# Sasakian structures on tangent sphere bundles of compact rank-one symmetric
spaces

J. C. González-Dávila  Departamento de Matemáticas, Estadística e
Investigación Operativa, University of La Laguna, 38200 La Laguna, Tenerife,
Spain<EMAIL_ADDRESS>
###### Abstract.

A positive answer is given to the existence of Sasakian structures on the
tangent sphere bundle of some Riemannian manifold whose sectional curvature is
not constant. Among other results, it is proved that the tangent sphere bundle
$T_{r}(G/K),$ for any $r>0,$ of a compact rank-one symmetric space $G/K,$ not
necessarily of constant sectional curvature, admits a unique K-contact
structure whose characteristic vector field is the standard field of $T(G/K)$.
Such a structure is in fact Sasakian and it can be expressed as an induced
structure from an almost Hermitian structure on the punctured tangent bundle
$T(G/K)\setminus\\{\mbox{zero section}\\}.$

###### Key words and phrases:

Sasakian structures, tangent sphere bundles, compact rank-one symmetric
spaces, projective Stiefel manifolds, restricted roots

###### 1991 Mathematics Subject Classification:

53C30, 53C35, 53D10.

Research partially supported by the AEI (Spain) and FEDER project
PID2019-105019GB-C21

##  1\. Introduction

The tangent sphere bundle of radius $r>0$ over a Riemannian manifold $(M,g)$
is the hypersurface $T_{r}M=\\{u\in TM\colon g(u,u)=r^{2}\\}$ of the tangent
bundle $TM.$ There is an extensive bibliography on the Riemannian geometry of
$T_{r}M$ equipped with the induced metric $\widetilde{g}^{S}$ from the Sasaki
metric $g^{S},$ specially about the unit tangent sphere bundle $T_{1}M.$ See
E. Boeckx and L. Vanhecke [6, 7] and G. Calvaruso [9] for surveys of $T_{1}M$
and O. Kowalski and M. Sekizawa [12, 13, 14] of $T_{r}M,$ for an arbitrary
radius $r>0.$ The induced metric on $T_{1}M$ from a $g$-natural metric, which
generalizes the Sasaki and the Cheeger-Gromoll metric, has been treated by K.
M. T. Abbassi, G. Calvaruso and M. Sarih (see, for example, [1, 2, 3]).

With respect to the standard almost complex structure $J,$ the Sasaki metric
on $TM$ is Hermitian, in fact it is almost Kähler [18]. Then $T_{r}M,$ as
hypersurface of $(TM,J,g^{S}),$ inherits an almost contact metric structure
$(\varphi,\xi,\eta,\widetilde{g}^{S})$ and the unit vector field $\xi=-JN,$
where $N$ is the outward normal unit vector field to the singular foliation
$\\{T_{r}M\\}_{r\geq 0},$ is defined on $TM\setminus\\{\mbox{\rm zero
section}\\}.$ We say that $\xi$ is the standard vector field of $TM.$

On $T_{1}M,$ the pair $(\frac{1}{2}\eta,\frac{1}{4}\widetilde{g}^{S})$ becomes
a contact metric structure and Tashiro [19] proved it is $K$-contact if and
only if the base manifold $(M,g)$ has constant sectional curvature 1.
Moreover, in this case, $T_{1}M$ is Sasakian. For $g$-natural metrics, $(M,g)$
must be of constant sectional curvature [1, Theorem 2] and then, Sasakian too.
Sasakian manifolds can be considered in many respects as the class analogous
to that of the Kähler manifolds for the odd dimensional case. Principal circle
bundles over Kähler manifolds, together with the standard structure on
${\mathbb{R}}^{2n+1},$ are the best known examples (see, for example, [5, Ch.
6]).

The main objective of this paper is to find Sasakian structures on tangent
sphere bundles $T_{r}M,$ for any radius $r>0,$ over Riemannian manifolds
$(M,g)$ with non-constant sectional curvature. As far as the author knows,
there are no examples satisfying these conditions in the literature.

The most natural Riemannian manifolds $(M,g)$ in this search should be complex
or quaternionic projective spaces, or even the Cayley plane since the tangent
sphere bundle of any compact rank-one symmetric space $G/K$ is, among other
properties, $G$-homogeneous (see [8, 15]) and its standard vector field $\xi$
is $G$-invariant. In order to achieve our objective, it is necessary to go
further and carry out a detailed study of compact rank-one symmetric spaces
and their tangent bundles and tangent sphere bundles. Such a study is included
in Sections $3$ and $4,$ while in Section $2$ some basic concepts about almost
contact metric structures and homogeneous manifolds are exposed.

Given a compact rank-one symmetric space $G/K,$ we set a Cartan decomposition
${\mathfrak{g}}={\mathfrak{m}}\oplus{\mathfrak{k}}$ and a Cartan subspace
${\mathfrak{a}}={\mathbb{R}}\\{X\\}$ of ${\mathfrak{m}}.$ Then the punctured
tangent bundle $D^{+}(G/K):=T(G/K)\setminus\\{\mbox{zero section}\\}$ can be
identified with $G/H\times{\mathbb{R}}^{+}$ and $T_{r}(G/K),$ for each $r>0,$
with the quotient $G/H$ via $G$-equivariant diffeomorphisms, where $H$ is the
closed subgroup $H=\\{k\in K\colon{\rm Ad}_{k}X=X\\}$ of $K$ (Proposition
4.1).

The homogeneity of $T_{r}(G/K)$ is discussed in Section $4,$ where the
quotient expressions $G/H,$ for each compact rank-one symmetric space,
together with the sphere-homogeneous fibration associated to the projection
$\pi^{T}\colon T_{r}(G/K)=G/H\to G/K$ are given. As a direct consequence, a
natural diffeomorphism is established between
$T_{r}{\mathbb{K}}{\mathbf{P}}^{n}$ and the ${\mathbb{K}}$-projective Stiefel
manifold $W_{2}({\mathbb{K}}^{n+1}),$ ${\mathbb{K}}$ being the field of real
numbers, complex numbers or quaternions.

On the other hand, we consider the system $\Sigma$ of restricted roots of
$({\mathfrak{g}},{\mathfrak{k}},{\mathfrak{a}}),$ which is either
$\Sigma=\\{\pm\varepsilon\\}$ or
$\Sigma=\\{\pm\varepsilon,\pm\varepsilon/2\\},$ where
$\varepsilon\in({\mathfrak{a}}^{\mathbb{C}})^{*},$ together with the subspaces

${\mathfrak{m}}_{\lambda}=\\{\xi\in{\mathfrak{m}}\colon{\rm
ad}^{2}_{X}\xi=\lambda^{2}(X)\xi\\},\quad{\mathfrak{k}}_{\lambda}=\\{\zeta\in{\mathfrak{k}}\colon{\rm
ad}^{2}_{X}\zeta=\lambda^{2}(X)\zeta\\},\quad\lambda\in\Sigma^{+}.$

Because the tangent $T_{o_{H}}(G/H)$ at the origin $o_{H}=\\{H\\}$ is
identified with
$\overline{\mathfrak{m}}=({\mathfrak{a}}\oplus{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{m}}_{\varepsilon/2})\oplus({\mathfrak{k}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon/2})$
and this decomposition is ${\rm Ad}(H)$-irreducible, all the $G$-invariant
metrics on $T_{r}(G/K)$ are obtained (Proposition 4.4). In particular, the
induced metric $\tilde{g}^{S}$ of Sasaki metric is analyzed in this context,
which allows obtaining, among other results, a version of Tashiro’s Theorem
for tangent sphere bundles of any radius (Corollary 5.5).

In the last section, Section 5, we use all the material developed in the
previous sections to prove our main result.

######  Theorem 1.1.

Let $G/K$ be a compact rank-one symmetric space and let $\xi$ be the standard
vector field of $T(G/K).$ Then, for each smooth function
$f\colon{\mathbb{R}}^{+}\to{\mathbb{R}}^{+},$ there exists a $G$-invariant
almost Hermitian structure on $T(G/K)\setminus\\{\mbox{zero section}\\}$ such
that its induced almost contact metric structure on $T_{r}(G/K),$ for all
$r>0,$ is $K$-contact and $f(r)^{-1}\xi$ is the characteristic vector field.

Such $K$-contact structure on $T_{r}(G/K),$ for each $r>0,$ is Sasakian and it
is given by $(\kappa^{-1}\xi,\tilde{\mathbf{g}}^{\kappa}),$ where
$\kappa=f(r)$ and the $G$-invariant metric $\tilde{\mathbf{g}}^{\kappa}$ is
determined by

$\tilde{\mathbf{g}}^{\kappa}_{o_{H}}(\cdot,\cdot)=\kappa^{2}\langle\cdot,\cdot\rangle_{\mathfrak{a}}+\frac{\kappa}{2}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}}+\frac{\kappa}{4}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon/2}\oplus{\mathfrak{k}}_{\varepsilon/2}}.$
(1.1)

Moreover, $(\kappa^{-1}\xi,\tilde{\mathbf{g}}^{\kappa})$ is the unique
$G$-invariant $K$-contact structure on $T_{r}(G/K)$ whose characteristic
vector field is $\kappa^{-1}\xi.$

We must point out that, from Proposition 5.2, the $G$-invariant almost
Hermitian structure in Theorem 1.1 cannot be differentially extended to all
$T(G/K)$ and, from Proposition 4.8, the standard contact structure
$(\frac{1}{2}\eta,\frac{1}{4}\tilde{g}^{S})$ on $T_{1}{\mathbb{S}}^{n}$ or on
$T_{1}{\mathbb{R}}{\mathbf{P}}^{n}$ is the Sasakian structure for
$\kappa=\frac{1}{2}.$

##  2\. Preliminaries

###  2.1. Almost contact metric structures

We briefly recall some basic concepts of almost contact metric structures. For
more information, see [5]. An odd-dimensional smooth manifold $M$ is called
almost contact if it admits a $(\varphi,\xi,\eta)$-structure, where $\varphi$
is a tensor field of type $(1,1),$ $\xi$ is a vector field and $\eta$ is a
$1$-form, such that

$\varphi^{2}=-{\rm I}+\eta\otimes\xi,\quad\eta(\xi)=1.$

Then $\varphi\xi=0$ and $\eta\circ\varphi=0.$ If $M$ is equipped with a
Riemannian metric $g$ such that

$g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y),$

for all $X,Y\in{\mathfrak{X}}(M),$ where ${\mathfrak{X}}(M)$ is the Lie
algebra of the vector fields on $M,$ $(M,\varphi,\xi,\eta,g)$ is said to be an
almost contact metric manifold and $g$ is called a compatible metric. We say
it is contact metric if $d\eta(X,Y)=g(X,\varphi Y).$ Then a contact metric
structure is determined by the pair $(\xi,g)$ (or $(\eta,g))$ and $\xi$ is
known as its characteristic vector field. If, in addition, $\xi$ is a Killing
vector field, then the manifold is called $K$-contact.

An almost contact structure $(\varphi,\xi,\eta)$ is said to be normal if the
$(1,1)$-tensor field ${\mathbf{N}}(X,Y)=[\varphi,\varphi](X,Y)+2{\rm
d}\eta(X,Y)\xi$ vanishes for all $X,Y\in{\mathfrak{X}}(M),$ where
$[\varphi,\varphi]$ is the Nijenhuis torsion of $\varphi,$

$[\varphi,\varphi](X,Y)=\varphi^{2}[X,Y]+[\varphi X,\varphi Y]-\varphi[\varphi
X,Y]-\varphi[X,\varphi Y].$ (2.2)

A contact metric structure $(\xi,g)$ which is normal is called a Sasakian
structure. A useful characterization for Sasakian manifolds is the following:
An almost contact metric manifold $(\varphi,\xi,\eta,g)$ is Sasakian if and
only if

$(\nabla_{X}\varphi)Y=g(X,Y)\xi-\eta(Y)X,\quad X,Y\in{\mathfrak{X}}(M),$ (2.3)

where $\nabla$ is the Levi-Civita connection of $(M,g).$ Any Sasakian manifold
is always $K$-contact. The converse holds for the three-dimensional case, but
it may not true in higher dimension.

###  2.2. Homogeneous manifolds

A connected homogeneous manifold $M$ can be described as a quotient manifold
$G/K,$ where $G$ is a Lie group, which is supposed to be connected, acting
transitively and effectively on $M$ and $K$ is the isotropy subgroup of $G$ at
some point $o\in M,$ the origin of $G/K.$ Denote by $\pi_{K}$ the projection
$\pi_{K}\colon G\to G/K,$ $\pi_{K}(a)=aK,$ and by $\tau_{b},$ for each $b\in
G,$ the translation $\tau_{b}\colon G/K\to G/K,$ $\tau_{b}(aK)=baK.$ If
moreover $g$ is a $G$-invariant Riemannian metric on $M=G/K,$ then $(M,g)$ is
said to be a homogeneous Riemannian manifold.

When $G$ is compact, there exists a positive-definite ${\rm Ad}(G)$-invariant
$2$-form $\langle\cdot,\cdot\rangle$ on the Lie algebra ${\mathfrak{g}}$ of
$G$ and we have the reductive decomposition
${\mathfrak{g}}={\mathfrak{m}}\oplus{\mathfrak{k}},$ ${\mathfrak{m}}$ being
the $\langle\cdot,\cdot\rangle$-orthogonal complement of the Lie algebra
${\mathfrak{k}}$ of $K.$ Then the restriction of $\langle\cdot,\cdot\rangle$
to ${\mathfrak{m}}$ determines a $G$-invariant metric $g$ on $M.$

The differential map $(\pi_{K})_{*e}$ of $\pi_{K}$ at the identity element $e$
of $G$ gives a linear isomorphism of ${\mathfrak{m}}$ onto $T_{o}(G/K).$ Here
and in what follows, $T_{o}(G/K)$ is identified with ${\mathfrak{m}}$ via
$(\pi_{K})_{*e}.$ It is clear that there exists a sufficiently small
neighborhood $O_{\mathfrak{m}}\subset{\mathfrak{m}}$ of zero in
${\mathfrak{m}}$ such that $\exp(O_{\mathfrak{m}})$ is a submanifold of $G$
and the mapping $\pi_{K\mid\exp(O_{\mathfrak{m}})}$ is a diffeomorphism onto a
neighborhood ${\mathcal{U}}_{o}$ of $o.$ Then, for each
$\xi\in{\mathfrak{m}},$ define $\xi^{\tau}$ as the vector field on
${\mathcal{U}}_{o}$ such that $\xi^{\tau}_{\exp xK}=(\tau_{\exp x})_{*o}\xi,$
for all $x\in O_{\mathfrak{m}}.$ Then $\xi^{\tau}=(\pi_{K})_{*}\xi^{\tt l},$
where $\xi^{\tt l}$ denotes the left $G$-invariant vector field on $G$ such
that $\xi^{\tt l}_{e}=\xi.$ Under the identification
$T_{o}(G/K)\cong{\mathfrak{m}},$ $\xi^{\tau}_{o}=\xi$ and
$[\xi_{1}^{\tau},\xi^{\tau}_{2}]_{o}=[\xi_{1},\xi_{2}]_{\mathfrak{m}},$ where
$[\cdot,\cdot]_{\mathfrak{m}}$ denotes the ${\mathfrak{m}}$-component of
$[\cdot,\cdot]$ (see [16]).

Let $\alpha\colon{\mathfrak{m}}\times{\mathfrak{m}}\to{\mathfrak{m}}$ be the
${\rm Ad}(K)$-invariant bilinear function that determines $\nabla,$ given by
$\alpha(\xi_{1},\xi_{2})=\nabla_{\xi_{1}}\xi_{2}^{\tau},$ for all
$\xi_{1},\xi_{2}\in{\mathfrak{m}}.$ Then, using the Koszul formula, we have

$\alpha(\xi_{1},\xi_{2})=\frac{1}{2}[\xi_{1},\xi_{2}]_{\mathfrak{m}}+{\mathfrak{U}}(\xi_{1},\xi_{2}),$
(2.4)

where ${\mathfrak{U}}$ is the symmetric bilinear function on
${\mathfrak{m}}\times{\mathfrak{m}}$ such that

$2\langle{\mathfrak{U}}(\xi_{1},\xi_{2}),\xi_{3}\rangle=\langle[\xi_{3},\xi_{1}]_{\mathfrak{m}},\xi_{2}\rangle+\langle[\xi_{3},\xi_{2}]_{\mathfrak{m}},\xi_{1}\rangle.$
(2.5)

When ${\mathfrak{U}}=0,$ $(G/K,g)$ is said to be naturally reductive. A
$G$-invariant vector field $X$ on $G/K$ is Killing if and only if
$\langle\alpha(\xi_{1},\xi),\xi_{2}\rangle+\langle\alpha({\xi_{2}},\xi),\xi_{1}\rangle=0,$
for all $\xi_{1},\xi_{2}\in{\mathfrak{m}},$ where
$\xi=X_{o}\in{\mathfrak{m}}.$ But, from (2.4) and (2.5), we get
$\langle\alpha(\xi_{1},\xi),\xi_{2}\rangle+\langle\alpha({\xi_{2}},\xi),\xi_{1}\rangle=-2\langle{\mathfrak{U}}(\xi_{1},\xi_{2}),\xi\rangle.$
Hence, $X$ is a Killing vector field if and only if
$\langle{\mathfrak{U}}(\cdot,\cdot),\xi\rangle=0.$ In particular, on naturally
reductive homogeneous manifolds any $G$-invariant vector field must be
Killing.

###  2.3. Tangent bundles

Let $M=G/K$ be a homogeneous manifold. On the trivial vector bundle
$G\times{\mathfrak{m}}$ consider two Lie group actions which commute on it:
the left $G$-action, $l_{b}\colon(a,x)\mapsto(ba,x)$ and the right $K$-action
$r_{k}\colon(a,x)\mapsto(ak,{\rm Ad}_{k^{-1}}x)$.

Let $\pi\colon G\times{\mathfrak{m}}\to G\times_{K}{\mathfrak{m}},$
$(a,x)\mapsto[(a,x)],$ be the natural projection for this right $K$-action.
Then $\pi$ is $G$-equivariant and, because the linear isotropy group
$\\{(\tau_{K})_{*o}\,:k\in K\\}$ acting on $T_{o}(G/K)$ corresponds under
$(\pi_{K})_{*e}$ with ${\rm Ad}(K)$ on ${\mathfrak{m}},$ the mapping $\phi$
given by

$\phi\colon G\times_{K}{\mathfrak{m}}\to
T(G/K),\quad[(a,x)]\mapsto(\tau_{a})_{*o}x,$ (2.6)

is a $G$-equivariant diffeomorphism.

