# All invariant contact metric 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.

All invariant contact metric structures on tangent sphere bundles of each
compact rank-one symmetric space are obtained explicitly, distinguishing for
the orthogonal case those that are $K$-contact, Sasakian or $3$-Sasakian. Only
the tangent sphere bundle $T_{r}{\mathbb{C}}{\bf P}^{n}$ $(r>0)$ of complex
projective spaces admits $3$-Sasakian metrics and there exists a unique
orthogonal Sasakian-Einstein metric on $T_{r}{\mathbb{C}}{\bf P}^{n}.$
Furthermore, there is a unique invariant contact metric that is Einstein, in
fact Sasakian-Einstein, on tangent sphere bundles of spheres and real
projective spaces.

Each invariant contact metric, Sasakian, Sasakian-Einstein or $3$-Sasakian
structure on the unit tangent sphere of any compact rank-one symmetric space
is extended, respectively, to an invariant almost Kähler, Kähler, Kähler
Ricci-flat or hyperKähler structure on the punctured tangent bundle.

###### Key words and phrases:

Homogeneous contact metric manifolds, Sasakian structures, $3$-Sasakian
structures, Einstein metrics, tangent sphere bundles, compact rank-one
symmetric spaces

###### 1991 Mathematics Subject Classification:

53C30, 53C35, 53D10.

Partially supported by grant PID2019-105019GB-C21 funded by MCIN/AEI/
10.13039/501100011033 and by ERDF ‘A way of making Europe’

##  1\. Introduction

Compact rank-one Riemannian symmetric spaces can be characterized as those
Riemannian symmetric spaces $(G/K,g)$ with strictly positive curvature. They
are Euclidean spheres, real, complex and quaternionic projective spaces and
the Cayley plane [2, Ch. 3]. The natural $G$-action on the tangent bundle
$T(G/K)$ of $G/K$ is of cohomogeneity one with principal orbits the tangent
sphere bundles $T_{r}(G/K)=\\{u\in T(G/K)\colon g(u,u)=r^{2}\\}$ of radius
$r>0$ (see [12]). Hence, fixed a Cartan decomposition
${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{m}}$ of $G/K$ and a (one-
dimensional) Cartan subspace ${\mathfrak{a}}$ of ${\mathfrak{m}},$ the
punctured tangent bundle $D(G/K):=T(G/K)\setminus\\{\mbox{zero section}\\}$ of
$T(G/K)$ is 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 of $K$ given by $H=\\{k\in K\colon{\rm
Ad}_{k}u=u,\;\mbox{\rm for all}\,u\in{\mathfrak{a}}\\}.$ The quotient
expressions $G/K$ as symmetric spaces and the Lie groups $H$ are listed in
Table I.

Then the tangent sphere bundle $T_{r}{\mathbb{S}}^{n}$ of the unit
$n$-dimensional sphere ${\mathbb{S}}^{n}\subset{\mathbb{R}}^{n+1}$ is
equivariantly diffeomorphic to the Stiefel manifold
$V_{2}({\mathbb{R}}^{n+1})$ of all orthonormal $2$-frames in
${\mathbb{R}}^{n+1}$ and $T_{r}{\mathbb{K}}{\bf P}^{n},$ where
${\mathbb{K}}{\bf P}^{n}$ is the $n$-dimensional projective space for
${\mathbb{K}}={\mathbb{R}},$ ${\mathbb{C}}$ or ${\mathbb{H}},$ to the
projective Stiefel manifold
$W_{2}({\mathbb{K}}^{n+1}):=V_{2}({\mathbb{K}}^{n+1})/S,$ with $S$ being the
subset of unit vectors of ${\mathbb{K}}$ $(S={\mathbb{S}}^{0},$
${\mathbb{S}}^{1}$ or ${\mathbb{S}}^{3}).$

