_{t}$
determines a $1$-parameter group $A$ that is isomorphic to $\mathbb{R}$ or
$\mathbb{S}^{1}$. As a consequence, $M$ is now a homogeneous space acted upon
by $A\times G$ by the formula $(t,g)\cdot p=\varphi_{t}(g\cdot
p)=g\cdot\varphi_{t}(p)$. In particular, $A$ is contained in the center of
$A\times G$, and $\xi$ is the fundamental vector field on $M$ determined by
$1\in A$.

From now on we fix $p\in M$. We have the reductive decomposition
$\mathfrak{a}+\mathfrak{m}=\mathfrak{h}+(\mathfrak{a}+\mathfrak{m})$ at $p$.
We determine the canonical connection $\tilde{\nabla}$ associated with this
decomposition. We define $\gamma(t)=\varphi_{t}(p)$, which is a unit speed
geodesic that intersects all the orbits of $G$ orthogonally.

######  Proposition 6.3.

If $X$, $Y\in\mathfrak{m}$, then we have
$\bigl{(}\tilde{\nabla}_{\xi}X^{*}\bigr{)}_{\gamma(t)}=\bigl{(}\tilde{\nabla}_{X^{*}}\xi\bigr{)}_{\gamma(t)}=\bigl{(}\tilde{\nabla}_{\xi}\xi\bigr{)}_{\gamma(t)}=0$,
and
$\bigl{(}\tilde{\nabla}_{X^{*}}Y^{*}\bigr{)}_{\gamma(t)}=-\varphi_{t*p}[X,Y]_{\mathfrak{m}}$.

###### Proof.

Since $A$ is contained in the center of $A\times G$ we have that
$\operatorname{Ad}(t)$, $t\in A$, acts as the identity of
$\mathfrak{a}+\mathfrak{g}$. Thus, $\operatorname{Ad}(t^{-1})\xi=\xi$ and
$\operatorname{Ad}(t^{-1})X=X$ for all $X\in\mathfrak{m}$. Since
$[\mathfrak{a},\mathfrak{a}+\mathfrak{g}]=0$, the first three equalities of
the statement follow from Lemma 6.2. Finally, if $X$, $Y\in\mathfrak{m}$,
Lemma 6.2 yields

$\displaystyle\bigl{(}\tilde{\nabla}_{X^{*}}Y^{*}\bigr{)}_{\gamma(t)}$
$\displaystyle{}=-\varphi_{t*p}\bigl{[}\bigl{(}\operatorname{Ad}(t^{-1})X\bigr{)}_{\mathfrak{a}+\mathfrak{m}},\operatorname{Ad}(t^{-1})Y\bigr{]}_{\mathfrak{a}+\mathfrak{m}}$
$\displaystyle{}=-\varphi_{t*p}\bigl{[}X_{\mathfrak{a}+\mathfrak{m}},Y\bigr{]}_{\mathfrak{a}+\mathfrak{m}}=-\varphi_{t*p}\bigl{[}X,Y\bigr{]}_{\mathfrak{m}},$

which finishes the proof. ∎

Finally, we calculate the cohomogeneity one structure
$S=\nabla-\tilde{\nabla}$, where $\nabla$ is the Levi-Civita connection of
$M$.

######  Proposition 6.4.

Let $X$, $Y$, $Z\in\mathfrak{m}$, and $a$, $b$, $c\in\mathbb{R}$. Then, the
difference tensor $S$ is given by

$\displaystyle
2g\big{(}S_{a\xi+X^{*}}(b\xi+Y^{*}),c\xi+Z^{*}\big{)}_{\gamma(t)}={}$
$\displaystyle\qquad{}a\frac{d}{dt}g\big{(}\varphi_{t*p}Y,\varphi_{t*p}Z\big{)}+b\frac{d}{dt}g\big{(}\varphi_{t*p}X,\varphi_{t*p}Z\big{)}-c\frac{d}{dt}g\big{(}\varphi_{t*p}X,\varphi_{t*p}Y\big{)}$
$\displaystyle\qquad{}+g\big{(}\varphi_{t*p}[X,Y]_{\mathfrak{m}},\varphi_{t*p}Z\big{)}-g\big{(}\varphi_{t*p}[X,Z]_{\mathfrak{m}},\varphi_{t*p}Y\big{)}-g\big{(}\varphi_{t*p}[Y,Z]_{\mathfrak{m}},\varphi_{t*p}X\big{)}.$