Next, we determine the tangent space $T_{[(a,x)]}G\times_{K}{\mathfrak{m}}$
for all $(a,x)\in G\times{\mathfrak{m}}.$ For each $\xi\in{\mathfrak{g}},$
$\xi=\xi_{\mathfrak{m}}+\xi_{\mathfrak{k}}\in{\mathfrak{m}}\oplus{\mathfrak{k}},$

$\pi_{*(a,x)}((\xi_{\mathfrak{k}})^{\tt l}_{a},0)=\frac{d}{dt}_{\mid
t=0}\pi(a\exp t\xi_{\mathfrak{k}},x)=\frac{d}{dt}_{\mid t=0}\pi(a,{\rm
Ad}_{\exp t\xi_{\mathfrak{k}}}x)=\pi_{*(a,x)}(0,[\xi_{\mathfrak{k}},x]_{x}).$

Then, since $\pi_{*(a,x)}(\xi^{\tt
l}_{g},u_{x})=\pi_{*(a,x)}(((\xi_{\mathfrak{m}})^{\tt
l}_{a},u_{x})+((\xi_{\mathfrak{k}})^{\tt l}_{a},0)),$ it follows that

$\pi_{*(a,x)}(\xi^{\tt l}_{a},u_{x})=\pi_{*(a,x)}((\xi_{\mathfrak{m}})^{\tt
l}_{a},u_{x}+[\xi_{\mathfrak{k}},x]_{x})$ (2.7)

and we have

$T_{[(a,x)]}G\times_{K}{\mathfrak{m}}=\\{\pi_{*(a,x)}(\xi^{\tt
l}_{a},u_{x})\colon(\xi,u)\in{\mathfrak{m}}\times{\mathfrak{m}}\\},\quad(a,x)\in
G\times{\mathfrak{m}}.$ (2.8)

##  3\. Compact rank-one symmetric spaces

###  3.1. Restricted roots of symmetric spaces of compact type

We review a few facts about complex simple Lie algebras and symmetric spaces.
See [11, Ch. III and Ch. VII] for more details. Suppose that ${\mathfrak{g}}$
is a compact simple Lie algebra and denote by ${\mathfrak{g}}^{\mathbb{C}}$
its complexification. If ${\mathfrak{t}}^{\mathbb{C}}$ is a Cartan subalgebra
of ${\mathfrak{g}}^{\mathbb{C}},$ we have the root space decomposition

${\mathfrak{g}}^{\mathbb{C}}={\mathfrak{t}}^{\mathbb{C}}\oplus\sum_{\alpha\in\Delta}{\mathfrak{g}}_{\alpha},$

where $\Delta$ is the root system of ${\mathfrak{g}}^{\mathbb{C}}$ with
respect to ${\mathfrak{t}}^{\mathbb{C}}$ and
${\mathfrak{g}}_{\alpha}=\\{\xi\in{\mathfrak{g}}^{\mathbb{C}}\;:\;{\rm
ad}_{t}\xi=\alpha(t)\xi,\;t\in{\mathfrak{t}}^{\mathbb{C}}\\}.$

Let $\Pi=\\{\alpha_{1},\dots,\alpha_{l}\\}$ be a basis of $\Delta.$ Because
the restriction of the Killing form ${\mathbf{B}}$ of
${\mathfrak{g}}^{\mathbb{C}}$ to
${\mathfrak{t}}^{\mathbb{C}}\times{\mathfrak{t}}^{\mathbb{C}}$ is
nondegenerate, there exists a unique element
$t_{\alpha}\in{\mathfrak{t}}^{\mathbb{C}}$ such that
${\mathbf{B}}(t,t_{\alpha})=\alpha(t),$ for all
$t\in{\mathfrak{t}}^{\mathbb{C}}$ and ${\mathfrak{t}}^{\mathbb{C}}$ is
expressed as
${\mathfrak{t}}^{\mathbb{C}}=\sum_{\alpha\in\Delta}{\mathbb{C}}t_{\alpha}.$
Put $<\alpha,\beta>={\mathbf{B}}(t_{\alpha},t_{\beta}).$

Choose root vectors $\\{E_{\alpha}\\}_{\alpha\in\Delta}$ such that
$E_{\alpha}$ and $E_{\beta}$ are orthogonal under ${\mathbf{B}},$ if
$\alpha+\beta\neq 0,$ and ${\mathbf{B}}(E_{\alpha},E_{-\alpha})=1$ and define
the number $N_{\alpha,\beta}$ by
$[E_{\alpha},E_{\beta}]=N_{\alpha,\beta}E_{\alpha+\beta},$ if
$\alpha+\beta\in\Delta$ and $N_{\alpha,\beta}=0$ if
$\alpha+\beta\not\in\Delta.$ Then

$N_{\alpha,\beta}=-N_{-\alpha,-\beta},\;\;\;N_{\alpha,\beta}=-N_{\beta,\alpha}$
(3.9)

and, if $\alpha,\beta,\gamma\in\Delta$ and $\alpha+\beta+\gamma=0,$

$N_{\alpha,\beta}=N_{\beta,\gamma}=N_{\gamma,\alpha}.$ (3.10)

Moreover, given an $\alpha$-series $\beta+n\alpha$ $(p\leq n\leq q)$
containing $\beta,$

$(N_{\alpha,\beta})^{2}=\frac{\textstyle q(1-p)}{\textstyle
2}<\alpha,\alpha>.$ (3.11)

Denote by $\Delta^{+}$ the set of positive roots of $\Delta$ with respect to
some lexicographic order in $\Pi.$ Then each $\alpha\in\Delta^{+}$ may be
written as $\alpha=\sum_{k=1}^{l}n_{k}(\alpha)\alpha_{k},$ where
$n_{k}(\alpha)\in{\mathbb{Z}},$ $n_{k}(\alpha)\geq 0,$ for all $k=1,\dots,l,$
and ${\mathfrak{g}}$ as subspace of ${\mathfrak{g}}^{\mathbb{C}}$ is given by

${\mathfrak{g}}={\mathfrak{t}}\oplus\displaystyle\sum_{\alpha\in\Delta^{+}}({\mathbb{R}}\;U^{0}_{\alpha}+{\mathbb{R}}\;U^{1}_{\alpha}),$

where ${\mathfrak{t}}=\sum_{\alpha\in\Delta}{\mathbb{R}}{\rm i}t_{\alpha},$
$U^{0}_{\alpha}=E_{\alpha}-E_{-\alpha}$ and
$U^{1}_{\alpha}={\mathrm{i}}(E_{\alpha}+E_{-\alpha}).$ Using [11, Theorem 5.5,
Ch. III], we obtain the following.

######  Lemma 3.1.

For all $\alpha,\beta\in\Delta^{+}$ and $a=0,1,$ the following equalities
hold:

  1. (i)

$[U^{a}_{\alpha},it_{\beta}]=(-1)^{a+1}<\alpha,\beta>U^{a+1}_{\alpha};$

  2. (ii)

$[U^{0}_{\alpha},U^{1}_{\alpha}]=2it_{\alpha};$

  3. (iii)

$[U^{a}_{\alpha},U^{b}_{\beta}]=(-1)^{ab}N_{\alpha,\beta}U^{a+b}_{\alpha+\beta}+(-1)^{a+b}N_{-\alpha,\beta}U^{a+b}_{\alpha-\beta},$
where $\alpha\neq\beta$ and $a\leq b.$

Let $\mu=\sum_{i=1}^{l}m_{i}\alpha_{i}$ be the maximal root of $\Delta$ and
consider $t_{j}\in{\mathfrak{t}}^{\mathbb{C}},$ $j=1,\dots,l,$ defined by
$\alpha_{k}(t_{j})=(1/m_{j})\delta_{jk},$ $j,k=1,\dots,l.$ Each inner
automorphism of order $2$ on ${\mathfrak{g}}^{\mathbb{C}}$ is conjugate in the
inner automorphism group of ${\mathfrak{g}}^{\mathbb{C}}$ to some
$\sigma=Ad_{\exp 2\pi{\rm i}t},$ where $t=(1/2)t_{k}$ with $m_{k}=1$
(Hermitian symmetric space) or $t=t_{k}$ with $m_{k}=2$ (non-Hermitian
symmetric space). Denote by $\Delta^{+}(t)$ the positive root system generated
by $\Pi(t),$ where $\Pi(t)=\\{\alpha_{j}\in\Pi\colon j\neq k\\},$ if
$m_{k}=1,$ and $\Pi(t)=\\{\alpha_{j}\in\Pi\colon j\neq k\\}\cup\\{-\mu\\},$ if
$m_{k}=2.$ Then, ${\mathfrak{t}}\subset{\mathfrak{k}},$ or equivalently ${\rm
rank}\;{\mathfrak{g}}={\rm rank}\;{\mathfrak{k}},$ and

${\mathfrak{k}}={\mathfrak{t}}\oplus\displaystyle\sum_{\alpha\in\Delta^{+}(t)}({\mathbb{R}}\;U^{0}_{\alpha}+{\mathbb{R}}\;U^{1}_{\alpha}).$
(3.12)

Because
${\mathbf{B}}(U^{a}_{\alpha},U^{b}_{\beta})=-2\delta_{\alpha\beta}\delta_{ab},$
it follows that the set $\\{U^{a}_{\alpha}\colon
a=0,1;\,\alpha\in\Delta^{+}\setminus\Delta^{+}(t)\\}$ is an orthonormal basis
for $({\mathfrak{m}},-\frac{1}{2}{\mathbf{B}}_{\mid{\mathfrak{m}}}).$

If $G/K$ is a symmetric spaces of compact type, then $G$ is a compact
semisimple Lie group and there exists an involutive automorphism
$\sigma\colon{\mathfrak{g}}\to{\mathfrak{g}}$ such that the corresponding $\pm
1$-eigenspaces
${\mathfrak{k}}=\\{\xi\in{\mathfrak{g}}\colon\sigma(\xi)=\xi\\}$ and
${\mathfrak{m}}=\\{\xi\in{\mathfrak{g}}\colon\sigma(\xi)=-\xi\\}$ determine a
reductive decomposition ${\mathfrak{g}}={\mathfrak{m}}\oplus{\mathfrak{k}}$
with $[{\mathfrak{m}},{\mathfrak{m}}]\subset{\mathfrak{k}}.$

Suppose that $G/K$ has rank ${\mathbf{r}}.$ Then, fixed a
${\mathbf{r}}$-dimensional Cartan subspace ${\mathfrak{a}}$ of
${\mathfrak{m}},$ there exists a $\sigma$-invariant Cartan subalgebra
${\mathfrak{t}}_{\mathfrak{a}}$ of ${\mathfrak{g}}$ containing
${\mathfrak{a}}.$ This implies that
${\mathfrak{t}}_{\mathfrak{a}}={\mathfrak{a}}\oplus{\mathfrak{t}}_{0},$ where
${\mathfrak{t}}_{0}={\mathfrak{t}}_{\mathfrak{a}}\cap{\mathfrak{k}},$ and the
complexification ${\mathfrak{t}}_{\mathfrak{a}}^{\mathbb{C}}$ is a Cartan
subalgebra of ${\mathfrak{g}}^{\mathbb{C}}.$

To the set
$\Sigma=\\{\lambda\in({\mathfrak{a}}^{\mathbb{C}})^{*}\colon\lambda=\alpha_{\mid{\mathfrak{a}}^{\mathbb{C}}},\,\alpha\in\Delta\setminus\Delta_{0}\\}$
of non-vanishing restrictions of roots to ${\mathfrak{a}}^{\mathbb{C}},$ where
$\Delta_{0}=\\{\alpha\in\Delta\colon\alpha_{\mid{\mathfrak{a}}^{\mathbb{C}}}=0\\},$
is known as the set of restricted roots of
$({\mathfrak{g}},{\mathfrak{k}},{\mathfrak{a}}).$ Denote by $\Sigma^{+}$ the
subset of positive restricted roots in $\Sigma$ determined $\Delta^{+}$ and by
$m_{\lambda}$ the multiplicity of $\lambda\in\Sigma^{+},$ that is,
$m_{\lambda}={\rm
card}\\{\alpha\in\Delta\colon\alpha_{\mid{\mathfrak{a}}^{\mathbb{C}}}=\lambda\\}.$
For each linear form $\lambda$ on ${\mathfrak{a}}^{\mathbb{C}}$ put

${\mathfrak{m}}_{\lambda}=\big{\\{}\eta\in{\mathfrak{m}}:\operatorname{ad}^{2}_{w}(\eta)=\lambda^{2}(w)\eta,\
\forall
w\in{\mathfrak{a}}\big{\\}},\quad{\mathfrak{k}}_{\lambda}=\big{\\{}\zeta\in{\mathfrak{k}}:\operatorname{ad}^{2}_{w}(\zeta)=\lambda^{2}(w)\zeta,\
\forall w\in{\mathfrak{a}}\big{\\}}.$

Then ${\mathfrak{m}}_{\lambda}={\mathfrak{m}}_{-\lambda}$,
${\mathfrak{k}}_{\lambda}={\mathfrak{k}}_{-\lambda}$,
${\mathfrak{m}}_{0}={\mathfrak{a}}$ and ${\mathfrak{k}}_{0}$ equals to the
centralizer ${\mathfrak{h}}$ of ${\mathfrak{a}}$ in ${\mathfrak{k}},$ that is,

${\mathfrak{h}}=\\{u\in{\mathfrak{k}}\;\colon[u,{\mathfrak{a}}]=0\\}.$

Clearly ${\mathfrak{t}}_{0}$ is a maximal abelian subalgebra of
${\mathfrak{h}}$ and ${\rm rank}\;{\mathfrak{h}}={\rm
rank}\;{\mathfrak{g}}-{\mathbf{r}}.$ Put
${\mathfrak{h}}={\mathfrak{h}}_{0}\oplus{\mathfrak{h}}_{1},$ where
${\mathfrak{h}}_{0}={\mathfrak{t}}_{0}$ and ${\mathfrak{h}}_{1}$ is the
orthogonal complement of ${\mathfrak{h}}_{0}$ in ${\mathfrak{h}}.$ By [11, Ch.
VII, Lemma 11.3], the following decompositions are direct and orthogonal:

${\mathfrak{m}}={\mathfrak{a}}\oplus\sum_{\lambda\in\Sigma^{+}}{\mathfrak{m}}_{\lambda},\qquad{\mathfrak{k}}={\mathfrak{h}}\oplus\sum_{\lambda\in\Sigma^{+}}{\mathfrak{k}}_{\lambda}.$
(3.13)

Define the linear function
$\lambda_{\mathbb{R}}\colon{\mathfrak{a}}\to{\mathbb{R}}$,
$\lambda\in\Sigma^{+}$, by the relation ${\rm i}\lambda_{\mathbb{R}}=\lambda$.
Note that, since the Lie algebra ${\mathfrak{g}}$ is compact,
$\lambda({\mathfrak{a}})\subset{\rm i}{\mathbb{R}}$. From [10, Lemma 4.2] we
have that for any vector $\xi_{\lambda}\in{\mathfrak{m}}_{\lambda}$,
$\lambda\in\Sigma^{+}$, there exists a unique vector
$\zeta_{\lambda}\in{\mathfrak{k}}_{\lambda}$ such that

$[u,\xi_{\lambda}]=-\lambda_{\mathbb{R}}(u)\zeta_{\lambda},\quad[u,\zeta_{\lambda}]=\lambda_{\mathbb{R}}(u)\xi_{\lambda},\qquad\text{for
all}\ u\in{\mathfrak{a}}.$ (3.14)

###  3.2. Restricted roots of compact rank-one symmetric spaces

Compact rank-one Riemannian symmetric spaces can be characterized as those
symmetric spaces with strictly positive curvature. They are Euclidean spheres,
real, complex and quaternionic projective spaces and the Cayley plane [4, Ch.
3]. Their quotient expressions $G/K$ as symmetric spaces appear in Table I.

Table I. Compact rank-one symmetric spaces
---
| $G/K$ | $\dim$ | $m_{\varepsilon}$ | $m_{\varepsilon/{\scriptscriptstyle 2}}$ | ${\mathfrak{h}}$
$\begin{matrix}\mathbb{S}^{n},(n\geq 2)\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr{\mathbb{R}}{\mathbf{P}}^{n},(n\geq 2)\end{matrix}$ |  $\begin{matrix}\mathrm{SO}(n{+}1)/\mathrm{SO}(n)\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\mathrm{SO}(n{+}1)/{\rm S}({\rm O}(1)\times{\rm O}(n))\end{matrix}$ | $n$ | $n{-}1$ | $0$ | $\mathfrak{so}(n{-}1)$
${\mathbb{C}}{\mathbf{P}}^{n}$, $(n\geq 2)$ | $\mathrm{SU}(n{+}1)/\mathrm{S}(\mathrm{U}(1){\times}\mathrm{U}(n))$ | $2n$ | $1$ | $2n{-}2$ | ${\mathbb{R}}\oplus\mathfrak{su}(n{-}1)$
${\mathbb{H}}{\mathbf{P}}^{n}$, $(n\geq 1)$ | $\mathrm{Sp}(n{+}1)/\mathrm{Sp}(1){\times}\mathrm{Sp}(n)$ | $4n$ | $3$ | $4n{-}4$ | $\mathfrak{sp}(1)\oplus\mathfrak{sp}(n{-}1)$
${\mathbb{C}\mathrm{a}}{\mathbf{P}}^{2}$ | $\mathrm{F_{4}}/\mathrm{Spin(9)}$ | $16$ | $7$ | 8 | $\mathfrak{so}(7)$

Except for odd-dimensional spheres
${\mathbb{S}}^{2n-1}=G/K=\operatorname{\textsl{SO}}(2n)/\operatorname{\textsl{SO}}(2n-1)$
$(n\geq 2),$ where ${\rm rank}\,{\mathfrak{g}}=n>{\rm
rank}\,{\mathfrak{k}}=n-1,$ the rest of all compact rank-one symmetric spaces
admit inner automorphisms. Note that $\dim{\mathfrak{a}}=1$ and, from Lemma
3.1 and (3.12), a compact symmetric space $G/K$ has rank one if and only if
$\alpha+\beta\neq 0$ or $\alpha-\beta\neq 0,$ for all
$\alpha,\beta\in\Delta^{+}\setminus\Delta^{+}(t),$ $\alpha\neq\beta.$

Each compact rank-one symmetric is described in detail below. As a
consequence, the following lemmas, Lemma 3.2 and Lemma 3.3, are proven.

######  Lemma 3.2.

The restricted root system $\Sigma$ of
$({\mathfrak{g}},{\mathfrak{k}},{\mathfrak{a}})$ is either
$\Sigma=\\{\pm\varepsilon\\}$ or
$\Sigma=\\{\pm\varepsilon,\pm\frac{1}{2}\varepsilon\\},$ where
$\varepsilon\in({\mathfrak{a}}^{\mathbb{C}})^{*}.$ The corresponding
multiplicities $m_{\varepsilon}$, $m_{\varepsilon/{\scriptscriptstyle 2}}$ of
these roots and the Lie algebra ${\mathfrak{h}}$ are listed in Table I.

Denote by $X$ the unique (basis) vector $X\in{\mathfrak{a}}$ such that
$\varepsilon_{\mathbb{R}}(X)=1,$ where $\varepsilon={\rm
i}\varepsilon_{\mathbb{R}}.$ Multiplying the inner product
$\langle\cdot,\cdot\rangle$ by a positive constant, we can assume that
$\langle X,X\rangle=1.$

In order to simplify notations we put
${\mathfrak{m}}^{+}={\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{m}}_{\varepsilon/2}$
and
${\mathfrak{k}}^{+}={\mathfrak{k}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon/2},$
where ${\mathfrak{m}}_{\varepsilon/2}=0$ and
${\mathfrak{k}}_{\varepsilon/2}=0$ if $\varepsilon/2$ is not in $\Sigma.$ Then
fixing in ${\mathfrak{m}}_{\varepsilon}$ and ${\mathfrak{m}}_{\varepsilon/2}$
some $\langle\cdot,\cdot\rangle$-orthonormal basis
$\\{\xi^{j}_{\varepsilon}\\}$ and $\\{\xi^{p}_{\varepsilon/2}\\},$
$j=1,\dots,m_{\varepsilon},$ $p=1,\dots,m_{\varepsilon/2},$ we take the unique
bases $\\{\zeta^{j}_{\varepsilon}\\}$ and $\\{\zeta^{p}_{\varepsilon/2}\\}$ of
${\mathfrak{k}}_{\varepsilon}$ and ${\mathfrak{k}}_{\varepsilon/2}$ satisfying
(3.14) for $\lambda=\varepsilon$ and $\lambda=\varepsilon/2.$ Clearly these
last two basis are also $\langle\cdot,\cdot\rangle$-orthonormal.

######  Lemma 3.3.

We have

$[\xi^{j}_{\varepsilon},\xi^{p}_{\varepsilon/2}]=[\zeta^{j}_{\varepsilon},\zeta^{p}_{\varepsilon/2}],\quad[\zeta^{j},\xi^{p}_{\varepsilon/2}]=-[\xi^{j}_{\varepsilon},\zeta^{p}_{\varepsilon/2}],$
(3.15)

for all $j=1,\dots,m_{\varepsilon}$ and $p=1,\dots,m_{\varepsilon/2}.$

Applying [11, Ch. VII, Lemma 11.4, p. 335]), it is immediate the following.

######  Lemma 3.4.

We have:

$\begin{array}[]{l}[{\mathfrak{h}},{\mathfrak{m}}_{\lambda}]\subset{\mathfrak{m}}_{\lambda},\quad[{\mathfrak{h}},{\mathfrak{k}}_{\lambda}]\subset{\mathfrak{k}}_{\lambda},\quad[{\mathfrak{a}},{\mathfrak{m}}_{\lambda}]\subset{\mathfrak{k}}_{\lambda},\quad[{\mathfrak{a}},{\mathfrak{k}}_{\lambda}]\subset{\mathfrak{m}}_{\lambda},\;\;\lambda=\varepsilon,\varepsilon/2,\\\\[4.79993pt]
[{\mathfrak{m}}_{\varepsilon},{\mathfrak{m}}_{\varepsilon}]\subset{\mathfrak{h}},\quad[{\mathfrak{m}}_{\varepsilon},{\mathfrak{m}}_{\varepsilon/2}]\subset{\mathfrak{k}}_{\varepsilon/2},\quad[{\mathfrak{m}}_{\varepsilon},{\mathfrak{k}}_{\varepsilon}]\subset{\mathfrak{a}},\quad[{\mathfrak{m}}_{\varepsilon},{\mathfrak{k}}_{\varepsilon/2}]\subset{\mathfrak{m}}_{\varepsilon/2},\\\\[4.79993pt]
[{\mathfrak{m}}_{\varepsilon/2},{\mathfrak{m}}_{\varepsilon/2}]\subset{\mathfrak{h}}\oplus{\mathfrak{k}}_{\varepsilon},\quad[{\mathfrak{m}}_{\varepsilon/2},{\mathfrak{k}}_{\varepsilon}]\subset{\mathfrak{m}}_{\varepsilon/2},\quad[{\mathfrak{m}}_{\varepsilon/2},{\mathfrak{k}}_{\varepsilon/2}]\subset{\mathfrak{a}}\oplus{\mathfrak{m}}_{\varepsilon},\\\\[4.79993pt]
[{\mathfrak{k}}_{\varepsilon},{\mathfrak{k}}_{\varepsilon}]\subset{\mathfrak{h}},\quad[{\mathfrak{k}}_{\varepsilon},{\mathfrak{k}}_{\varepsilon/2}]\subset{\mathfrak{k}}_{\varepsilon/2},\quad[{\mathfrak{k}}_{\varepsilon/2},{\mathfrak{k}}_{\varepsilon/2}]\subset{\mathfrak{h}}\oplus{\mathfrak{k}}_{\varepsilon}.\end{array}$
(3.16)

Case 1: ${\mathbb{S}}^{n},$ ${\mathbb{R}}{\mathbf{P}}^{n}$ $(n\geq 2).$ Denote
by $E_{jk}$ the $(n+1)\times(n+1)$-matrix with $1$ in the entry of the $j$th
rwo and the $k$th column and $0$ elsewhere. Then the set of matrices
$\\{A_{jk}=E_{jk}-E_{kj}\colon 1\leq j<k\leq n+1\\}$ is an orthonormal basis
of the Lie algebra ${\mathfrak{s}}{\mathfrak{o}}(n+1)$ of
$\operatorname{\textsl{SO}}(n+1)$ with respect to the invariant trace form
$\langle A,B\rangle=-\frac{1}{2}{\rm traceAB},$ for all
$A,B\in{\mathfrak{s}}{\mathfrak{o}}(n+1).$ Here,
${\mathfrak{k}}={\mathbb{R}}\\{A_{jk}\\}_{2\leq j<k\leq n+1}$ is isomorphic to
${\mathfrak{s}}{\mathfrak{o}}(n)$ and
${\mathfrak{m}}={\mathbb{R}}\\{A_{1k}\\}_{2\leq k\leq n+1}$ is its orthogonal
complement in ${\mathfrak{s}}{\mathfrak{o}}(n+1).$ We take as Cartan subspace
to ${\mathfrak{a}}={\mathbb{R}}X,$ where $X=A_{12}.$ Then the centralizer
${\mathfrak{h}}$ of ${\mathfrak{a}}$ in ${\mathfrak{k}}$ is given by

${\mathfrak{h}}={\mathbb{R}}\\{A_{jk}\\}_{3\leq j<k\leq n+1}.$ (3.17)

Because ${\rm ad}^{2}_{X}A_{1k}=-A_{1k},$ for $k=3,\dots,n+1,$ it follows that
${\mathfrak{m}}_{\varepsilon}={\mathbb{R}}\\{A_{1k}\\}_{k=3,\dots,n+1}.$ This
implies that $\Sigma^{+}=\\{\varepsilon\\},$ $m_{\varepsilon}=n-1$ and
$m_{\varepsilon/2}=0.$

Case 2: ${\mathbb{C}}{\mathbf{P}}^{n}$ $(n\geq 2).$ Consider
${\mathfrak{g}}={\mathfrak{s}}{\mathfrak{u}}(n+1)$ as compact real form of the
complex simple Lie algebra
${\mathfrak{a}}_{n}={\mathfrak{s}}{\mathfrak{l}}(n+1,{\mathbb{C}}).$ Let
$\Pi=\\{\alpha_{1},\dots,\alpha_{n}\\}$ be a system of simple roots. Then the
set $\Delta^{+}$ of positive roots is expressed as

$\Delta^{+}=\\{\alpha_{pq}=\alpha_{p}+\dots+\alpha_{q}\;\colon 1\leq p\leq
q\leq n\\}.$

We get
$\langle\alpha_{1},\alpha_{1}\rangle=\dots=\langle\alpha_{n},\alpha_{n}\rangle=-\frac{1}{2}\langle\alpha_{k},\alpha_{k+1}\rangle,$
for all $k=1,\dots,n-1,$ being zero the rest of the products of basic
elements.

The complex projective space ${\mathbb{C}}{\mathbf{P}}^{n}$ is obtained for
the inner automorphism $\sigma={\rm Ad}_{\exp 2\pi it},$ where
$t=\frac{1}{2}t_{1}$ (or, equivalently, $t=\frac{1}{2}t_{n}).$ Then
$\Pi(t)=\\{\alpha_{2},\dots,\alpha_{n}\\},$
$\Delta^{+}(t)=\\{\alpha_{pq}\colon 2\leq p\leq q\leq n\\}$ and
$\Delta^{+}\setminus\Delta^{+}(t)=\\{\alpha_{1q}\colon 1\leq q\leq n\\}.$ We
choose as Cartan subspace to ${\mathfrak{a}}={\mathbb{R}}U^{0}_{\alpha_{1}}.$

From Lemma 3.1 it follows that
$[U^{0}_{\alpha_{1}},it_{\alpha_{1}}+it_{\alpha_{2}}]=0$ and
$[U^{0}_{\alpha_{1}},it_{\alpha_{k}}]=0,$ for all $k=3,\dots,n.$ Therefore,
the Cartan subalgebra ${\mathfrak{t}}^{\mathbb{C}}_{\mathfrak{a}}$ of
${\mathfrak{a}}_{n}$ is the complexification of
${\mathfrak{t}}_{\mathfrak{a}}={\mathfrak{a}}\oplus{\mathfrak{t}}_{0},$ where
${\mathfrak{t}}_{0}={\mathbb{R}}\\{it_{\alpha_{1}}+2it_{\alpha_{2}},\;it_{\alpha_{k}}\;(k=3,\dots,n)\\}.$

Put $X=\beta U^{0}_{\alpha_{1}},$ where
$\beta=(2\langle\alpha_{1},\alpha_{1}\rangle)^{-1}.$ Then, using Lemma 3.1 and
(3.9), (3.10) and (3.11), we have

${\rm ad}^{2}_{X}U^{1}_{\alpha_{1}}=-U^{1}_{\alpha_{1}},\quad{\rm
ad}^{2}_{X}U^{a}_{\alpha_{1(j+1)}}=-\frac{1}{4}U^{a}_{\alpha_{1(j+1)}},$

for all $a=0,1$ and $j=1,\dots,n-1.$ Hence,
${\mathfrak{m}}_{\varepsilon}={\mathbb{R}}U^{1}_{\alpha_{1}}$ and
${\mathfrak{m}}_{\varepsilon/2}={\mathbb{R}}\\{U^{a}_{\alpha_{1(j+1)}}\colon
a=0,1;\;1\leq j\leq n-1\\}.$ Since
$\dim{\mathfrak{m}}_{\varepsilon/2}=2(n-1),$ it follows that
$\Sigma^{+}=\\{\varepsilon,\frac{1}{2}\varepsilon\\}.$

We choose the inner product $\langle\cdot,\cdot\rangle$ such that on
${\mathfrak{m}}$ it is proportional to ${\mathbf{B}}_{\mid{\mathfrak{m}}}$ and
$\langle X,X\rangle=1.$ Then,
$\langle\cdot,\cdot\rangle=-\langle\alpha_{1},\alpha_{1}\rangle{\mathbf{B}}(\cdot,\cdot).$
Hence $\xi_{\varepsilon}=\beta U^{1}_{\alpha_{1}}$ is a unit vector of
${\mathfrak{m}}_{\varepsilon}$ and the set of vectors

$\\{\xi^{j,a}_{\varepsilon/2}=\beta
U^{a}_{\alpha_{1(j+1)}}\colon\;a=0,1,\;j=1,\dots,n-1\\}$

forms an orthonormal basis for ${\mathfrak{m}}_{\varepsilon/2}.$ Applying
Lemma 3.1 and (3.14),
${\mathfrak{k}}_{\varepsilon}={\mathbb{R}}\zeta_{\varepsilon}$ and
${\mathfrak{k}}_{\varepsilon/2}={\mathbb{R}}\\{\zeta^{j,a}_{\varepsilon/2}\colon\,a=0,1,\;j=1,\dots,n-1\\},$
where $\zeta_{\varepsilon}=-{\rm
i}\langle\alpha_{1},\alpha_{1}\rangle^{-1}t_{\alpha_{1}}$ and
$\zeta^{j,a}_{\varepsilon/2}=\beta U^{a}_{\alpha_{2(j+1)}}.$

Let
$\Delta^{+}_{1}(t)=\\{\alpha\in\Delta^{+}(t)\colon\alpha\pm\alpha_{1}\not\in\Delta\\}=\\{\alpha_{pq}\colon
3\leq p\leq q\leq n\\}.$ Then
${\mathfrak{h}}={\mathfrak{h}}_{0}\oplus{\mathfrak{h}}_{1},$ where
${\mathfrak{h}}_{0}={\mathfrak{t}}_{0}$ and
${\mathfrak{h}}_{1}=\sum_{\alpha\in\Delta^{+}_{1}(t)}({\mathbb{R}}U^{0}_{\alpha}+{\mathbb{R}}U^{1}_{\alpha}).$
From Lemma 3.1, the vector $Z_{1}$ given by

$Z_{1}=(n-1)(it_{\alpha_{1}}+2it_{\alpha_{2}})+2((n-2)it_{\alpha_{3}}+(n-3)it_{\alpha_{4}}+\dots+it_{\alpha_{n}})$
(3.18)

belongs to the center ${\mathfrak{z}}({\mathfrak{h}})$ of ${\mathfrak{h}}$ and

${\mathfrak{h}}={\mathbb{R}}Z_{1}\oplus{\mathfrak{h}}_{s},$ (3.19)

where ${\mathfrak{h}}_{s}={\mathbb{R}}\\{{\rm i}t_{\alpha_{k}}\colon
k=3,\dots,n\\}\oplus{\mathfrak{h}}_{1}={\mathfrak{s}}{\mathfrak{u}}(n-1)$ is
the maximal semisimple ideal $[{\mathfrak{h}},{\mathfrak{h}}]$ of
${\mathfrak{h}}.$ By direct calculation, using (3.14) and (3.16), the nonzero
brackets of basic elements of ${\mathfrak{m}}^{+}\oplus{\mathfrak{k}}^{+}$
satisfy the following equalities:

$\begin{array}[]{lcl}[\xi_{\varepsilon},\xi^{j,a}_{\varepsilon/2}]=[\zeta_{\varepsilon},\zeta^{j,a}_{\varepsilon/2}]=\frac{(-1)^{a}}{2}\zeta^{j,a+1}_{\varepsilon/2},&&[\xi_{\varepsilon},\zeta^{j,a}_{\varepsilon/2}]=-[\zeta_{\varepsilon},\xi^{j,a}_{\varepsilon/2}]=\frac{(-1)^{a}}{2}\xi^{j,a+1}_{\varepsilon/2},\\\\[6.0pt]
[\xi^{j}_{\varepsilon/2},\zeta^{j}_{\varepsilon/2}]=[\xi^{j,1}_{\varepsilon/2},\zeta^{j,1}_{\varepsilon/2}]=-\frac{1}{2}X,&&[\xi^{j}_{\varepsilon/2},\zeta^{j,1}_{\varepsilon/2}]=-[\xi^{j,1}_{\varepsilon/2},\zeta^{j}_{\varepsilon/2}]=\frac{1}{2}\xi_{\varepsilon},\\\\[6.0pt]
[\xi_{\varepsilon},\zeta_{\varepsilon}]=-X,&&[\xi^{j}_{\varepsilon/2},\xi^{j,1}_{\varepsilon/2}]_{\overline{\mathfrak{m}}}=-[\zeta^{j}_{\varepsilon/2},\zeta^{j,1}_{\varepsilon/2}]_{\overline{\mathfrak{m}}}=-\frac{1}{2}\zeta_{\varepsilon}.\end{array}$
(3.20)

Case 3: ${\mathbb{H}}{\mathbf{P}}^{n}$ $(n\geq 1).$ The Lie algebra
${\mathfrak{g}}=\operatorname{\mathfrak{sp}}(n+1)$ of
$\operatorname{\textsl{Sp}}(n+1)$ is a compact real form of
${\mathfrak{c}}_{n+1}=\operatorname{\mathfrak{sp}}(n+1,{\mathbb{C}}),$
$\textstyle{\stackrel{{\scriptstyle
2}}{{\stackrel{{\scriptstyle\circ}}{{\alpha_{1}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
2}}{{\stackrel{{\scriptstyle\circ}}{{\alpha_{2}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\;\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
2}}{{\stackrel{{\scriptstyle\circ}}{{\alpha_{n}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
1}}{{\stackrel{{\scriptstyle\circ}}{{\alpha_{n+1}}}}}}$ and

$\Delta^{+}=\\{\alpha_{pq}\;(1\leq p\leq q\leq
n+1),\widetilde{{\alpha}_{pq}}\;(1\leq p\leq q\leq n)\\}$

is a set of positive roots, where
$\widetilde{{\alpha}_{pq}}=\alpha_{p}+\dots+2\alpha_{q}+\dots+2\alpha_{n}+\alpha_{n+1},$
for $1\leq p<q\leq n,$ and
$\widetilde{\alpha_{pp}}=2\alpha_{p}+\dots+2\alpha_{n}+\alpha_{n+1},$ for
$1\leq p\leq n.$ The maximal root is $\mu=\widetilde{\alpha_{11}}$ and the
inner product $\langle\cdot,\cdot\rangle$ on
$\Pi=\\{\alpha_{1},\dots,\alpha_{n+1}\\}$ satisfies

$\langle\alpha_{1},\alpha_{1}\rangle=\dots=\langle\alpha_{n},\alpha_{n}\rangle=\frac{1}{2}\langle\alpha_{n+1},\alpha_{n+1}\rangle,\quad\langle\alpha_{1},\alpha_{1}\rangle=-\langle\alpha_{n},\alpha_{n+1}\rangle=-\frac{1}{2}\langle\alpha_{j},\alpha_{j+1}\rangle,$

$1\leq j\leq n-1,$ being zero the rest of products of basic elements.