###### Proof.

Since $\xi$ is a geodesic vector field, $\nabla_{\xi}\xi=0$. Moreover, since
$Y^{*}$, $Y\in\mathfrak{m}$, is Killing, we have
$g\big{(}\nabla_{\xi}Y^{*},\xi\big{)}=0$ and
$g\big{(}\nabla_{\xi}Y^{*},Z^{*}\big{)}=-g\big{(}\nabla_{Z^{*}}Y^{*},\xi\big{)}$.
Also, $g\big{(}\xi,\xi\big{)}=1$ implies
$g\big{(}\nabla_{X^{*}}\xi,\xi\big{)}=0$, and $g\big{(}X^{*},\xi\big{)}=0$
implies
$g\big{(}\nabla_{X^{*}}\xi,Z^{*}\big{)}=-g\big{(}\nabla_{X^{*}}Z^{*},\xi\big{)}$.
Altogether this gives

$\displaystyle 2g\big{(}\nabla_{a\xi+X^{*}}(b\xi+Y^{*}),c\xi+Z^{*}\big{)}={}$
$\displaystyle-2ag\big{(}\nabla_{Z^{*}}Y^{*},\xi\big{)}-2bg\big{(}\nabla_{X^{*}}Z^{*},\xi\big{)}$
(6.1)
$\displaystyle+2cg\big{(}\nabla_{X^{*}}Y^{*},\xi\big{)}+2g\big{(}\nabla_{X^{*}}Y^{*},Z^{*}\big{)}.$

We calculate $g\big{(}\nabla_{X^{*}}Y^{*},\xi\big{)}_{\gamma(t)}$ for $X$,
$Y\in\mathfrak{m}$. First note that
$X^{*}_{\gamma(t)}=X^{*}_{\varphi_{t}(p)}=\varphi_{t*p}X^{*}_{p}$ because
$\varphi_{t}$ commutes with the action of $G$. The fact that $\xi$ is
geodesic, $X^{*}$ and $Y^{*}$ are Killing vector fields, and the Levi-Civita
connection is torsion-free yields

$\displaystyle\frac{d}{dt}g\big{(}\varphi_{t*p}X,\varphi_{t*p}Y\big{)}$
$\displaystyle{}=\frac{d}{dt}g\big{(}X^{*}_{\gamma(t)},Y^{*}_{\gamma(t)}\big{)}$
$\displaystyle{}=g\big{(}\nabla_{\xi_{\gamma(t)}}X^{*},Y^{*}_{\gamma(t)}\big{)}+g\big{(}X^{*},\nabla_{\xi_{\gamma(t)}}Y^{*}_{\gamma(t)}\big{)}$
$\displaystyle{}=-g\big{(}\nabla_{Y^{*}_{\gamma(t)}}X^{*},\xi_{\gamma(t)}\big{)}-g\big{(}\nabla_{X^{*}_{\gamma(t)}}Y^{*},\xi_{\gamma(t)}\big{)}$
$\displaystyle{}=-g\big{(}2\nabla_{X^{*}_{\gamma(t)}}Y^{*}-[X^{*},Y^{*}]_{\gamma(t)},\xi_{\gamma(t)}\big{)}$
$\displaystyle{}=-2g\big{(}\nabla_{X^{*}_{\gamma(t)}}Y^{*},\xi_{\gamma(t)}\big{)}.$