The quaternionic projective space ${\mathbb{H}}{\mathbf{P}}^{n}$ appears for
$\sigma={\rm Ad}_{\exp 2\pi{\rm i}t},$ where $t=t_{1}.$ Then,
$\Pi(t)=\\{-\mu,\alpha_{2},\dots,\alpha_{n+1}\\}.$ Hence,
$\Delta^{+}(t)=\\{\alpha_{p,q}\;(2\leq p\leq q\leq
n+1),\;\widetilde{\alpha_{p,q}}\;(2\leq p\leq q\leq n),\mu\\}$ and
$\Delta^{+}\setminus\Delta^{+}(t)=\\{\alpha_{1q}\;(1\leq q\leq
n+1),\widetilde{\alpha_{1q}}\;(2\leq q\leq n)\\}.$

From (3.12),
${\mathfrak{k}}={\mathfrak{s}}{\mathfrak{p}}(1)\oplus{\mathfrak{s}}{\mathfrak{p}}(n),$
where ${\mathfrak{s}}{\mathfrak{p}}(1)\cong{\mathbb{R}}\\{{\rm
i}t_{\alpha_{\mu}},U^{0}_{\mu},U^{1}_{\mu}\\}.$ By Lemma 3.1, because
$\mu\pm\alpha\notin\Delta$ and $\langle\mu,\alpha\rangle=0,$ for all
$\alpha\in\Delta^{+}(t),$ this subalgebra is the center
${\mathfrak{z}}({\mathfrak{k}})$ of ${\mathfrak{k}}.$

We choose as Cartan subalgebra to
${\mathfrak{a}}={\mathbb{R}}U^{0}_{\alpha_{1}}.$ From Lemma 3.1, it follows
that the Cartan subalgebra ${\mathfrak{t}}^{\mathbb{C}}_{\mathfrak{a}}$ of
${\mathfrak{c}}_{n+1}$ containing ${\mathfrak{a}}$ is the complexification of
${\mathfrak{t}}_{\mathfrak{a}}={\mathfrak{a}}\oplus{\mathfrak{t}}_{0},$ where
${\mathfrak{t}}_{0}={\mathbb{R}}\\{{\rm i}t_{\alpha_{1}}+2{\rm
i}t_{\alpha_{2}},{\rm i}t_{\alpha_{k}}\;(k=3,\dots,n+1)\\}.$ Put $X=\beta
U^{0}_{\alpha_{1}},$ where
$\beta=(2\langle\alpha_{1},\alpha_{1}\rangle)^{-1/2}.$ Applying again Lemma
3.1 and using (3.9), (3.10) and (3.11), we have

$\begin{array}[]{lclclcl}{\rm
ad}^{2}_{X}U^{1}_{\alpha_{1}}&=&-U^{1}_{\alpha_{1}},&&{\rm
ad}^{2}_{X}U^{a}_{\widetilde{\alpha_{12}}}&=&-U^{a}_{\widetilde{\alpha_{12}}},\\\\[4.79993pt]
{\rm
ad}^{2}_{X}U^{a}_{\alpha_{1(j+1)}}&=&-\frac{1}{4}U^{a}_{\alpha_{1(j+1)}},&&{\rm
ad}^{2}_{X}U^{a}_{\widetilde{\alpha_{1(k+2)}}}&=&-\frac{1}{4}U^{a}_{\widetilde{\alpha_{1(k+2)}}},\end{array}$

for $a=0,1,$ $j=1,\dots,n$ and $k=1,\dots,n-2.$ We choose the inner product
$\langle\cdot,\cdot\rangle=-\langle\alpha_{1},\alpha_{1}\rangle{\mathbf{B}}(\cdot,\cdot).$
Then $\\{\xi_{\varepsilon}^{1},\xi^{2}_{\varepsilon},\xi^{3}_{\varepsilon}\\}$
is an $\langle\cdot,\cdot\rangle$-orthonormal basis of
${\mathfrak{m}}_{\varepsilon},$ where $\xi^{1}_{\varepsilon}=\beta
U^{1}_{\alpha_{1}}$ and $\xi^{2+a}_{\varepsilon}=\beta
U^{a}_{\widetilde{\alpha_{12}}},$ $a=0,1,$ and the set of vectors

$\xi^{j,a}_{\varepsilon/2}=\beta
U^{a}_{\alpha_{1(j+1)}}\;\;(j=1,\dots,n),\quad\xi^{n+k,a}_{\varepsilon/2}=\beta
U^{a}_{\widetilde{\alpha_{1(k+2)}}}\;\;(k=1,\dots,n-2)$

form an orthonormal basis for ${\mathfrak{m}}_{\varepsilon/2}.$ From Lemma 3.1
and (3.14), we have that the vectors

$\zeta^{1}_{\varepsilon}=-\frac{1}{\langle\alpha_{1},\alpha_{1}\rangle}{\rm
i}{t_{\alpha_{1}}},\quad\zeta^{2+a}_{\varepsilon}=\frac{\sqrt{2}\beta}{2}(U^{a}_{\widetilde{\alpha_{22}}}-U^{a}_{\mu}),\;\;(a=0,1)$

form an orthonormal basis of ${\mathfrak{k}}_{\varepsilon}$ and, of
${\mathfrak{k}}_{\varepsilon/2},$ the following set of vectors

$\zeta^{j,a}_{\varepsilon/2}=\beta
U^{a}_{\alpha_{2(j+1)}},\quad\zeta^{n+k,a}_{\varepsilon/2}=\beta
U^{a}_{\widetilde{\alpha_{2(k+2)}}}.$

Let
$\Delta^{+}_{1}(t)=\\{\alpha\in\Delta^{+}(t)\colon\alpha\pm\alpha_{1}\not\in\Delta\\}=\\{\alpha_{pq}\;(3\leq
p\leq q\leq n+1),\;\widetilde{\alpha_{pq}}\;(3\leq p\leq q\leq n)\\}.$ Then
${\mathfrak{h}}={\mathfrak{h}}_{0}\oplus{\mathfrak{h}}_{1},$ where
${\mathfrak{h}}_{0}={\mathfrak{t}}_{0}$ and ${\mathfrak{h}}_{1}$ is the
subspace

${\mathfrak{h}}_{1}={\mathbb{R}}\\{U^{0}_{\mu}+U^{0}_{\widetilde{\alpha_{22}}},U^{1}_{\mu}+U^{1}_{\widetilde{\alpha_{22}}}\\}\oplus\sum_{\alpha\in\Delta^{+}_{1}(t)}({\mathbb{R}}U^{0}_{\alpha}+{\mathbb{R}}U^{1}_{\alpha}).$

Hence, ${\mathfrak{h}}={\mathbb{R}}\\{{\rm i}t_{\alpha_{1}}+2{\rm
i}t_{\alpha_{2}},U^{a}_{\mu}+U^{a}_{\widetilde{\alpha_{22}}};\;a=0,1\\}\oplus{\mathfrak{s}}{\mathfrak{p}}(n-1).$
By Lemma 3.1 and since $\mu\pm\widetilde{\alpha_{22}}\notin\Delta,$
$\langle\mu,\widetilde{\alpha_{22}}\rangle=0$ and
$\langle\mu,\mu\rangle=\langle\widetilde{\alpha_{22}},\widetilde{\alpha_{22}}\rangle=2\langle\alpha_{1},\alpha_{1}\rangle,$
${\mathbb{R}}\\{{\rm i}t_{\mu}+{\rm
i}t_{\widetilde{\alpha_{22}}},U^{a}_{\mu}+U^{a}_{\widetilde{\alpha_{22}}}\\}$
is a subalgebra isomorphic to ${\mathfrak{s}}{\mathfrak{p}}(1),$ which using
that
$\langle\mu,\alpha\rangle=\langle\widetilde{\alpha_{22}},\alpha\rangle=0,$
$\mu\pm\alpha\notin\Delta$ and $\widetilde{\alpha_{22}}\pm\alpha\notin\Delta$
for all $\alpha\in\Delta^{+}_{1}(t),$ it is the center
${\mathfrak{z}}({\mathfrak{h}})$ of ${\mathfrak{h}}.$ On the other hand, ${\rm
i}t_{\mu}+{\rm i}t_{\widetilde{\alpha_{22}}}=2(({\rm i}t_{\alpha_{1}}+2{\rm
i}t_{\alpha_{2}})+2({\rm i}t_{\alpha_{3}}+\dots+{\rm i}t_{\alpha_{n}})+{\rm
i}t_{\alpha_{n+1}}).$ This implies that
${\mathfrak{h}}={\mathfrak{s}}{\mathfrak{p}}(1)\oplus{\mathfrak{s}}{\mathfrak{p}}(n-1).$

Using (3.9), (3.10), (3.11) and Lemma 3.1, we obtain

$\begin{array}[]{l}[\xi^{1}_{\varepsilon},\xi^{l,a}_{\varepsilon/2}]=[\zeta^{1}_{\varepsilon},\zeta^{l,a}_{\varepsilon/2}]=\frac{(-1)^{a}}{2}\zeta^{l,a+1}_{\varepsilon/2},\quad[\zeta^{1}_{\varepsilon},\xi^{l,a}_{\varepsilon/2}]=-[\xi^{1}_{\varepsilon},\zeta^{l,a}_{\varepsilon/2}]=\frac{(-1)^{a+1}}{2}\xi^{l,a+1}_{\varepsilon/2},\\\\[6.0pt]
[\xi^{2+b}_{\varepsilon},\xi^{k,a}_{\varepsilon/2}]=[\zeta^{2+b}_{\varepsilon},\zeta^{k,a}_{\varepsilon/2}]=\frac{(-1)^{c}}{2}\zeta^{n+k,a+b}_{\varepsilon/2},\quad[\xi^{2+b}_{\varepsilon},\xi^{n+k,a}_{\varepsilon/2}]=[\zeta^{2+b}_{\varepsilon},\zeta^{n+k,a}_{\varepsilon/2}]=\frac{(-1)^{c+1}}{2}\zeta^{k,a+b}_{\varepsilon/2},\\\\[6.0pt]
[\zeta^{2+b}_{\varepsilon/2},\xi^{k,a}_{\varepsilon/2}]=-[\xi^{2+b}_{\varepsilon},\zeta^{k,a}_{\varepsilon/2}]=\frac{(-1)^{c+1}}{2}\xi^{n+k,a+b}_{\varepsilon/2},\;\,[\zeta^{2+b}_{\varepsilon},\xi^{n+k,a}_{\varepsilon/2}]=-[\xi^{2+b}_{\varepsilon},\zeta^{n+k,a}_{\varepsilon/2}]=\frac{(-1)^{c}}{2}\xi^{k,a+b}_{\varepsilon/2},\\\\[6.0pt]
[\xi^{2+b}_{\varepsilon},\xi^{n-1,a}_{\varepsilon/2}]=[\zeta^{2+b}_{\varepsilon},\zeta^{n-1,a}_{\varepsilon/2}]=\frac{(-1)^{c}}{2}\zeta^{n,a+b}_{\varepsilon/2},\quad[\xi^{2+b}_{\varepsilon},\xi^{n,a}_{\varepsilon/2}]=[\zeta^{2+b}_{\varepsilon},\zeta^{n,a}_{\varepsilon/2}]=\frac{(-1)^{c+1}}{2}\zeta^{n-1,a+b}_{\varepsilon/2}\\\\[6.0pt]
[\zeta^{2+b}_{\varepsilon},\xi^{n-1,a}_{\varepsilon/2}]=-[\xi^{2+b}_{\varepsilon},\zeta^{n-1,a}_{\varepsilon/2}]=\frac{(-1)^{c+1}}{2}\xi^{n,a+b}_{\varepsilon},\;\,[\zeta^{2+b}_{\varepsilon},\xi^{n,a}_{\varepsilon/2}]=-[\xi^{2+b}_{\varepsilon},\zeta^{n,a}_{\varepsilon/2}]=\frac{(-1)^{c}}{2}\xi^{n-1,a+b}_{\varepsilon/2},\end{array}$

where $1\leq l\leq 2n-2,$ $1\leq k\leq n-2,$ $a,b=0,1$ and $c=a(b+1).$

Case 4: ${\mathbb{C}}a{\mathbf{P}}^{2}.$ On ${\mathfrak{f}}_{4},$
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\stackrel{{\scriptstyle
4}}{{\stackrel{{\scriptstyle\circ}}{{\alpha_{3}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
134.89024pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\stackrel{{\scriptstyle
2}}{{\stackrel{{\scriptstyle\circ}}{{\alpha_{4}}}}}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$
a set of positive roots is given by

$\begin{array}[]{lcl}\Delta^{+}&=&\\{\alpha_{p,q}\;(1\leq p\leq q\leq
4),\;\alpha_{2,3}+\alpha_{3},\alpha_{1,q}+\alpha_{p,3}\;(p=2,3;\;q=3,4),\alpha_{2,4}+\alpha_{3,q}\;(q=3,4),\\\
&&\alpha_{14}+\alpha_{p4}\;(p=2,3),\alpha_{1,4}+\alpha_{2,q}+\alpha_{3}\;(q=3,4),\alpha_{1,4}+\alpha_{2,4}+\alpha_{p,3}+\alpha_{3}\;(p=2,3),\mu\\},\end{array}$

where $\mu=2\alpha_{1,4}+\alpha_{2,3}+\alpha_{3}$ is the maximal root. We get
the following relationships for the nonzero products on
$\Pi=\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}:$

$1=\langle\alpha_{3},\alpha_{3}\rangle=\langle\alpha_{4},\alpha_{4}\rangle=\frac{1}{2}\langle\alpha_{1},\alpha_{1}\rangle=\frac{1}{2}\langle\alpha_{2},\alpha_{2}\rangle=-\langle\alpha_{1},\alpha_{2}\rangle=-\langle\alpha_{2},\alpha_{3}\rangle=-2\langle\alpha_{3},\alpha_{4}\rangle.$

The Cayley plane ${\mathbb{C}}a{\mathbf{P}}^{2}={\rm F}_{4}/{\rm Spin}(9)$ is
obtained for $\sigma={\rm Ad}_{\exp 2\pi{\rm i}t},$ where $t=t_{4}.$ Then
$\Pi(t)=\\{-\mu,\alpha_{1},\alpha_{2},\alpha_{3}\\}.$ Putting
$\beta_{1}=-\mu,$ $\beta_{2}=\alpha_{1},$ $\beta_{3}=\alpha_{2}$ and
$\beta_{4}=\alpha_{3},$ we have

$\Delta^{+}(t)=\\{\beta_{pq}\;(1\leq p\leq q\leq
4),\;\widetilde{\beta_{pq}}=\beta_{p}+\dots+2\beta_{q}+\dots+2\beta_{4}\;(1\leq
p<q\leq 4)\\},$

which coincides with a positive root set for ${\mathfrak{b}}_{4}:$
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
12.95029pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\stackrel{{\scriptstyle
1}}{{\stackrel{{\scriptstyle\circ}}{{\beta_{1}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
35.71257pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\stackrel{{\scriptstyle
2}}{{\stackrel{{\scriptstyle\circ}}{{\beta_{2}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
84.37543pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\stackrel{{\scriptstyle
2}}{{\stackrel{{\scriptstyle\circ}}{{\beta_{3}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{\hbox{\kern
133.03828pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>0.0pt\raise
1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 0.0pt\raise
1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 133.03828pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\stackrel{{\scriptstyle
2}}{{\stackrel{{\scriptstyle\circ}}{{\beta_{4}}}}}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$
and

$\Delta^{+}\setminus\Delta^{+}(t)=\\{\alpha_{4},\alpha_{3,4},\alpha_{2,4},\alpha_{1,4},\alpha_{1,4}+\alpha_{p,3}\;(p=2,3),\alpha_{2,4}+\alpha_{3},\alpha_{14}+\alpha_{2,3}+\alpha_{3}\\}.$

We choose as Cartan subalgebra to
${\mathfrak{a}}={\mathbb{R}}U^{0}_{\alpha_{4}}.$ Then, from Lemma 3.1,
${\mathfrak{t}}^{\mathbb{C}}_{\mathfrak{a}}$ is the complexification of
${\mathfrak{t}}_{\mathfrak{a}}={\mathfrak{a}}\oplus{\mathfrak{t}}_{0},$ where
${\mathfrak{t}}_{0}={\mathbb{R}}\\{{\rm i}t_{\alpha_{1}},{\rm
i}t_{\alpha_{2}},{\rm i}t_{\alpha_{4}}+2{\rm
i}t_{\alpha_{3}}\\}={\mathbb{R}}\\{{\rm i}t_{\beta_{1}},{\rm
i}t_{\beta_{2}},{\rm i}t_{\beta_{3}}\\}.$ Note that
$2(\alpha_{4}+2\alpha_{3})=-(\beta_{1}+2\beta_{2}+3\beta_{3}).$

Put $X=\frac{\sqrt{2}}{2}U^{0}_{\alpha_{4}}$ and take
$\langle\cdot,\cdot\rangle=-{\mathbf{B}}.$ Then applying again Lemma 3.1 and
(3.9), (3.10) and (3.11), the following basis is orthonormal for
${\mathfrak{m}}_{\varepsilon}$ and ${\mathfrak{m}}_{\varepsilon/2},$
respectively:

$\begin{array}[]{l}\\{\xi^{1}_{\varepsilon}=\frac{\sqrt{2}}{2}U^{1}_{\alpha_{4}},\xi^{p,a}_{\varepsilon}=\frac{\sqrt{2}}{2}U^{a}_{\alpha_{14}+\alpha_{p3}}\;(p=2,3),\xi^{4,a}_{\varepsilon}=\frac{\sqrt{2}}{2}U^{a}_{\alpha_{24}+\alpha_{3}}\\}\\\\[4.79993pt]
\\{\xi^{q,a}_{\varepsilon/2}=\frac{\sqrt{2}}{2}U^{a}_{\alpha_{q4}}\;(q=1,2,3),\;\xi^{4,a}_{\varepsilon/2}=\frac{\sqrt{2}}{2}U^{a}_{\alpha_{14}+\alpha_{23}+\alpha_{3}}\\}.\end{array}$

The corresponding orthonormal basis for ${\mathfrak{k}}_{\varepsilon}$ is
$\\{\zeta^{1}_{\varepsilon},\zeta^{p,a}_{\varepsilon}\;(p=2,3),\zeta^{4,a}_{\varepsilon}\\}$
and, for ${\mathfrak{k}}_{\varepsilon/2},$
$\\{\zeta^{q,a}_{\varepsilon/2}\;(q=1,2,3),\zeta^{4,a}_{\varepsilon/2}\\},$
where

$\begin{array}[]{l}\zeta^{1}_{\varepsilon}=-{\rm
i}t_{\alpha_{4}}=\frac{1}{2}({\rm i}t_{\beta_{1}}+2{\rm i}t_{\beta_{2}}+3{\rm
i}t_{\beta_{3}}+4{\rm i}t_{\beta_{4}}),\\\\[4.79993pt]
\zeta^{p,a}_{\varepsilon}=\frac{1}{2}(U^{a}_{\alpha_{13}+\alpha_{p3}}-U^{a}_{\alpha_{14}+\alpha_{p4}})=\frac{1}{2}(U^{a}_{\widetilde{\beta_{2(p+1)}}}+(-1)^{a}U^{a}_{\widetilde{\beta_{1(6-p)}}}),\\\\[4.79993pt]
\zeta^{4,a}_{\varepsilon}=\frac{1}{2}(U^{a}_{\alpha_{23}+\alpha_{3}}-U^{a}_{\alpha_{24}+\alpha_{34}})=\frac{1}{2}(U^{a}_{\widetilde{\beta_{34}}}+(-1)^{a}U^{a}_{\widetilde{\beta_{12}}}),\\\\[4.79993pt]
\zeta^{q,a}_{\varepsilon/2}=\frac{\sqrt{2}}{2}U^{a}_{\alpha_{q3}}=\frac{\sqrt{2}}{2}U^{a}_{\beta_{(q+1)4}},\quad\zeta^{4,a}_{\varepsilon/2}=\frac{\sqrt{2}}{2}U^{a}_{\alpha_{14}+\alpha_{24}+\alpha_{3}}=\frac{\sqrt{2}}{2}(-1)^{a+1}U^{a}_{\beta_{14}}.\end{array}$

The subalgebra ${\mathfrak{h}}\subset{\mathfrak{k}}$ is of type
${\mathfrak{b}}_{3},$ given by
${\mathfrak{h}}={\mathfrak{h}}_{0}\oplus{\mathfrak{h}}_{1},$ where
${\mathfrak{h}}_{0}={\mathfrak{t}}_{0}$ and

${\mathfrak{h}}_{1}={\mathbb{R}}\\{U^{a}_{\widetilde{\beta_{2(p+1)}}}+(-1)^{a+1}U^{a}_{\widetilde{\beta_{1(6-p)}}}\;(p=2,3),U^{a}_{\widetilde{\beta_{34}}}+(-1)^{a+1}U^{a}_{\widetilde{\beta_{12}}},U^{a}_{\beta_{st}}\;\;(1\leq
s\leq t\leq 3)\\}.$

This proves Lemma 3.2 and, by direct calculation, also Lemma 3.3 for this
case.

##  4\. Tangent sphere bundles of compact rank-one symmetric spaces

###  4.1. Tangent sphere bundles as homogeneous manifolds

Let $(M=G/K,g)$ be a compact rank-one Riemannian symmetric space and let
${\mathfrak{a}}={\mathbb{R}}\\{X\\}$ be a Cartan subspace of
${\mathfrak{m}}=T_{o}M,$ where $X$ is the unique vector in ${\mathfrak{a}}$
such that ${\varepsilon}_{\mathbb{R}}(X)=1$ and the $G$-invariant Riemannian
metric $g$ is determined by the inner product $\langle\cdot,\cdot\rangle$ such
that $\langle X,X\rangle=1.$ Consider the Weyl chamber $W^{+}$ in
${\mathfrak{a}}$ containing $X$ given by

$W^{+}=\\{w\in{\mathfrak{a}}\colon\varepsilon_{\mathbb{R}}(w)>0\\}=\\{tX\;:\;t\in{\mathbb{R}}^{+}\\},$

which is naturally identified with ${\mathbb{R}}^{+}.$ Since each nonzero
${\rm Ad}(K)$-orbit in ${\mathfrak{m}}$ intersects ${\mathfrak{a}}$ and also
$W^{+},$ the open dense subset of regular points ${\mathfrak{m}}^{R}:={\rm
Ad}(K)(W^{+})$ of ${\mathfrak{m}}$ is ${\mathfrak{m}}\setminus\\{0\\}$ and
$\phi(G\times_{K}{\mathfrak{m}}^{R})$ is the punctured tangent bundle
$D^{+}(G/K).$ This means that the action of the subgroup ${\rm Ad}(K)$ on the
unit sphere ${\mathcal{S}}_{\mathfrak{m}}(1)$ of ${\mathfrak{m}}$ is
transitive. Such property characterizes the class of compact rank-one
symmetric spaces. In fact, if ${\rm rank}\,G/K\geq 2,$ the hyperplane
$\lambda(w)=0$ for each restricted root $\lambda$ intersects
${\mathcal{S}}_{\mathfrak{m}}(1)$ and it implies that the ${\rm Ad}(K)$-action
cannot be transitive on ${\mathcal{S}}_{\mathfrak{m}}(1).$

Let $H$ be the closed subgroup of $K$ defined by

$H=\\{k\in K\colon{\rm Ad}_{k}X=X\\}.$

Then $H=G_{\mathfrak{a}}\cap K,$ where $G_{\mathfrak{a}}$ is the centralizer
of ${\mathfrak{a}},$ and ${\mathfrak{h}}$ is the Lie algebra of $H.$ Moreover,
${\mathcal{S}}_{\mathfrak{m}}(1),$ under the $K$-action determined by the
isotropy representation, is expressed as the quotient manifold $K/H.$

Consider the projection $\pi_{H}\times{\rm id}:G\times{\mathbb{R}}^{+}\to
G/H\times{\mathbb{R}}^{+},$ $(a,t)\mapsto(aH,t),$ and the mapping

$f^{+}\colon G/H\times{\mathbb{R}}^{+}\to
G\times_{K}({\mathfrak{m}}\setminus\\{0\\}),\quad(aH,t)\mapsto[(a,tX)],$
(4.21)

which is a well-defined $G$-equivariant diffeomorphism and
$(f^{+})^{-1}([(a,{\rm Ad}_{k}w)])=(akH,t),$ for all $a\in G,$ $k\in K$ and
$w=tX\in W^{+}.$ Then, $G/H\times{\mathbb{R}}^{+}$ is identified via
$\phi\circ f^{+}$ with $D^{+}(G/K).$

######  Proposition 4.1.

The natural action of $G$ on $T(G/K)\cong G\times_{K}{\mathfrak{m}}$ is of
cohomogeneity one. The principal orbits are the tangent sphere bundles
$T_{r}(G/K)$ of radius $r>0,$ which are diffeomorphic to the quotient manifold
$G/H.$ The zero section is the unique singular orbit. Moreover, the natural
projection $\pi^{T}\colon T_{r}(G/K)\to G/K$ determines the homogeneous
fibration

${\mathcal{S}}_{\mathfrak{m}}(r)=K/H\to T_{r}(G/K)=G/H\to G/K,\quad aH\to aK.$
(4.22)

###### Proof.

From the transitivity of the isotropy action on
${\mathcal{S}}_{\mathfrak{m}}(1),$ for each
$u\in{\mathfrak{m}}\setminus\\{0\\}$ there exists $k\in K$ such that $u=r{\rm
Ad}_{k}X,$ where $\langle u,u\rangle=r^{2},$ and the $G$-orbit
$G\cdot[(a,u)]=G\times_{K}\\{u\\}$ of $[(a,u)]$ in $G\times_{K}{\mathfrak{m}}$
can be expressed as

$G\cdot[(a,u)]=\\{[(a,r{\rm Ad}_{k}X)]\colon a\in
G\\}=G\times_{K}{\mathcal{S}}_{\mathfrak{m}}(r).$

Let $\iota_{r}\colon G/H\to G/H\times{\mathbb{R}}^{+}$ be the imbedding given
by $\iota_{r}(aH)=(aH,r).$ Then $f^{+}\circ\iota_{r}$ is a $G$-equivariant
diffeomorphism from $G/H$ onto $G\cdot[(a,u)].$

Taking into account that for each $(a,x)\in G\times{\mathfrak{m}},$ we get
$\langle
x,x\rangle=g_{\tau_{g}(o)}((\tau_{a})_{*o}x,(\tau_{a})_{*o}x)=g_{\tau_{a}(o)}(\phi[(a,x)],\phi[(a,x)]),$
the tangent sphere bundle $T_{r}(G/K)$ is given by

$T_{r}(G/K)=\phi(G\times_{K}{\mathcal{S}}_{\mathfrak{m}}(r)).$ (4.23)

Hence, using that $\phi$ is $G$-equivariant, the $G$-orbit of
$(\tau_{a})_{*o}u$ is $\phi(G\cdot[(a,u)]=T_{r}(G/K)$ and the orbit
$\phi(G\cdot[(a,0)])$ of the zero vector in $T_{aK}(G/K)$ is clearly the zero
section in $T(G/K).$ Therefore, the first part of the proposition is proved.
For the second, note that by definition, $H$ is the isotropy subgroup of the
${\rm Ad}(K)$-action on ${\mathcal{S}}_{\mathfrak{m}}(r)$ at $rX.$ ∎

The compact Lie groups acting effectively and transitively on some sphere
${\mathbb{S}}^{n}\subset{\mathbb{R}}^{n+1}$ have been classified by D.
Montgomery and H. Samelson and A. Borel (see [17] for more information). They
are given, together with their isotropy subgroups and isotropy representations
in Table II, where the superscript indicates the dimension of the
corresponding irreducible subspace.

Table II. Compact transitive groups on spheres
---
$G$ | $K$ | $\dim G/K$ | isotropy repr.
$(1)\;\operatorname{\textsl{SO}}(n)$ | $\operatorname{\textsl{SO}}(n-1)$ | $n-1$ | irred.
$(2)\;\operatorname{\textsl{SU}}(n)$ | $\operatorname{\textsl{SU}}(n-1)$ | $2n-1$ | ${\mathfrak{m}}={\mathfrak{m}}_{1}^{1}\oplus{\mathfrak{m}}_{2}^{2n-2}$
$(3)\;\operatorname{\textsl{U}}(n)$ | $\operatorname{\textsl{U}}(n-1)$ | $2n-1$ | ${\mathfrak{m}}={\mathfrak{m}}_{1}^{1}\oplus{\mathfrak{m}}^{2n-2}_{2}$
$(4)\;\operatorname{\textsl{Sp}}(n)$ | $\operatorname{\textsl{Sp}}(n-1)$ | $4n-1$ | ${\mathfrak{m}}={\mathfrak{m}}_{1}^{3}\oplus{\mathfrak{m}}_{2}^{4n-4}$
$(5)\;\operatorname{\textsl{Sp}}(n)\times\operatorname{\textsl{Sp}}(1)$ | $\operatorname{\textsl{Sp}}(n-1)\times\operatorname{\textsl{Sp}}(1)$ | $4n-1$ | ${\mathfrak{m}}={\mathfrak{m}}_{1}^{3}\oplus{\mathfrak{m}}_{2}^{4n-4}$
$(6)\;\operatorname{\textsl{Sp}}(n)\times\operatorname{\textsl{U}}(1)$ | $\operatorname{\textsl{Sp}}(n-1)\times\operatorname{\textsl{U}}(1)$ | $4n-1$ | ${\mathfrak{m}}={\mathfrak{m}}_{1}^{1}\oplus{\mathfrak{m}}_{2}^{2}\oplus{\mathfrak{m}}_{3}^{4n-4}$
$(7)\;\operatorname{\textsl{Spin}}(9)$ | $\operatorname{\textsl{Spin}}(7)$ | $15$ | ${\mathfrak{m}}={\mathfrak{m}}_{1}^{7}\oplus{\mathfrak{m}}_{2}^{8}$
$(8)\;\operatorname{\textsl{Spin}}(7)$ | $\operatorname{\textsl{G}}_{2}$ | $7$ | irred.
$(9)\;\operatorname{\textsl{G}}_{2}$ | $\operatorname{\textsl{SU}}(3)$ | $6$ | irred.

######  Proposition 4.2.

Tangent sphere bundles of compact rank-one symmetric spaces and their
$\pi^{T}$-homogeneous fibrations admit the following quotient expressions:

$\begin{array}[]{l}{\mathbb{S}}^{n-1}=\operatorname{\textsl{SO}}(n)/\operatorname{\textsl{SO}}(n-1)\to
T_{r}{\mathbb{S}}^{n}=\operatorname{\textsl{SO}}(n+1)/\operatorname{\textsl{SO}}(n-1)\to{\mathbb{S}}^{n},\\\\[4.79993pt]
{\mathbb{S}}^{n-1}={\rm O}(n)/{\rm O}(n-1)\to
T_{r}{\mathbb{R}}{\mathbf{P}}^{n}=\operatorname{\textsl{SO}}(n+1)/S({\rm
O}(1)\times{\rm O}(n-1))\to{\mathbb{R}}{\mathbf{P}}^{n},\\\\[4.79993pt]
{\mathbb{S}}^{2n-1}=\operatorname{\textsl{SU}}(n)/\operatorname{\textsl{SU}}(n-1)\to
T_{r}{\mathbb{C}}{\mathbf{P}}^{n}=\operatorname{\textsl{SU}}(n+1)/S(\operatorname{\textsl{U}}(1)\times\operatorname{\textsl{U}}(n-1))\to{\mathbb{C}}{\mathbf{P}}^{n},\\\\[4.79993pt]
{\mathbb{S}}^{4n-1}=\operatorname{\textsl{Sp}}(n)\times\operatorname{\textsl{Sp}}(1)/\operatorname{\textsl{Sp}}(n-1)\times\operatorname{\textsl{Sp}}(1)\to
T_{r}{\mathbb{H}}{\mathbf{P}}^{n}=\operatorname{\textsl{Sp}}(n+1)/\operatorname{\textsl{Sp}}(n-1)\times\operatorname{\textsl{Sp}}(1)\to{\mathbb{H}}{\mathbf{P}}^{n},\\\\[4.79993pt]
{\mathbb{S}}^{15}=\operatorname{\textsl{Spin}}(9)/\operatorname{\textsl{Spin}}(7)\to
T_{r}{\mathbb{C}}a{\mathbf{P}}^{2}={\rm
F}_{4}/\operatorname{\textsl{Spin}}(7)\to{\mathbb{C}}a{\mathbf{P}}^{2}.\end{array}$

###### Proof.

Applying Proposition 4.1, Table II and the cases studied in Section 3.2, it is
explicitly obtained the subgroup $H$ in each compact rank-one symmetric space:

Case 1: ${\mathbb{S}}^{n}$ and ${\mathbb{R}}{\mathbf{P}}^{n}$ $(n\geq 2).$ For
${\mathbb{S}}^{n}=G/K=\operatorname{\textsl{SO}}(n+1)/\operatorname{\textsl{SO}}(n),$
using (3.17), the identity connected component $H_{0}$ of $H$ is
$I_{2}\times\operatorname{\textsl{SO}}(n-1)\subset\operatorname{\textsl{SO}}(n+1).$
Then, $K/H_{0}=\operatorname{\textsl{SO}}(n)/\operatorname{\textsl{SO}}(n-1)$
is the $(n-1)$-sphere and hence, $H=H_{0}.$

For ${\mathbb{R}}{\mathbf{P}}^{n}=G/K=\operatorname{\textsl{SO}}(n+1)/S({\rm
O}(1)\times{\rm O}(n)),$ taking again $X$ as $A_{12},$ we also have
$H_{0}=I_{2}\times\operatorname{\textsl{SO}}(n-1).$ Since $K$ has two
connected components: $1\times SO(n)$ and $-1\times O^{-}(n),$ where
$O^{-}(n)$ denotes the matrices in $O(n)$ with determinant $-1,$ $H$ is not
connected. Consider the natural isomorphism $\psi\colon{\rm O}(n)\to S({\rm
O}(1)\times{\rm O}(n)),$ $\psi(A)=({\rm det}\,A)\times
A\in\operatorname{\textsl{SO}}(n+1),$ $A\in{\rm O}(n).$ Then, from (4.22),
${\mathbb{S}}^{n-1}=K/H={\rm O}(n)/{\rm O}(n-1).$ Hence, $H=\psi({\rm
O}(n-1))=S({\rm O}(1)\times{\rm O}(n-1))\subset S({\rm O}(1)\times{\rm
O}(n)).$

Case 2: ${\mathbb{C}}{\mathbf{P}}^{n}$ $(n\geq 2).$ Here,
$K=S(\operatorname{\textsl{U}}(1)\times\operatorname{\textsl{U}}(n))\subset\operatorname{\textsl{SU}}(n+1).$
The center ${\mathfrak{z}}({\mathfrak{k}})$ of the Lie algebra
${\mathfrak{k}}$ of $K$ is one-dimensional and

$Z_{0}={\rm diag}({\rm i}b_{0},{\rm i}(b_{0}-1),\dots,{\rm i}(b_{0}-1)),\quad
b_{0}=\frac{n}{n+1},$

is a generator. From (3.19) it is obtained that $H_{0}=\\{\exp tZ_{1}\colon
t\in{\mathbb{R}}\\}\cdot SU(n-1),$ where $Z_{1}$ is given in (3.18). Hence the
matrix

${\rm diag}({\rm i}b_{1},{\rm i}b_{1},{\rm i}(b_{1}-1),\dots,{\rm
i}(b_{1}-1)),\quad b_{1}=\frac{n-1}{n+1},$

is a generator of ${\mathfrak{z}}({\mathfrak{h}})$ and

$H_{0}=\\{e^{itb_{1}}I_{2}\times
e^{it(b_{1}-1)}A\in\operatorname{\textsl{SU}}(n+1)\colon
t\in{\mathbb{R}},\;A\in\operatorname{\textsl{SU}}(n-1)\\}.$

Because $\widetilde{Z_{0}}:={\rm diag}(0,{\rm i}\widetilde{b_{0}},{\rm
i}(\widetilde{b_{0}}-1),\dots,{\rm i}(\widetilde{b_{0}}-1)),$ with
$\widetilde{b_{0}}=\frac{b_{1}}{b_{0}}=\frac{n-1}{n},$ belongs to the center
of the Lie algebra of $1\times{\rm
S}(\operatorname{\textsl{U}}(1)\times\operatorname{\textsl{U}}(n-1))\subset\operatorname{\textsl{SU}}(n+1),$
$H_{0}$ can be identified with this subgroup via the isomorphism $(\exp
tZ_{1})A\mapsto(\exp t\widetilde{Z_{0}})A,$
$A\in\operatorname{\textsl{SU}}(n-1),$ or equivalently, with ${\rm
S}(\operatorname{\textsl{U}}(1)\times\operatorname{\textsl{U}}(n-1))$ via the
isomorphism

$e^{itb_{1}}I_{2}\times e^{it(b_{1}-1)}A\mapsto e^{it\widetilde{b_{0}}}\times
1\times e^{it(\widetilde{b_{0}}-1)}A,\quad
t\in{\mathbb{R}},\;A\in\operatorname{\textsl{SU}}(n-1).$

Then
$K/H_{0}=\operatorname{\textsl{SU}}(n)/\operatorname{\textsl{SU}}(n-1)={\mathbb{S}}^{2n-1}$
and $H=H_{0}.$

Case 3: ${\mathbb{H}}{\mathbf{P}}^{n}$ $(n\geq 1).$ According with Section
3.2, the centers of ${\mathfrak{h}}$ and of ${\mathfrak{k}}$ are isomorphic to
${\mathfrak{s}}{\mathfrak{p}}(1).$ They are given by

${\mathfrak{z}}({\mathfrak{h}})={\mathbb{R}}\\{{\rm i}t_{\mu}+{\rm
i}t_{\widetilde{\alpha_{22}}},U^{a}_{\mu}+U^{a}_{\widetilde{\alpha_{22}}}\colon
a=0,1\\},\quad{\mathfrak{z}}({\mathfrak{k}})={\mathbb{R}}\\{{\rm
i}t_{\mu},U^{a}_{\mu}\colon a=0,1\\}.$