The last addend of (6.1) can be calculated from [Bes87, 7.27] as

$g\big{(}\nabla_{X^{*}}Y^{*},Z^{*}\big{)}=g\big{(}[X^{*},Y^{*}],Z^{*}\big{)}+g\big{(}[X^{*},Z^{*}],Y^{*}\big{)}+g\big{(}[Y^{*},Z^{*}],X^{*}\big{)}.$

Since $X^{*}_{\varphi_{t}(p)}=\varphi_{t*p}X_{p}$, we see that $X^{*}$ is
$\varphi_{t}$-related to itself. Thus,
$[X^{*},Y^{*}]_{\gamma(t)}=[X^{*},Y^{*}]_{\varphi_{t}(p)}=\varphi_{t*p}[X^{*},Y^{*}]_{p}=-\varphi_{t*p}[X,Y]_{\mathfrak{m}}$.

Substituting in (6.1) we obtain the Levi-Civita connection of $M$ along
$\gamma$. This, together with Proposition 6.3, finishes the proof of
Proposition 6.4. ∎

######  Remark 6.5.

Let $A_{\gamma(t)}$ denote the shape operator of $G\cdot\gamma(t)$ at
$\gamma(t)$ with respect to the normal vector $\xi_{\gamma(t)}$, that is,
$A_{\gamma(t)}(v)=-\nabla_{v}\xi$, $v\in
T_{\gamma(t)}\bigl{(}G\cdot\gamma(t)\bigr{)}$. Then,

$g\big{(}A_{\gamma(t)}X^{*}_{\gamma(t)},Y^{*}_{\gamma(t)}\big{)}=-g\big{(}\nabla_{X^{*}}Y^{*},\xi\big{)}_{\gamma(t)}=\frac{1}{2}\frac{d}{dt}g\big{(}\varphi_{t*p}X,\varphi_{t*p}Y\big{)}.$

Furthermore, $X^{*}_{\gamma(t)}$ is a Jacobi vector field along $\gamma$ with
$\frac{d}{dt}_{|_{0}}X^{*}_{\gamma(t)}=-A_{p}X^{*}_{p}$. Thus, if one is able
to solve the Jacobi equation along $\gamma$, then $X^{*}_{\gamma(t)}$ can be
calculated explicitly, and so can the shape operator $A_{\gamma(t)}$.

######  Definition 6.6.

The cohomogeneity one structure $S$ given in Proposition 6.4 is called _the
canonical cohomogeneity one structure_.

From Proposition 6.3, the canonical cohomogeneity one structure satisfies

$\tilde{T}(\xi,\ \cdot\ )=0,$

where $\tilde{T}$ is the torsion of $\tilde{\nabla}=\nabla-S$.

######  Proposition 6.7.

Let $S_{1}$ and $S_{2}$ be two cohomogeneity one structures on $(M,g)$
associated with $\xi$ such that $\tilde{T}_{1}(\xi,\ \cdot\ )=0$ and
$\tilde{T}_{2}(\xi,\ \cdot\ )=0$. If
$\Pi_{\mathcal{T}(V)}(S_{1})=\Pi_{\mathcal{T}(V)}(S_{2})$, then $S_{1}=S_{2}$.

###### Proof.

We notice that the distribution $\mathcal{D}=\\{X:\>g(X,\xi)=0\\}$ is
independent of the choice of $S_{1}$ and $S_{2}$. Therefore, it makes sense to
consider $\Pi_{\mathcal{T}(V)}(S_{1})=\Pi_{\mathcal{T}(V)}(S_{2})$. This means
that $(S_{1})_{X}Y=(S_{2})_{X}Y$ for all $X$, $Y\in\mathcal{D}$. For $i=1$,
$2$, as the Levi-Civita connection is torsion-free,

$\tilde{T}_{i}(\xi,A)=-(S_{i})_{\xi}A+(S_{i})_{A}\xi,\quad\forall A\in TM.$

We apply $\tilde{T}_{i}(\xi,\cdot)=0$ and get $(S_{i})_{\xi}A=(S_{i})_{A}\xi$.
For all $a$, $b\in{\mathbb{R}}$, we apply Proposition 4.10,