Hence,
$H_{0}=\operatorname{\textsl{Sp}}(1)\cdot\operatorname{\textsl{Sp}}(n-1)$ and
$K/H_{0}=\operatorname{\textsl{Sp}}(n)\times\operatorname{\textsl{Sp}}(1)/\operatorname{\textsl{Sp}}(n-1)\times\operatorname{\textsl{Sp}}(1)={\mathbb{S}}^{4n-1}.$
This means that $H=H_{0}.$

Case 4: ${\mathbb{C}}a{\mathbf{P}}^{2}.$ In Section 3.2 it has been proven
that ${\mathfrak{h}}={\mathfrak{s}}{\mathfrak{o}}(7).$ Since
$K=\operatorname{\textsl{Spin}}(9),$ $H_{0}=\operatorname{\textsl{Spin}}(7).$
Then, $K/H_{0}={\mathbb{S}}^{15}$ and so, $H=H_{0}.$ ∎

Let ${\mathbb{K}}$ be the field of real numbers, complex numbers or
quaternions. On ${\mathbb{K}}^{n+1},$ $n\geq 0,$ consider its standard inner
product. The Stiefel manifold $V_{2}({\mathbb{K}}^{n+1})$ of all orthonormal
$2$\- frames in ${\mathbb{K}}^{n+1}$ is a homogeneous manifold with quotient
expression: $\operatorname{\textsl{SO}}(n+1)/\operatorname{\textsl{SO}}(n-1),$
$\operatorname{\textsl{SU}}(n+1)/\operatorname{\textsl{SU}}(n-1)$ or
$\operatorname{\textsl{Sp}}(n+1)/\operatorname{\textsl{Sp}}(n-1)$ depending on
whether ${\mathbb{K}}={\mathbb{R}},$ ${\mathbb{C}}$ or ${\mathbb{H}},$ and the
projective Stiefel manifold $W_{2}({\mathbb{K}}^{n+1})$ is the quotient space
of the free $S$-action on $V_{2}({\mathbb{K}}^{n+1}),$ where $S$ is the subset
of unit vectors of ${\mathbb{K}}$ $(S={\mathbb{S}}^{0},$ ${\mathbb{S}}^{1}$ or
${\mathbb{S}}^{3}).$

Clearly, the tangent sphere bundle $T_{r}{\mathbb{S}}^{n}$ of
${\mathbb{S}}^{n}\subset{\mathbb{R}}^{n+1}$ is naturally identified with
$V_{2}({\mathbb{R}}^{n+1}),$ by considering each
$u=u_{x}\in{T_{r}}_{x}{\mathbb{S}}^{n}$ as the pair $(x,\frac{1}{r}u)$ in
${\mathbb{R}}^{n+1},$ and $T_{r}{\mathbb{R}}{\mathbf{P}}^{n}$ with
$W_{2}({\mathbb{R}}^{n+1}),$ by identification of $u$ with its opposite. As a
direct consequence from Proposition 4.2, we have the following.

######  Corollary 4.3.

$T_{r}{\mathbb{K}}{\mathbf{P}}^{n}=W_{2}({\mathbb{K}}^{n+1}),$ for
${\mathbb{K}}={\mathbb{R}},$ ${\mathbb{C}}$ or ${\mathbb{H}}.$

###  4.2. The set of all invariant metrics on $T_{r}(G/K)$

According with Proposition 4.1, the tangent sphere bundle $T_{r}(G/K)$ can be
expressed as the homogeneous manifold $G/H.$ Let $\pi_{H}\colon G\to G/H$ be
the natural projection and put
$\overline{\mathfrak{m}}={\mathfrak{m}}\oplus{\mathfrak{k}}^{+}.$ Then
${\mathfrak{g}}=\overline{\mathfrak{m}}\oplus{\mathfrak{h}}$ is a reductive
decomposition associated to $G/H$ and $T_{o_{H}}G/H$, $o_{H}=\\{H\\}$ being
the origin of $G/H,$ is identified with $\overline{\mathfrak{m}}$ via
$(\pi_{H})_{*e}.$ As in Section 3.2, we choose the ${\rm Ad}(G)$-invariant
inner product $\langle\cdot,\cdot\rangle$ on ${\mathfrak{g}}$ satisfying
$\langle X,X\rangle=1,$ where $X\in{\mathfrak{a}}$ and
$\varepsilon_{\mathbb{R}}(X)=1.$ Next, using the identification
$\overline{\mathfrak{m}}\cong T_{o_{H}}(G/H),$ all the $G$-invariant metrics
$\tilde{\mathbf{g}}$ on $T_{r}(G/K)=G/H$ are obtained.

######  Proposition 4.4.

Any $G$-invariant metric on $T_{r}(G/K)$ is determined by an ${\rm
Ad}(H)$-invariant inner product
$\langle\cdot,\cdot\rangle^{(a;a_{\lambda};b_{\lambda})},$
$\lambda\in\Sigma^{+},$ on $\overline{\mathfrak{m}}$ of the form

$\langle\cdot,\cdot\rangle^{(a,a_{\varepsilon},a_{\varepsilon/2},b_{\varepsilon},b_{\varepsilon/2})}=a^{2}\langle\cdot,\cdot\rangle_{\mathfrak{a}}+a_{\varepsilon}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon}}+a_{\varepsilon/2}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon/2}}+b_{\varepsilon}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon}}+b_{\varepsilon/2}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon/2}},$
(4.24)

where $a,a_{\varepsilon},a_{\varepsilon/2},b_{\varepsilon},$ and
$b_{\varepsilon/2}$ are positive constants.

###### Proof.

By definition of $H,$ the vector $X\in{\mathfrak{a}}$ is ${\rm
Ad}(H)$-invariant and we get ${\rm ad}^{2}_{X}\circ{\rm Ad}_{h}={\rm
Ad}_{h}\circ{\rm ad}^{2}_{X},$ for each $h\in H.$ Hence the subspaces
${\mathfrak{m}}_{\varepsilon},{\mathfrak{m}}_{\varepsilon/2},{\mathfrak{k}}_{\varepsilon}$
and ${\mathfrak{k}}_{\varepsilon/2}$ are all ${\rm Ad}(H)$-invariant. Now,
using the isotropy representation of spheres in Table II, it follows from
Proposition 4.2 that the subspaces ${\mathfrak{k}}_{\varepsilon}$ and
${\mathfrak{k}}_{\varepsilon/2}$ of ${\mathfrak{k}}$ are ${\rm
Ad}(H)$-irreducible. Hence, applying (3.14), the representation ${\rm Ad}(H)$
of $H$ is also irreducible on ${\mathfrak{m}}_{\varepsilon}$ and
${\mathfrak{m}}_{\varepsilon/2}.$ This proves the result. ∎

In what follows we denote by
$\tilde{\mathbf{g}}^{(a;a_{\lambda};b_{\lambda})}$ the $G$-invariant metric
$\tilde{\mathbf{g}}$ in (4.24). If $\varepsilon/2$ is not in $\Sigma,$ the
numbers $a_{\varepsilon/2}$ and $b_{\varepsilon/2}$ can take any value and we
will simply write
$\tilde{\mathbf{g}}=\tilde{\mathbf{g}}^{(a,a_{\varepsilon},b_{\varepsilon})}.$

Let
${\mathfrak{U}}\colon\overline{\mathfrak{m}}\times\overline{\mathfrak{m}}\to\overline{\mathfrak{m}}$
be the symmetric bilinear function defined in (2.5) for the Euclidean space
$(\overline{\mathfrak{m}},\langle\cdot,\cdot\rangle^{(a;a_{\lambda};b_{\lambda})}).$
The following lemma is obtained using (3.14), (3.16) and (4.24).

######  Lemma 4.5.

We have:

$\begin{array}[]{l}{\mathfrak{U}}({\mathfrak{a}},{\mathfrak{a}})={\mathfrak{U}}({\mathfrak{m}}_{\varepsilon},{\mathfrak{m}}_{\varepsilon})={\mathfrak{U}}({\mathfrak{k}}_{\varepsilon},{\mathfrak{k}}_{\varepsilon})=0,\\\\[4.79993pt]
{\mathfrak{U}}(X,\xi^{j}_{\varepsilon})=\frac{a^{2}-a_{\varepsilon}}{2b_{\varepsilon}}\zeta^{j}_{\varepsilon},\quad{\mathfrak{U}}(X,\zeta^{j}_{\varepsilon})=\frac{b_{\varepsilon}-a^{2}}{2a_{\varepsilon}}\xi^{j}_{\varepsilon},\quad{\mathfrak{U}}(X,\xi^{p}_{\varepsilon/2})=\frac{a^{2}-a_{\varepsilon/2}}{4b_{\varepsilon/2}}\zeta^{p}_{\varepsilon/2},\\\\[4.79993pt]
{\mathfrak{U}}(X,\zeta^{p}_{\varepsilon/2})=\frac{b_{\varepsilon/2}-a^{2}}{4a_{\varepsilon/2}}\xi^{p}_{\varepsilon/2},\quad{\mathfrak{U}}(\xi^{j}_{\varepsilon},\zeta^{k}_{\varepsilon})=\frac{a_{\varepsilon}-b_{\varepsilon}}{2a}\delta_{jk}X,\quad{\mathfrak{U}}(\xi^{j}_{\varepsilon},\xi^{p}_{\varepsilon/2})=\frac{a_{\varepsilon/2}-a_{\varepsilon}}{2b_{\varepsilon/2}}[\xi^{j}_{\varepsilon},\xi^{p}_{\varepsilon/2}],\\\\[4.79993pt]
{\mathfrak{U}}(\xi^{j}_{\varepsilon},\zeta^{p}_{\varepsilon/2})=\frac{b_{\varepsilon/2}-a_{\varepsilon}}{2a_{\varepsilon/2}}[\xi^{j}_{\varepsilon},\zeta^{p}_{\varepsilon/2}],\quad{\mathfrak{U}}(\xi^{p}_{\varepsilon/2},\zeta^{j}_{\varepsilon})=\frac{b_{\varepsilon}-a_{\varepsilon/2}}{2a_{\varepsilon/2}}[\xi^{p}_{\varepsilon/2},\zeta^{j}_{\varepsilon}],\\\\[4.79993pt]
{\mathfrak{U}}(\zeta^{j}_{\varepsilon},\zeta^{p}_{\varepsilon/2})=\frac{b_{\varepsilon/2}-b_{\varepsilon}}{2b_{\varepsilon/2}}[\zeta^{j}_{\varepsilon},\zeta^{p}_{\varepsilon/2}],\quad{\mathfrak{U}}(\xi^{p}_{\varepsilon/2},\zeta^{q}_{\varepsilon/2})=\frac{a_{\varepsilon/2}-b_{\varepsilon/2}}{2}\Big{(}\frac{\delta_{pq}}{2a}X-\frac{1}{a_{\varepsilon}}[\xi^{p}_{\varepsilon/2},\zeta^{q}_{\varepsilon/2}]_{{\mathfrak{m}}_{\varepsilon}}\Big{)},\end{array}$

for all $j,k=1,\dots,m_{\varepsilon}$ and $p,q=1,\dots,m_{\varepsilon/2}.$

Since $X$ is ${\rm Ad}(H)$-invariant, it determines a $G$-invariant vector
field on $T_{r}(G/K)$ that we also denote by $X.$ As a direct consequence from
this lemma, the following is proved.

######  Proposition 4.6.

We have:

  1. (i)

$X$ on $(T_{r}(G/K),\tilde{\mathbf{g}})$ is a Killing vector field if and only
if $a_{\lambda}=b_{\lambda},$ for all $\lambda\in\Sigma^{+}.$

  2. (ii)

$(T_{r}(G/K)=G/H,\tilde{\mathbf{g}})$ is naturally reductive (with associated
reductive decomposition
${\mathfrak{g}}=\overline{\mathfrak{m}}\oplus{\mathfrak{h}})$ if and only if
$\tilde{\mathbf{g}}$ is proportional to
$\langle\cdot,\cdot\rangle_{\overline{\mathfrak{m}}}.$

  3. (iii)

The projection $\pi^{T}\colon(T_{r}(G/K),\tilde{\mathbf{g}})\to(G/K,g)$ is a
Riemannian submersion if and only if $a=a_{\varepsilon}=a_{\varepsilon/2}=1.$

###  4.3. The standard almost contact metric structure

An affine connection $\nabla$ on arbitrary smooth manifold $M$ defines a
distribution ${\mathcal{H}}\colon u\in TM\to{\mathcal{H}}_{u}\subset T_{u}TM,$
called the horizontal distribution, where ${\mathcal{H}}_{u}$ is the space of
all horizontal lifts of tangent vectors in $T_{p}M,$ $p=\pi^{T}(u),$ obtained
by parallel translation with respect to $\nabla.$ (See, for example [5, Ch. 9]
and references inside for more information). Then $\nabla$ induces the direct
decomposition $T_{u}TM={\mathcal{H}}_{u}\oplus{\mathcal{V}}_{u},$ where
${\mathcal{V}}_{u}$ is the vertical space ${\mathcal{V}}_{u}={\rm
Ker}(\pi^{T})_{*u}.$ The standard almost complex structure $J^{\nabla}$ on
$TM$ induced by $\nabla,$ is defined by

$J^{\nabla}X^{\tt h}=X^{\tt v},\quad J^{\nabla}X^{\tt v}=-X^{\tt h},\quad
X\in{\mathfrak{X}}(M),$ (4.25)

where $X^{\tt h}$ and $X^{\tt v}$ are the horizontal and vertical lift of $X,$
respectively. Given a Riemannian metric $g$ on $M,$ the Sasaki metric $g^{S}$
with respect to the pair $(g,\nabla)$ is given by

$g^{S}(\eta_{1},\eta_{2})=(g((\pi^{T})_{*}\eta_{1},(\pi^{T})_{*}\eta_{2})+g(K(\eta_{1}),K(\eta_{2})))\circ\pi^{T},\quad\eta_{1},\eta_{2}\in
TT(M),$ (4.26)

where $K$ is the connection map of $\nabla.$ Then $g^{S}(X^{\tt h},Y^{\tt
h})=g^{S}(X^{\tt v},Y^{\tt v})=g(X,Y)\circ\pi^{T}$ and $g^{S}(X^{\tt v},Y^{\tt
h})=0,$ for all $X,Y\in{\mathfrak{X}}(M).$ This implies that $g^{S}$ is a
Hermitian metric with respect to $J^{\nabla}.$ If $\nabla$ is the Levi-Civita
connection of $(M,g),$ the pair $(J,g^{S}),$ where $J=J^{\nabla},$ known as
the standard almost Hermitian structure, is almost Kähler.

Now suppose that $M$ is a compact rank-one symmetric space $G/K,$ $g$ is the
$G$-invariant Riemannian metric determined at the origin by the inner product
$\langle\cdot,\cdot\rangle$ on ${\mathfrak{m}}$ and $\nabla$ is its Levi-
Civita connection. Then $(J,g^{S})$ is defined on $G/H\times{\mathbb{R}}^{+}$
by its identification with $D^{+}(G/K)$ via $\phi\circ f^{+},$ i.e.
$J=(\phi\circ f^{+})^{-1}_{*}J(\phi\circ f^{+})_{*}$ and $g^{S}=(\phi\circ
f^{+})^{*}g^{S}.$ Moreover, the tangent space
$T_{(o_{H},t)}(G/H\times{\mathbb{R}}^{+})$ is taken as
$\overline{\mathfrak{m}}\times T_{t}{\mathbb{R}},$ via $(\pi_{H}\times{\rm
id})_{*(e,t)},$ for each $t\in{\mathbb{R}}^{+},$ and the vectors

$\\{(X,0),(0,\frac{\partial}{\partial
t}),(\xi^{j}_{\varepsilon},0),(\xi^{p}_{\varepsilon/2},0),(\zeta^{j}_{\varepsilon},0),(\zeta^{p}_{\varepsilon/2},0);\;j=1,\dots,m_{\varepsilon},\;p=1,\dots,m_{\varepsilon/2}\\}$
(4.27)

is a basis of $T_{(o_{H},t)}(G/H\times{\mathbb{R}}^{+}).$ From (4.21), using
(2.7), we have

$(f^{+})_{*(o_{H},t)}(\xi,\frac{\partial}{\partial
t})=\pi_{*(e,tX)}(\xi,X_{tX})=\pi_{*(e,tX)}(\xi_{\mathfrak{m}},X_{tX}+t[\xi_{\mathfrak{k}},X]_{tX}),$

for all $\xi\in\overline{\mathfrak{m}}.$ Then from (3.14), for all
$s=1,\dots,m_{\lambda},$ $\lambda\in\Sigma^{+},$ we get

$\begin{array}[]{l}(f^{+})_{*(o_{H},t)}(X,0)=\pi_{*(e,tX)}(X,0),\;\;(f^{+})_{*(o_{H},t)}(0,\frac{\partial}{\partial
t})=\pi_{*(e,tX)}(0,X_{tX}),\\\\[4.79993pt]
(f^{+})_{*(o_{H},t)}(\xi^{s}_{\lambda},0)=\pi_{*(e,tX)}(\xi^{s}_{\lambda},0),\;\;(f^{+})_{*(o_{H},t)}(\zeta^{s}_{\lambda},0)=\pi_{*(e,tX)}(0,-\lambda_{\mathbb{R}}(t)(\xi^{s}_{\lambda})_{tX}),\end{array}$
(4.28)

where $\lambda_{\mathbb{R}}$ is taken as the mapping
$\lambda_{\mathbb{R}}\colon{\mathbb{R}}^{+}\to{\mathbb{R}}^{+},$
$\lambda_{\mathbb{R}}(t)=\lambda_{\mathbb{R}}(tX).$ (Then,
$\varepsilon_{\mathbb{R}}={\rm id}_{\mid{\mathbb{R}}^{+}}.)$

######  Proposition 4.7.

The standard almost Hermitian structure $(J,g^{S})$ on
$G/H\times{\mathbb{R}}^{+}$ is $G$-invariant and it is determined at
$(o_{H},t),$ $t\in{\mathbb{R}}^{+},$ by