$\displaystyle g((S_{i})_{a\xi+X}\xi,c\xi+Y)$
$\displaystyle{}\overset{(4)}{=}g((S_{i})_{X}\xi,c\xi+Y)$
$\displaystyle{}\overset{(2)}{=}g((S_{i})_{X}\xi,Y)$
$\displaystyle{}\overset{(3)}{=}-\mathrm{II}(X,Y),$

which is independent of the choice of $i$. Then $S_{1}=S_{2}$. ∎

######  Remark 6.8.

The canonical cohomogeneity one structure $S$ is the unique (up to homogeneous
structure in the leaves) cohomogeneity one structure satisfying,

$\tilde{T}(\xi,\ \cdot\ )=0$

where $\tilde{T}$ is the torsion of $\tilde{\nabla}=\nabla-S$.

##  7 Examples

In this section we examine examples and techniques to construct cohomogeneity
one structures.

### Parallel hyperplanes in euclidean spaces

Let $({\mathbb{R}}^{n},g_{{\mathbb{R}}^{n}})$ be the Euclidean space. If $H$
is a closed subgroup of $\operatorname{SO}(n-1)$, then
$G=H\ltimes{\mathbb{R}}^{n-1}$ acts transitively on
$({\mathbb{R}}^{n-1},g_{{\mathbb{R}}^{n-1}})$. Consequently, $G$ acts with
cohomogeneity one on ${\mathbb{R}}^{n}$. For $p=(p_{1},\ \ldots\ ,p_{n})$, the
orbit space $G\cdot p=\\{(x_{1},\ \ldots\
x_{n-1},p_{n}):\>x_{i}\in{\mathbb{R}}\\}$ gives a foliation without singular
orbits. The unit vector field orthogonal to the leaves is $\xi_{p}=(0,\
\ldots\ 0,1)$ and its flow is $\varphi_{t}(p)=(p_{1},\ \ldots\ ,p_{n}+t)$.
Since $\varphi_{t}$ is isometric and ${\mathbb{R}}^{n-1}$ is an abelian ideal
of ${\mathbb{R}}^{n}$, the first and second rows of Proposition 6.4 are zero.
Thus, $S=0$.

### Concentric spheres in euclidean spaces

The space $({\mathbb{R}}^{l}-\\{0\\},g_{{\mathbb{R}}^{l}})$ equipped with the
Eucliden metric is isometric to the warped product
$({\mathbb{R}}^{+}\times_{f}\mathbb{S}^{l-1},g)$,
$g=dr^{2}+f(r)^{2}g_{\mathbb{S}^{l-1}}$, where $r$ is the distance to the
origin, $f(r)=r$, and $g_{\mathbb{S}^{l-1}}$ is the round metric of the
sphere. Let
$\pi_{2}\colon{\mathbb{R}}^{+}\times_{f}\mathbb{S}^{l-1}\longrightarrow\mathbb{S}^{l-1}$
be the projection to the second coordinate. Consider a Lie group $G$ of
isometries of $(\mathbb{S}^{l-1},g_{\mathbb{S}^{l-1}})$ and the canonical
connection $\tilde{\nabla}^{\mathbb{S}^{l-1}}$ associated with it. We also
consider the pseudo-group $({\mathbb{R}}^{+},+)$ acting transitively on
${\mathbb{R}}^{+}$. The Lie pseudo-group ${\mathbb{R}}^{+}\times G$ acts
transitively on ${\mathbb{R}}^{+}\times_{f}\mathbb{S}^{l-1}$ and there is an
AS-connection $\tilde{\nabla}$ (non-necessarily Riemannian and non necessarily
complete) such that

$\tilde{\nabla}\tilde{T}=0,\qquad\tilde{\nabla}\tilde{R}=0.$

Actually,

$\tilde{\nabla}=\nabla^{{\mathbb{R}}^{+}}\oplus\tilde{\nabla}^{\mathbb{S}^{l-1}},$

where $\nabla^{{\mathbb{R}}^{+}}$ is the Levi-Civita connection for
${\mathbb{R}}^{+}$.

The Lie group $G$ acts with cohomogeneity one on
${\mathbb{R}}^{+}\times\mathbb{S}^{l-1}={\mathbb{R}}^{+}\times G/H$ and its
action gives a foliation of spheres $\mathbb{S}^{l-1}(r)$ of radius $r>0$. The
normal vector field for each one of these submanifolds is,
$\eta(r,p)=\partial/\partial r$ (in Euclidean coordinates,
$\eta(x)=\frac{x}{||x||}$) and the associated COAS-1 connection is
$\tilde{\nabla}$. In terms of the warped elementes (see [ONe83, p. 206, Prop.
35] we have

$\nabla=\nabla^{{\mathbb{R}}^{+}}\oplus\nabla^{\mathbb{S}^{l-1}}-\frac{f^{2}\pi_{2}{}^{*}(g_{\mathbb{S}^{l-1}})}{f}\mathrm{grad}(f)+\sum_{i=1}^{l-1}\frac{\eta(f)}{f}(\eta^{*}\otimes
de^{i}+de^{i}\otimes\eta^{*})\frac{\partial}{\partial e^{i}},$

where $\nabla^{\mathbb{S}^{l-1}}$ is the Levi-Civita connection for the sphere
$\mathbb{S}^{l-1}$ and $\left\\{\frac{\partial}{\partial e^{1}},\ \dots\
,\frac{\partial}{\partial e^{l-1}}\right\\}$ is an orthonormal basis of
$T_{x}\mathbb{S}^{l-1}$ and $\left\\{de^{1},\ \dots\ ,de^{l-1}\right\\}$ is
its dual basis. As $\eta(f)=1$, we have

$\nabla=\nabla^{{\mathbb{R}}^{+}}\oplus\nabla^{\mathbb{S}^{l-1}}-\frac{f^{2}\pi_{2}{}^{*}(g_{\mathbb{S}^{l-1}})}{||x||}\eta+\frac{1}{||x||}\sum_{i=1}^{l-1}\left(\eta^{*}\otimes
de^{i}+de^{i}\otimes\eta^{*}\right)\frac{\partial}{\partial e^{i}}.$

Hence, the sphere foliation of $({\mathbb{R}}^{n-k-1}-\\{0\\},g)$ induced by
the Lie group $G$ possesses a cohomogeneity-one structure, represented by,

$S=0\oplus
S^{\mathbb{S}^{l-1}}-\frac{f^{2}\pi_{2}{}^{*}(g_{\mathbb{S}^{l-1}})}{||x||}\eta+\frac{1}{||x||}\sum_{i=1}^{l-1}\left(\eta^{*}\otimes
de^{i}+de^{i}\otimes\eta^{*}\right)\frac{\partial}{\partial e^{i}},$ (7.1)

where
$S^{\mathbb{S}^{l-1}}=\nabla^{\mathbb{S}^{l-1}}-\tilde{\nabla}^{\mathbb{S}^{l-1}}$
is the homogeneous structure for the action of $G$ on $\mathbb{S}^{l-1}=G/H$.
In other words,

$\displaystyle S_{B}C={}$ $\displaystyle
S^{\mathbb{S}^{l-1}}_{(\pi_{2})_{*}B}(\pi_{2})_{*}C-\frac{g((\pi_{2})_{*}B,(\pi_{2})_{*}C))}{r}\
\frac{\partial}{\partial r}$
$\displaystyle{}+\frac{1}{r}\left(g\left(\frac{\partial}{\partial
r},B\right)(\pi_{2})_{*}C+g\left(\frac{\partial}{\partial
r},C\right)(\pi_{2})_{*}B\right).$

In a broader sense, let $(M_{1},g_{2},\tilde{\nabla}_{1})$ and
$(M_{2},g_{2},\tilde{\nabla}_{2})$ be two Riemannian AS-manifolds, that is,