$\begin{array}[]{l}J_{(o_{H},t)}(X,0)=(0,\frac{\partial}{\partial t}),\qquad
J_{(o_{H},t)}(0,\frac{\partial}{\partial t})=(-X,0),\\\\[4.79993pt]
J_{(o_{H},t)}(\xi^{s}_{\lambda},0)=(-\frac{1}{\lambda_{\mathbb{R}}(t)}\zeta_{\lambda}^{s},0),\quad
J_{(o_{H},t)}(\zeta^{s}_{\lambda},0)=(\lambda_{\mathbb{R}}(t)\xi^{s}_{\lambda},0);\\\\[4.79993pt]
g^{S}_{(o_{H},t)}((X,0),(X,0))=g^{S}_{(o_{H},t)}((0,\frac{\partial}{\partial
t}),(0,\frac{\partial}{\partial t}))=1,\\\\[4.79993pt]
g^{S}_{(o_{H},t)}((\xi^{s}_{\lambda},0),(\xi^{s}_{\lambda},0))=\frac{1}{\lambda^{2}_{\mathbb{R}}(t)}g^{S}_{(o_{H},t)}((\zeta^{s}_{\lambda},0),(\zeta^{s}_{\lambda},0))=1,\end{array}$

where $s=1,\dots,m_{\lambda},$ $\lambda\in\Sigma^{+},$ being zero the rest of
components of $g^{S}.$

###### Proof.

Denote by $\widetilde{\pi}$ the projection $\widetilde{\pi}\colon
G\times_{K}{\mathfrak{m}}\to G/K.$ Then, $\widetilde{\pi}=\pi^{T}\circ\phi.$
Let $\widetilde{\mathcal{V}}$ and $\widetilde{\mathcal{H}}$ be the
distributions $\widetilde{\mathcal{V}}:=(\phi^{-1})_{*}({\mathcal{V}})={\rm
Ker}\;\widetilde{\pi}_{*}$ and
$\widetilde{\mathcal{H}}:=(\phi^{-1})_{*}({\mathcal{H}})$ in
$T(G\times_{K}{\mathfrak{m}}).$ Taking into account that
$(\widetilde{\pi}\circ\pi)(a,x)=\pi_{K}(a),$ for all $(a,x)\in
G\times{\mathfrak{m}},$ and using (2.8), we have

$\widetilde{\mathcal{V}}_{[(a,x)]}=\\{\pi_{*(a,x)}(0,u_{x})\;:\;u\in{\mathfrak{m}}\\},\quad(a,x)\in
G\times{\mathfrak{m}}.$ (4.29)

Because $\nabla$ is $G$-invariant, ${\mathcal{V}},$ $\widetilde{\mathcal{V}},$
${\mathcal{H}}$ and $\widetilde{\mathcal{H}}$ are $G$-invariant distributions.
Moreover,

$\xi^{\tt v}_{[(e,x)]}=\pi_{*(e,x)}(0,\xi_{x}),\quad\xi\in{\mathfrak{m}}.$
(4.30)

For each $\eta\in T_{u}TM$ denote by $\eta^{\tt ver}$ (resp., $\eta^{\tt
hor})$ the ${\mathcal{V}}_{u}$-component (resp.,
${\mathcal{H}}_{u}$-component) of $\eta$ with respect to the decomposition
$T_{u}T(G/K)={\mathcal{H}}_{u}\oplus{\mathcal{V}}_{u}.$ Recall that the
connection map $K$ is defined by $K_{u}(\eta)=\iota_{u}(\eta^{\tt ver}),$
where $\iota_{u}$ is the projection $\iota_{u}\colon T_{u}T(G/K)\to
T_{p}(G/K)$ such that $\iota_{u}(\eta)=0,$ for all $\eta\in{\mathcal{H}}_{u}$
and $\iota_{u}(v^{\tt v}_{u})=v.$ This satisfies
$K_{X_{p}}(X_{*p}u)=\nabla_{u}X,$ for all $X\in{\mathfrak{X}}(G/K)$ and $u\in
T_{p}(G/K).$

For each $x\in{\mathfrak{m}},$ consider $x^{\tau}$ as a local section of
$T(G/K).$ Then

$(x^{\tau})_{*o}\xi=\frac{d}{dt}_{\mid t=0}x^{\tau}(\exp
t\xi)=\frac{d}{dt}_{\mid t=0}(\tau_{\exp
t\xi_{*o}})x=\phi_{*[(e,x)]}(\pi_{*(e,x)}(\xi,0)),\quad\xi\in{\mathfrak{m}}.$
(4.31)

Since, using (2.4), $K_{x}((x^{\tau})_{*o}\xi)=\nabla_{\xi}x^{\tau}=0,$ we
have $((x^{\tau})_{*o}\xi)^{\tt ver}=0_{x}\in T_{x}T_{o}M.$ Then from (4.31),
the vertical component $(\pi_{*(e,x)}(\xi,0))^{\tt ver}$ of
$\pi_{*(e,x)}(\xi,0)$ in the decomposition
$T_{[(e,x)]}G\times_{K}{\mathfrak{m}}=\widetilde{\mathcal{H}}_{[(e,x)]}\oplus\widetilde{\mathcal{V}}_{[(e,x)]}$
is the zero vector. Hence, applying (4.29),

$(\pi_{*(e,x)}(\xi,u_{x}))^{\tt ver}=(\pi_{*(e,x)}(0,u_{x}))^{\tt
ver}=\pi_{*(e,x)}(0,u_{x}),\;\;\mbox{\rm for
all}\;(\xi,u)\in{\mathfrak{m}}\times{\mathfrak{m}}.$

Because
$\widetilde{\pi}_{*[(e,x)]}(\pi_{*(e,x)}(\xi,u_{x}))=\xi\in{\mathfrak{m}}\cong
T_{o}M,$

$\xi^{\tt h}_{[(e,x)]}=(\pi_{*(e,x)}(\xi,u_{x}))^{\tt
hor}_{[(e,x)]}=\pi_{*(e,x)}(\xi,0_{x})$ (4.32)

and $\widetilde{\mathcal{H}}_{[(a,x)]}=\\{\pi_{*(a,x)}(\xi^{\tt
l}_{a},0_{x})\colon\xi\in{\mathfrak{m}}\\}.$ Then, applying in (4.25) and
(4.26), (4.30) and (4.32), $(J,g^{S})$ on $G\times_{K}{\mathfrak{m}}$ is
determined by

$\begin{array}[]{l}J_{[(e,x)]}\pi_{*(e,x)}(\xi,u_{x})=\pi_{*(e,x)}(-u,\xi_{x}),\\\\[4.79993pt]
g^{S}(\pi_{*(e,x)}(\xi,u_{x}),\pi_{*(e,x)}(\eta,v_{x}))=\langle\xi,\eta\rangle+\langle
u,v\rangle,\end{array}$ (4.33)

for all $x,\xi,\eta,u,v\in{\mathfrak{m}}.$ Hence the result is proved using
(4.28). ∎

From Proposition 4.7, $(0,\frac{\partial}{\partial t})$ is a $G$-invariant
unit vector field on the Riemannian manifold
$(G/H\times{\mathbb{R}}^{+},g^{S})$ and it is normal to the submanifolds
$\iota_{r}(G/H)=G/H\times\\{r\\}$ of $G/H\times{\mathbb{R}}^{+},$ for each
$r>0.$ Hence we have the following.

######  Proposition 4.8.

The standard almost contact metric structure
$(\varphi,\xi,\eta,\tilde{g}^{S}=\iota^{*}_{r}g^{S})$ on the tangent sphere
bundle $T_{r}(G/K)=(\phi\circ f^{+})(G/H\times\\{r\\}),$ for each $r>0,$ is
$G$-invariant and it is determined at $o_{H}$ by $\xi_{o_{H}}=X,$
$\eta_{o_{H}}=\langle X,\cdot\rangle,$ $\varphi_{o_{H}}X=0$ and

$\begin{array}[]{l}\varphi_{o_{H}}\xi^{s}_{\lambda}=-\frac{1}{\lambda_{\mathbb{R}}(r)}\zeta^{s}_{\lambda},\quad\varphi_{o_{H}}\zeta^{s}_{\lambda}=\lambda_{\mathbb{R}}(r)\xi^{s}_{\lambda},\hskip
66.86414pts=1,\dots,m_{\lambda},\;\lambda\in\Sigma^{+};\\\\[4.79993pt]
\tilde{g}^{S}_{o_{H}}=\left\\{\begin{array}[]{lcl}\tilde{\mathbf{g}}_{o_{H}}^{(1,1,r^{2})}=\langle\cdot,\cdot\rangle_{\mathfrak{m}}+r^{2}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon}}&\mbox{if}&G/K={\mathbb{S}}^{n}\;\mbox{or}\;\;{\mathbb{R}}{\mathbf{P}}^{n},\\\\[4.79993pt]
\tilde{\mathbf{g}}^{(1;1;\lambda^{2}_{\mathbb{R}}(r))}_{o_{H}}=\langle\cdot,\cdot\rangle_{\mathfrak{m}}+r^{2}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon}}+\frac{r^{2}}{4}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon/2}}&\mbox{if}&G/K={\mathbb{C}}{\mathbf{P}}^{n},\;{\mathbb{H}}{\mathbf{P}}^{n}\;\mbox{or}\;\;{\mathbb{C}}a{\mathbf{P}}^{n}.\end{array}\right.\end{array}$

Moreover, the standard field $\xi$ of $T(G/K)$ is the $G$-invariant vector
field determined by $\xi_{(o_{H},t)}=(X,0),$ for all $t\in{\mathbb{R}}^{+}.$

From Propositions 4.6 and 4.8, the projection
$\pi^{T}\colon(T_{r}(G/K),\widetilde{\mathbf{g}})\to(G/K,{\mathbf{g}})$ is a
Riemannian submersion and the following is proved.

######  Corollary 4.9.

The standard vector field $\xi$ of $(T_{r}(G/K),\tilde{g}^{S})$ is Killing if
and only if $r=1$ and $G/K={\mathbb{S}}^{n}$ or
${\mathbb{R}}{\mathbf{P}}^{n}.$ Then $(G/H,\tilde{g}^{S})$ is naturally
reductive.

##  5\. Invariant Sasakian structures on $T_{r}(G/K)$

We first consider all $G$-invariant Riemannian metrics ${\mathbf{g}}$ on
$G/H\times{\mathbb{R}}^{+}\cong D^{+}(G/K)$ such that the vector field
$(0,\frac{\partial}{\partial t})$ on $G/H\times{\mathbb{R}}^{+}$ is orthogonal
to the hypersurface $\iota_{r}(G/H)=G/H\times\\{r\\}\cong T_{r}(G/K),$ for
each $r>0.$ From Proposition 4.4, ${\mathbf{g}}$ is determined at the points
$(o_{H},t)\in G/H\times{\mathbb{R}}^{+}$ and with respect to the basis given
in (4.27), by

$\begin{array}[]{lcl}{\mathbf{g}}_{(o_{H},t)}((X,0),(X,0))=a^{2}(t),&{\mathbf{g}}_{(o_{H},t)}((0,\frac{\partial}{\partial
t}),(0,\frac{\partial}{\partial t}))=b^{2}(t),\\\\[4.79993pt]
{\mathbf{g}}_{(o_{H},t)}((\xi^{s}_{\lambda},0),(\xi^{s}_{\lambda},0))=a_{\lambda}(t),&{\mathbf{g}}_{(o_{H},t)}((\zeta^{s}_{\lambda},0),(\zeta^{s}_{\lambda},0))=b_{\lambda}(t),\end{array}$
(5.34)

where $a,b,a_{\lambda},b_{\lambda}\colon{\mathbb{R}}^{+}\to{\mathbb{R}}^{+}$
are smooth functions, for each $\lambda\in\Sigma^{+},$ being zero for the rest
of components.

Next, we introduce a family of $G$-invariant almost complex structures on
$D^{+}(G/K)$ containing the standard structure $J_{\mid D^{+}(G/K)}.$ Let
$q\colon{\mathbb{R}}^{+}\to{\mathbb{R}}^{+}$ be a smooth function and let
${\mathcal{J}}^{q}_{t}$ be the $(1,1)$-tensor on
$T_{(o_{H},t)}(G/H\times{\mathbb{R}}^{+})\cong\overline{\mathfrak{m}}\times
T_{t}{\mathbb{R}},$ for each $t\in{\mathbb{R}}^{+},$ given by

$\begin{array}[]{lcl}{\mathcal{J}}^{q}_{t}(X,0)=(0,\frac{\partial}{\partial
t}),&&{\mathcal{J}}^{q}_{t}(0,\frac{\partial}{\partial
t})=(-X,0),\\\\[4.79993pt]
{\mathcal{J}}^{q}_{t}(\xi^{s}_{\lambda},0)=(-\frac{1}{q_{\lambda}(t)}\zeta^{s}_{\lambda},0),&&{\mathcal{J}}^{q}_{t}(\zeta^{s}_{\lambda},0)=(q_{\lambda}(t)\xi^{s}_{\lambda},0),\end{array}$
(5.35)

for all $s=1,\dots,m_{\lambda},$ $\lambda\in\Sigma^{+},$ where
$q_{\lambda}=q\circ\lambda_{\mathbb{R}}.$ Because the subspaces
${\mathfrak{m}}_{\lambda}$ and ${\mathfrak{k}}_{\lambda}$ are ${\rm
Ad}(H)$-invariant, ${\mathcal{J}}^{q}$ on
$({\mathfrak{m}}^{+}\oplus{\mathfrak{k}}^{+})\times\\{0\\}$ is ${\rm
Ad}(H)$-invariant and ${\mathcal{J}}^{q}$ determines a (unique) $G$-invariant
almost complex structure $J^{q}$ on $G/H\times{\mathbb{R}}^{+}$ such that
$J^{q}_{(o_{H},t)}={\mathcal{J}}^{q}_{t}.$

From Proposition 4.7, $J^{q},$ $q$ being the identity map, is the standard
almost complex structure $J$ on $D^{+}(G/K)$ and the Sasaki metric $g^{S}$
coincides with ${\mathbf{g}}$ in (5.34), where the functions $a,$ $b$ and
$a_{\lambda}$ are constant, with $a=b=a_{\lambda}=1,$ and
$b_{\lambda}=\lambda^{2}_{\mathbb{R}}.$

######  Lemma 5.1.

$J^{q}:=(f^{+})_{*}J^{q}(f^{+})^{-1}_{*}$ on
$G\times_{K}({\mathfrak{m}}\setminus\\{0\\})$ is given by

$\begin{array}[]{lcl}J^{q}_{[(a,x)]}\pi_{*}(\xi^{\tt
l}_{a},u_{x})&=&\pi_{*}(-\langle u,{\rm Ad}_{k}X\rangle X^{\tt
l}_{ak}-\sum_{\lambda\in\Sigma^{+}}\frac{q_{\lambda}(t)}{\lambda_{\mathbb{R}}(t)}\sum_{s=1}^{m_{\lambda}}\langle
u,{\rm Ad}_{k}\xi^{s}_{\lambda}\rangle(\xi^{s}_{\lambda})^{\tt
l}_{ak},\\\\[4.79993pt] &&\hskip 42.67912pt\langle\xi,{\rm Ad}_{k}X\rangle
X_{tX}+\sum_{\lambda\in\Sigma^{+}}\frac{\lambda_{\mathbb{R}}(t)}{q_{\lambda}(t)}\sum_{s=1}^{m_{\lambda}}\langle\xi,{\rm
Ad}_{k}\xi^{s}_{\lambda}\rangle(\xi^{s}_{\lambda})_{tX}),\end{array}$

for all $a\in G,$ $x=t{\rm Ad}_{k}X,$ $t\in{\mathbb{R}}^{+}$ and $k\in K,$ and
$(\xi,u)\in{\mathfrak{m}}\times{\mathfrak{m}}.$

###### Proof.

Using (2.7), (2.8) and (4.28), together with (5.35), we have

$\begin{array}[]{lcl}(J^{q})_{[(a,tX)]}\pi_{*}(X^{\tt
l}_{a},0)=\pi_{*}(0,X_{tX}),&&(J^{q})_{[(a,tX)]}\pi_{*}(0,X_{x})=\pi_{*}(-X^{\tt
l}_{a},0),\\\\[4.79993pt] (J^{q})_{[(a,tX)]}\pi_{*}((\xi^{s}_{\lambda})^{\tt
l}_{a},0)=\pi_{*}(0,\frac{\lambda_{\mathbb{R}}(t)}{q_{\lambda}(t)}(\xi^{s}_{\lambda})_{tX}),&&(J^{q})_{[(a,tX)]}\pi_{*}(0,(\xi^{s}_{\lambda})_{tX})=\pi_{*}(-\frac{q_{\lambda}(t)}{\lambda_{\mathbb{R}}(t)}(\xi^{s}_{\lambda})^{\tt
l}_{a},0).\end{array}$

Then, for each $(\xi,u)\in{\mathfrak{m}}\times{\mathfrak{m}},$

$\begin{array}[]{lcl}(J^{q})_{[(a,tX)]}\pi_{*}(\xi^{\tt
l}_{a},u_{tX})&=&\pi_{*}(-\langle u,X\rangle X^{\tt
l}_{a}-\sum_{\lambda\in\Sigma^{+}}\frac{q_{\lambda}(t)}{\lambda_{\mathbb{R}}(t)}\sum_{s=1}^{m_{\lambda}}\langle
u,\xi^{s}_{\lambda}\rangle(\xi^{s}_{\lambda})^{\tt l}_{a},\\\\[4.79993pt]
&&\hskip 42.67912pt\langle\xi,X\rangle
X_{tX}+\sum_{\lambda\in\Sigma^{+}}\frac{\lambda_{\mathbb{R}}(t)}{q_{\lambda}(t)}\sum_{s=1}^{m_{\lambda}}\langle\xi,\xi^{s}_{\lambda}\rangle(\xi^{s}_{\lambda})_{tX}).\end{array}$
(5.36)

On the other hand,

$\begin{array}[]{lcl}\pi_{*}(\xi^{\tt l}_{a},u_{x})&=&\frac{d}{ds}_{\mid
s=0}\pi(a\exp s\xi,{\rm Ad}_{k}(tX+s{\rm Ad}_{k^{-1}}u))\\\\[4.79993pt]
&=&\frac{d}{ds}_{\mid s=0}\pi(a(\exp s\xi)k,tX+s{\rm
Ad}_{k^{-1}}u)=\pi_{*}(({\rm Ad}_{k^{-1}}\xi)^{\tt l}_{ak},({\rm
Ad}_{k^{-1}}u)_{tX}).\end{array}$

Hence, $J^{q}_{[(a,x)]}\pi_{*}(\xi^{\tt
l}_{a},u_{x})=J^{q}_{[(ak,tX)]}\pi_{*}(({\rm Ad}_{k^{-1}}\xi)^{\tt
l}_{ak},({\rm Ad}_{k^{-1}}u)_{tX}).$ Now the result follows from (5.36),
taking into account that $\langle\cdot,\cdot\rangle$ is ${\rm
Ad}(K)$-invariant. ∎

######  Proposition 5.2.

The almost complex structure $J^{q}$ on $D^{+}(G/K)$ can be extended to a
$G$-invariant almost complex structure on whole $T(G/K)$ if and only if

$0<\lim_{t\to 0^{+}}\frac{q(t)}{t}<+\infty.$ (5.37)

###### Proof.

From Lemma 5.1, since ${\rm
Ad}(K)({\mathbb{R}}^{+})={\mathfrak{m}}\setminus\\{0\\},$ the existence of an
extension of $J^{q}$ is determined by the existence with (finite) positive
value of $\lim_{t\to
0}\frac{q_{\lambda}(t)}{\lambda_{\mathbb{R}}(t)}=\lim_{t\to 0}\frac{q(t)}{t}.$
∎

From (5.34) and (5.35), the following is immediate.