$\begin{array}[]{cccccccccccc}\tilde{\nabla}_{1}\tilde{R}_{1}&=&0,&&\tilde{\nabla}_{1}\tilde{T}_{1}&=&0,&&\tilde{\nabla}_{1}g_{1}&=&0\\\
\tilde{\nabla}_{2}\tilde{R}_{2}&=&0,&&\tilde{\nabla}_{2}\tilde{T}_{2}&=&0,&&\tilde{\nabla}_{2}g_{2}&=&0,\end{array}$

where $\tilde{R}_{i}$ and $\tilde{T}_{i}$ are the curvature and torsion of
$\tilde{\nabla}_{i}$ with $i=1,2$. On the one hand, the product manifold
$M_{1}\times M_{2}$ is a general AS-manifold with AS-connection

$\tilde{\nabla}=\tilde{\nabla}_{1}\oplus\tilde{\nabla}_{2}.$

On the other hand, for a positive function $f\colon
M_{1}\longrightarrow{\mathbb{R}}^{+}$, then we can consider the warped product
$\left(M_{1}\times_{f}M_{2},\,g=\pi_{1}{}^{*}(g_{1})+(f\circ\pi_{1})^{2}\pi_{2}{}^{*}(g_{2})\right)$
where $\pi_{i}$ is the projection onto $M_{i}$ with $i=1$ or $2$. We write $f$
instead of $f\circ\pi_{1}$. In particular, if the dimension of $M_{1}$ is $1$,
the warped product is a regular (locally) cohomogeneity one and the Levi-
Civita connection in local coordinates is given by (again, [ONe83, p. 206,
Prop. 35])

$\nabla=\nabla_{1}\oplus\nabla_{2}-\frac{f^{2}\pi_{2}{}^{*}(g_{2})}{f}\mathrm{grad}(f)+\sum_{i=1}^{l-1}\frac{\eta(f)}{f}(\eta^{*}\otimes
de^{i}+de^{i}\otimes\eta^{*})\frac{\partial}{\partial e^{i}}$

where $\eta$ is a unit vector field in $(M_{1},g_{1})$,
$\left\\{\frac{\partial}{\partial e^{1}},\ \dots\ ,\frac{\partial}{\partial
e^{l-1}}\right\\}$ is a local orthonormal basis of $M_{2}$, and
$\left\\{de^{1},\ \dots\ ,de^{l-1}\right\\}$ is its dual basis. Finally, the
(1, 2)-tensor

$S=\nabla-\tilde{\nabla}$

is a cohomogeneity one structure.

### The horosphere foliation in the real hyperbolic space

The real hyperbolic space $\mathbb{R}\mathrm{H}(n)$ with the hyperbolic metric
of constant scalar curvature equal to $-1$ is isometric to the warped product
$({\mathbb{R}}\times_{f}{\mathbb{R}}^{n-1},g=dt^{2}+f(t)^{2}g_{{\mathbb{R}}^{n-1}})$,
with $f(t)=e^{-t}$. Following the procedure outlined above, we can introduce
the AS-connection $\tilde{\nabla}$, defined as

$\tilde{\nabla}=\nabla^{{\mathbb{R}}}\oplus\nabla^{{\mathbb{R}}^{n-1}},$

where $\nabla^{{\mathbb{R}}}$ and $\nabla^{{\mathbb{R}}^{n-1}}$ are the Levi-
Civita connections of ${\mathbb{R}}$ and ${\mathbb{R}}^{n-1}$, respectively.
Note that $\tilde{\nabla}$ corresponds to the Levi-Civita connection for the
euclidean metric in ${\mathbb{R}}^{n}$. However, $\tilde{\nabla}$ is not
metric with respect $g$, even though it satisfies

$\tilde{\nabla}\tilde{R}=0,\quad\tilde{\nabla}\tilde{T}=0,\quad\tilde{\nabla}\xi=0,\quad\tilde{\nabla}_{X}g=0,\quad\tilde{T}(X,Y)\in\mathcal{L}(\xi)^{\perp},$

where $\xi=\partial/\partial t$ and $\tilde{R}$ and $\tilde{T}$ are the
curvature and torsion of $\tilde{\nabla}$. In other words, $\tilde{\nabla}$ is
an AS-connection of Theorem 3.2 and $S=\nabla-\tilde{\nabla}$ is a
cohomogeneity one structure, where $\nabla$ is the Levi-Civita connection of
$\mathbb{R}\mathrm{H}(n)$. We now compute $S$ explicitly.