######  Lemma 5.3.

The metric ${\mathbf{g}}$ on $G/H\times{\mathbb{R}}^{+}$ is Hermitian with
respect to $J^{q}$ if and only if $a=b$ and $b_{\lambda}=q^{2}_{\lambda}\cdot
a_{\lambda},$ for $\lambda\in\Sigma^{+}.$

Moreover, the almost contact metric structure on $G/H\cong G/H\times\\{r\\},$
induced from the Hermitian structure $(J^{q},{\mathbf{g}}),$ is the
$G$-invariant structure
$(\varphi^{q},\frac{1}{a(r)}\xi,a(r)\eta,\tilde{\mathbf{g}}=(\iota_{r})^{*}{\mathbf{g}}),$
such that at $o_{H}$ and with respect to the basis
$\\{X,\xi^{s}_{\lambda},\zeta^{s}_{\lambda};\;s=1,\dots,m_{\lambda},\lambda\in\Sigma^{+}\\}$
of $\overline{\mathfrak{m}},$

$\begin{array}[]{l}\varphi^{q}_{o_{H}}\xi^{s}_{\lambda}=-\frac{1}{q_{\lambda}(r)}\zeta^{s}_{\lambda},\quad\varphi^{q}_{o_{H}}\zeta^{s}_{\lambda}=q_{\lambda}(r)\xi^{s}_{\lambda},\quad\tilde{\mathbf{g}}_{o_{H}}=\tilde{\mathbf{g}}_{o_{H}}^{(a^{2}(r);a_{\lambda}(r);(q_{\lambda}(r))^{2}a_{\lambda}(r))}.\end{array}$
(5.38)

######  Proposition 5.4.

$(\varphi^{q},\frac{1}{a(r)}\xi,a(r)\eta,\tilde{\mathbf{g}})$ on $T_{r}(G/K),$
for each $r>0,$ is contact metric if and only if

$a_{\lambda}(r)=\frac{a(r)\lambda_{\mathbb{R}}(r)}{2rq_{\lambda}(r)}.$

Moreover, it is $K$-contact if and only if $q_{\lambda}(r)=1,$ for all
$\lambda\in\Sigma^{+}.$

###### Proof.

For $u,v\in\overline{\mathfrak{m}},$
$d\eta_{o_{H}}(u,v)=-\frac{1}{2}\eta_{o_{H}}([u,v]_{\overline{\mathfrak{m}}})=-\frac{1}{2}\langle[X,u],v\rangle.$
Then, using (3.14),

$d\eta_{o_{H}}(\xi^{s}_{\lambda},\zeta^{s}_{\lambda})=\frac{\lambda_{\mathbb{R}}(r)}{2r},\quad
s=1,\dots,m_{\lambda},\;\lambda\in\Sigma^{+},$

being zero the rest of components of $d\eta_{o_{H}}.$ On the other hand, from
(5.38), the fundamental $2$-form $\Phi$ of
$(\varphi^{q},\frac{1}{a(r)}\xi,a(r)\eta,\tilde{\mathbf{g}})$ satisfies
$\Phi_{o_{H}}(\xi^{s}_{\lambda},\zeta^{s}_{\lambda})=q_{\lambda}(r)a_{\lambda}(r).$
Hence, using Proposition 4.6 (i), the result is proved. ∎

As a consequence from Propositions 4.8 and 5.4, we have the following version
of Tashiro’s Theorem [19] for tangent sphere bundles of any radius.

######  Corollary 5.5.

The standard almost contact metric structure
$(\varphi,\xi,\eta,\tilde{g}^{S})$ on $T_{r}(G/K)$ is contact metric if and
only if $r=\frac{1}{2}.$ Moreover, for all $r>0,$ the rectified almost contact
metric structure
$(\varphi^{\prime}=\varphi,\xi^{\prime}=2r\xi,\eta^{\prime}=\frac{1}{2r}\eta,\widetilde{\mathbf{g}}^{\prime}=\frac{1}{4r^{2}}\tilde{g}^{S})$
is always contact metric and it is $K$-contact if and only if $r=1$ and
$G/K={\mathbb{S}}^{n}$ or ${\mathbb{R}}{\mathbf{P}}^{n};$ in that case
$(\varphi^{\prime},\xi^{\prime},\eta^{\prime},\widetilde{\mathbf{g}}^{\prime})$
is Sasakian.

######  Remark 5.6.

$({\mathbb{S}}^{n},{\mathbf{g}})$ and
$({\mathbb{R}}{\mathbf{P}}^{n},{\mathbf{g}}),$ described in Case 1, have
constant curvature $1.$

We conclude with the proof of our main result.

Proof of Theorem 1.1. Denote by $J^{1}$ the almost complex structure on
$D^{+}(G/K)$ such that $J^{1}_{(o_{H},t)}={\mathcal{J}}^{q}_{t},$ for all
$t\in{\mathbb{R}}^{+},$ $q$ being the constant map $q=1.$ Note that, from
Proposition 5.2, such a structure cannot be extended to an almost complex
structure on whole $T(G/K).$

For each smooth function $f\colon{\mathbb{R}}^{+}\to{\mathbb{R}}^{+},$
consider the compatible $G$-invariant metric ${\mathbf{g}}^{f}$ to $J^{1},$
such that at $(o_{H},t)\in G/H\times{\mathbb{R}}^{+}$ it is given as in (5.34)
for $a=b=f$ and
$a_{\lambda}(t)=b_{\lambda}(t)=\frac{f(t)\lambda_{\mathbb{R}}(t)}{2t},$ for
all $t\in{\mathbb{R}}^{+}$ and $\lambda\in\Sigma^{+},$ i.e.

$\begin{array}[]{l}{\mathbf{g}}^{f}_{(o_{H},t)}((X,0),(X,0))={\mathbf{g}}^{f}_{(o_{H},t)}((0,\frac{\partial}{\partial
t}),(0,\frac{\partial}{\partial t}))=f^{2}(t),\\\\[4.79993pt]
{\mathbf{g}}^{f}_{(o_{H},t)}((\xi^{s}_{\lambda},0),(\xi^{s}_{\lambda},0))={\mathbf{g}}^{f}_{(o_{H},t)}((\zeta^{s}_{\lambda},0),(\zeta^{s}_{\lambda},0))=\frac{f(t)\lambda_{\mathbb{R}}(t)}{2t},\quad
s=1,\dots,m_{\lambda},\;\lambda\in\Sigma^{+},\end{array}$

being zero for the rest of components. The induced metric
$\tilde{\mathbf{g}}^{\kappa},$ $\kappa=f(r),$ from ${\mathbf{g}}^{f}$ on
$T_{r}(G/K)=G/H,$ for each $r>0,$ is the $G$-invariant metric satisfying (1.1)
and, from Proposition 5.4, the pair
$(\kappa^{-1}\xi,\tilde{\mathbf{g}}^{\kappa})$ is a $K$-contact structure.

Next, we show that $(\kappa^{-1}\xi,\tilde{\mathbf{g}}^{\kappa})$ is the
unique $G$-invariant $K$-contact structure on $T_{r}(G/K)$ whose
characteristic vector field is $\kappa^{-1}\xi.$ Let
$\tilde{\mathbf{g}}=\tilde{\mathbf{g}}^{(a;a_{\lambda};b_{\lambda})}$ be a
$G$-invariant Riemannian metric as in (4.24) such that
$(\kappa^{-1}\xi,\tilde{\mathbf{g}})$ is $K$-contact. Then, $a=\kappa$ and,
from Proposition 4.6, $b_{\lambda}=a_{\lambda},$ for all
$\lambda\in\Sigma^{+}.$ Moreover there exists a $G$-invariant $(1,1)$-tensor
field $\varphi$ such that
$(\varphi,\kappa^{-1}\xi,\kappa\eta,\tilde{\mathbf{g}})$ is an almost contact
metric structure verifying

$\tilde{\mathbf{g}}_{o_{H}}(\varphi u,v)=\kappa
d\eta_{o_{H}}(u,v)=-\frac{\kappa}{2}\langle[X,u],v\rangle,$

for all $u,v\in\overline{\mathfrak{m}}.$ Hence, using (3.14),
$\varphi\xi^{s}_{\lambda}=\frac{\kappa\lambda_{\mathbb{R}}(r)}{2ra_{\lambda}}\zeta^{s}_{\lambda},$
for all $s=1,\dots,m_{\lambda},$ $\lambda\in\Sigma^{+}.$ Then,
$a_{\lambda}=\frac{\kappa\lambda_{\mathbb{R}}(r)}{2r},$ because
$\tilde{\mathbf{g}}$ is a $(\varphi,\kappa^{-1}\xi,\kappa\eta)$-compatible
metric. Therefore, $\tilde{\mathbf{g}}=\tilde{\mathbf{g}}^{\kappa}$ and
$\varphi=\varphi^{1}.$ Applying (3.14) in (5.38), $\varphi^{1}$ satisfies

${\varphi^{1}_{o_{H}}}_{\mid{\mathfrak{m}}_{\lambda}\oplus{\mathfrak{k}}_{\lambda}}=\frac{1}{\lambda_{\mathbb{R}}(X)}{\rm
ad}_{X},\quad\lambda\in\Sigma^{+}.$ (5.39)

Now, we only need to show that
$(\varphi^{1},\kappa^{-1}\xi,\kappa\eta,\tilde{\mathbf{g}}^{\kappa})$ is
normal. By definition (2.2), the $G$-invariant tensor field
${\mathbf{N}}=[\varphi^{1},\varphi^{1}]+2d\eta\otimes\xi$ on $G/H$ is given by

${\mathbf{N}}_{o_{H}}(u,v)=-[u,v]_{\overline{\mathfrak{m}}}+[\varphi^{1}_{o_{H}}u,\varphi^{1}_{o_{H}}v]_{\overline{\mathfrak{m}}}-\varphi^{1}_{o_{H}}[\varphi^{1}_{o_{H}}u,v]_{\overline{\mathfrak{m}}}-\varphi^{1}_{o_{H}}[u,\varphi^{1}_{o_{H}}v]_{\overline{\mathfrak{m}}},$
(5.40)

for all $u,v\in\overline{\mathfrak{m}}.$ Hence, using that by application of
(5.39) we have

${\rm ad}_{X}\circ\varphi^{1}_{o_{H}}=\varphi^{1}_{o_{H}}\circ{\rm ad}_{X},$

we obtain that ${\mathbf{N}}^{1}_{o_{H}}(X,u)=0,$ for all
$u\in\overline{\mathfrak{m}},$ and

${\mathbf{N}}_{o_{H}}(u,\varphi^{1}_{o_{H}}v)={\mathbf{N}}_{o_{H}}(\varphi^{1}_{o_{H}}u,v),\quad
u,v\in\overline{\mathfrak{m}}.$ (5.41)

If $u,v\in{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}$
then, from (3.16), $[u,v]_{\overline{\mathfrak{m}}}\in{\mathfrak{a}}$ and so,
$\varphi^{1}_{o_{H}}[u,v]_{\overline{\mathfrak{m}}}=0.$ Using (5.39), the
Jacobi identity and taking into account that ${\rm
ad}_{X}\overline{\mathfrak{m}}\subset\overline{\mathfrak{m}},$ it follows that

$[\varphi^{1}_{o_{H}}u,v]_{\overline{\mathfrak{m}}}+[u,\varphi^{1}_{o_{H}}v]_{\overline{\mathfrak{m}}}=0,\quad[\varphi^{1}_{o_{H}}u,\varphi^{1}_{o_{H}}v]_{\overline{\mathfrak{m}}}=[u,v]_{\overline{\mathfrak{m}}}.$
(5.42)

Hence (5.40) implies that ${\mathbf{N}}_{o_{H}}(u,v)=0.$

If $u,v\in{\mathfrak{m}}_{\varepsilon/2}\oplus{\mathfrak{k}}_{\varepsilon/2},$
from (3.16),
$[u,v]_{\overline{\mathfrak{m}}}\in{\mathfrak{a}}\oplus{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}.$
Then, using (5.39) and the Jacobi identity,

$\varphi^{1}_{o_{H}}[u,v]_{\overline{\mathfrak{m}}}=\varphi^{1}_{o_{H}}[u,v]_{{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}}=\frac{1}{2}([\varphi^{1}_{o_{H}}u,v]_{\overline{\mathfrak{m}}}+[u,\varphi^{1}_{o_{H}}v]_{\overline{\mathfrak{m}}}).$

Therefore,
$[u,v]_{{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}}=-\frac{1}{2}\varphi^{1}_{o_{H}}([\varphi^{1}_{o_{H}}u,v]_{\overline{\mathfrak{m}}}+[u,\varphi^{1}_{o_{H}}v]_{\overline{\mathfrak{m}}})$
and we get

$[u,v]_{{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}}=-[\varphi^{1}_{o_{H}}u,\varphi^{1}_{o_{H}}v]_{{\mathfrak{m}}{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}},\quad[\varphi^{1}_{o_{H}}u,v]_{{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}}=[u,\varphi^{1}_{o_{H}}v]_{{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}}.$
(5.43)

Because, applying (5.39) and (5.40),

$\langle{\mathbf{N}}_{o_{H}}(u,v),X\rangle=\langle u,{\rm
ad}_{X}v\rangle-\langle\varphi^{1}_{o_{H}}u,{\rm
ad}_{X}\varphi^{1}_{o_{H}}v\rangle=0,$

we have
${\mathbf{N}}(u,v)={\mathbf{N}}(u,v)_{\mid{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}}=0.$

Finally, consider the case
$u\in{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}$ and
$v\in{\mathfrak{m}}_{\varepsilon/2}\oplus{\mathfrak{k}}_{\varepsilon/2}$ or,
equivalently using (5.41), the case
$u\in{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}$ and
$v\in{\mathfrak{m}}_{\varepsilon/2}.$ From (3.16),
$[u,v]\in{\mathfrak{k}}_{\varepsilon/2}\oplus{\mathfrak{m}}_{\varepsilon/2}$
and then,

$\varphi^{1}_{o_{H}}[u,v]=2{\rm ad}_{X}[u,v].$

Applying again the Jacobi identity, we obtain
$[u,v]=-\varphi^{1}_{o_{H}}(2[\varphi^{1}_{o_{H}}u,v]+[u,\varphi^{1}_{o_{H}}v]).$
Hence, (5.40) can be expressed as

${\mathbf{N}}_{o_{H}}(u,v)=\varphi^{1}_{o_{H}}([\varphi^{1}_{o_{H}}u,v]+[u,\varphi^{1}_{o_{H}}v]).$

Therefore, we have ${\mathbf{N}}_{o_{H}}(u,v)=0$ if and only if

$[\varphi^{1}_{o_{H}}u,\varphi^{1}_{o_{H}}v]=[u,v],\quad\mbox{for
all}\;u\in{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon},\;v\in{\mathfrak{m}}_{\varepsilon/2}.$
(5.44)

From (5.38), equality (5.44) is equivalent to (3.15) and the result follows
from Lemma 3.3.

## References

  * [1] Abbassi, K.M.T., Calvaruso, G., g-natural contact metrics on unit tangent sphere bundles, Monaths. Math. 151 (2) (2006), 89-109.
  * [2] Abbassi, K.M.T., Calvaruso, G., The curvature tensor of g-natural metrics on unit tangent sphere bundles, Int. J. Contemp. Math. Sci. 6 (3) (2008), 245-258.
  * [3] Abbassi, K.M.T., Sarih, M., On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. (Brno) 41 (1) (2005), 71-92.
  * [4] Besse, A.L., _Manifolds all of whose Geodesics are Closed_ , Springer, Berlin, Heidelberg, 1978.
  * [5] Blair, D.E., Riemannian geometry of contact and symplectic manifolds, Progress in Math. 203, Birk-häuser, 2002.
  * [6] Boeckx, E.,Vanhecke L., Harmonic and minimal vector fields on tangent and unit tangent bundles, Diff. Geom. Appl. 13 (2000), 77-93.
  * [7] Boeckx, E., Vanhecke L., Unit tangent sphere bundles with constant scalar curvature, Czechoslovak Math. J. 51 (126) (2001), 523-544.
  * [8] Boeckx, E., Perrone, D., Vanhecke, L., Unit tangent sphere bundles and two-point homogeneous spaces, Period. Math. Hungar. 36 (2-3) (1998), 79-95.
  * [9] Calvaruso, G., Contact metric geometry of the unit tangent sphere bundle, Complex, contact and symmetric manifolds, (Kowalski, O. et al, ed.), 234, Progress in Mathematics, 2005, 41-57.
  * [10] Gadea, P.M., González-Dávila, J.C., Mykytyuk, I.V., Invariant Ricci-flat Kähler metrics on tangent bundles of compact symmetric spaces, preprint, http://arxiv.org/abs/1903.00044v1.
  * [11] Helgason, S., Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York, San Francisco, London (Pure and Applied Mathematics, a Series of Monographs and Textbooks), 1978\.
  * [12] Kowalski, O., Sekizawa, M., Geometry of tangent sphere bundles with arbitrary constant radius, Proceedings of the Symposium Contemporary Mathematics (Bokan, N., ed.), Faculty of Mathematics, University of Belgrade, 2000, 219-228.
  * [13] Kowalski, O., Sekizawa, M., On tangent sphere bundles with small or large constant radius, Ann. Global Anal. Geom. 18 (2000), 207-219.
  * [14] Kowalski, O., Sekizawa, M., On the scalar curvature of tangent sphere bundles with arbitrary constant radius, Bull. Greek Math. Soc. 44 (2000), 17-30.
  * [15] Musso, E., Tricerri, F., Riemannian metrics on tangent bundles, Ann. Mat. Pura Appl. 150 (1988), 1-20.
  * [16] Nomizu, K., Invariant affine connections on homogeneous spaces, Amer. J. Math. 76, No. 1 (1954), 33-65.
  * [17] Onishchik, A.L., Transitive compact transformation groups, Mat. Sb. 60 (1963), 447-485 [Russian]; Am. Math. Soc. Transl. 55 (1966), 153-194.
  * [18] Tachibana, S., Okumura, M., On the almost complex structure of tangent bundles of Riemannian spaces, Tohoku Math. J. 14 (1962), 156-161.
  * [19] Tashiro, Y., On contact structures of tangent sphere bundles, Tohoku Math. J. 21 (1969), 117-143.