The Levi-Civita connection for a warped product is

$\nabla=\nabla^{{\mathbb{R}}}\oplus\nabla^{\mathbb{R}^{n-1}}-\frac{f^{2}\pi_{2}{}^{*}(g_{{\mathbb{R}}^{n-1}})}{f}\mathrm{grad}(f)+\sum_{i=1}^{n-1}\frac{\xi(f)}{f}\left(\xi^{*}\otimes
de^{i}+de^{i}\otimes\xi^{*}\right)\frac{\partial}{\partial e^{i}},$

that is

$\nabla=\nabla^{{\mathbb{R}}}\oplus\nabla^{\mathbb{R}^{n-1}}+f^{2}\pi_{2}{}^{*}(g_{{\mathbb{R}}^{n-1}})\
\xi-\sum_{i=1}^{n-1}\left(\xi^{*}\otimes
de^{i}+de^{i}\otimes\xi^{*}\right)\frac{\partial}{\partial e^{i}}.$

The cohomogeneity-1 structure $S=\nabla-\tilde{\nabla}$ is thus

$\displaystyle S$ $\displaystyle=f^{2}\pi_{2}{}^{*}(g_{{\mathbb{R}}^{n-1}})\
\xi-\sum_{i=1}^{l-1}\left(\xi^{*}\otimes
de^{i}+de^{i}\otimes\xi^{*}\right)\frac{\partial}{\partial e^{i}}$
$\displaystyle=g((\pi_{2})_{*}X,(\pi_{2})_{*}Y)\xi-g(\xi,X)(\pi_{2})_{*}Y-g(\xi,Y)(\pi_{2})_{*}X,$

and putting $\left((\pi_{2})_{*}X\right)=X-g(X,\xi)\xi$ and
$(\pi_{2})_{*}Y=Y-g(Y,\xi)\xi$, we finally get

$S_{X}Y=g(X,Y)\xi-g(\xi,X)Y-g(\xi,Y)X+g(X,\xi)g(\xi,Y)\xi.$

### A non-canonical cohomogeneity one structure

The real hyperbolic space $\mathbb{R}\mathrm{H}(n)$, in the coordinates above,
is characterized by the existence of a homogeneous structure $\bar{S}$ of
linear type, see [TV83, Thm. 5.2]. This has the expression,

$\bar{S}_{X}Y=g(X,Y)\xi-g(Y,\xi)X$

where $\xi=\frac{\partial}{\partial t}$ and satisfies,

$\tilde{\bar{\nabla}}\tilde{\bar{R}}=0,\quad\tilde{\bar{\nabla}}\tilde{\bar{T}}=0,\quad\tilde{\bar{\nabla}}\bar{S}=0,\quad\tilde{\bar{\nabla}}g=0,$

where $\bar{S}=\nabla-\tilde{\bar{\nabla}}$, the Levi-Civita connection is
$\nabla$, and the curvature and torsion of $\tilde{\bar{\nabla}}$ are
$\tilde{\bar{R}}$ and $\tilde{\bar{T}}$, respectively. We have

$-\tilde{\bar{T}}_{A}B=\bar{S}_{A}B-\bar{S}_{B}A=g(A,\xi)B-g(B,\xi)A.$

Then, for all $X$, $Y\in\mathcal{D}=\\{X:g(X,\xi)=0\\}$,

$\tilde{\bar{T}}_{X}Y=0,\quad\tilde{\bar{T}}_{\xi}Y=-Y\neq 0,$

which means that $\bar{S}$ is a cohomogeneity one structure, but not a
canonical cohomogeneity one structure.

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