In [13] the author has obtained the set of those $G$-invariant contact metric
and Sasakian structures on $T_{r}(G/K)=G/H$ whose characteristic vector field
is the standard vector field $\xi^{S}$ of $T(G/K),$ that is, the unit vector
field $\xi^{S}:=-J^{S}N$ defined on $D(G/K),$ where $J^{S}$ is the standard
almost complex structure and $N$ is the outward normal unit vector field to
the singular foliation $\\{T_{r}(G/K)\\}_{r>0}$ with respect to the Sasaki
metric $g^{S}$ on $T(G/K).$

Table I. Compact rank-one symmetric spaces
---
| $G/K$ | $\dim$ | $m_{\varepsilon}$ | $m_{\varepsilon/{\scriptscriptstyle 2}}$ | $H$
$\mathbb{S}^{n}\,(n\geq 2)$ | $\mathrm{SO}(n{+}1)/\mathrm{SO}(n)$ | $n$ | $n{-}1$ | $0$ | ${\mathrm{S}}{\mathrm{O}}(n{-}1)$
${\mathbb{R}}{\mathbf{P}}^{n}\,(n\geq 2)$ | $\mathrm{SO}(n{+}1)/{\rm S}({\rm O}(1)\times{\rm O}(n))$ | $n$ | $n{-}1$ | $0$ | ${\mathrm{S}}({\mathrm{O}}(1)\times{\mathrm{O}}(n{-}1))$
${\mathbb{C}}{\mathbf{P}}^{n}$ $(n\geq 2)$ | $\mathrm{SU}(n{+}1)/\mathrm{S}(\mathrm{U}(1){\times}\mathrm{U}(n))$ | $2n$ | $1$ | $2n{-}2$ | ${\mathrm{S}}({\mathrm{U}}(1)\times{\mathrm{U}}(n{-}1))$
${\mathbb{H}}{\mathbf{P}}^{n}$ $(n\geq 1)$ | $\mathrm{Sp}(n{+}1)/\mathrm{Sp}(1){\times}\mathrm{Sp}(n)$ | $4n$ | $3$ | $4n{-}4$ | ${\mathrm{S}}{\mathrm{p}}(1)\times{\mathrm{S}}{\mathrm{p}}(n{-}1)$
${\mathbb{C}\mathrm{a}}{\mathbf{P}}^{2}$ | $\mathrm{F_{4}}/\mathrm{Spin(9)}$ | $16$ | $7$ | 8 | $\mathrm{Spin}(7)$

A first objective of this paper is focused on explicitly obtaining all the
$G$-invariant contact metric structures in $T_{r}(G/K)$ and on determining for
the orthogonal case those that are $K$-contact, Sasakian or $3$-Sasakian. This
leads us to consider the space of $G$-invariant vector fields in $T_{r}(G/K).$
Such space is one-dimensional except for $G/K={\mathbb{C}}{\bf P}^{n}$ whose
dimension is three. From this, the characteristic vector field of
$G$-invariant contact metric structures on $T_{r}G/K,$ for
$G/K\in\\{{\mathbb{S}}^{n},{\mathbb{R}}{\bf P}^{n}(n\geq 3),{\mathbb{H}}{\bf
P}^{n}(n\geq 1),{\mathbb{C}}{\rm a}{\bf P}^{2}\\},$ is proportional to the
standard field and, for $G/K\in\\{{\mathbb{C}}{\bf P}^{n}\,(n\geq 1)\\},$ the
choice of possible characteristic vector fields produces seven different types
of invariant contact metric structures, listed as AI-AIII, BI-BIII and C in
Theorem 5.3. Furthermore, those orthogonal structures that are $K$-contact
(see Theorem 5.5 for their classification) are indeed Sasakian (Theorem 6.2)
and, on $T_{r}{\mathbb{C}}{\bf P}^{n},$ the set of these $G$-invariant
Sasakian structures form three families, depending on a positive parameter, of
type AI, AII and AIII, respectively.

Boyer, Galicki and Mann in [8] obtained the classification of homogeneous
3-Sasakian manifolds (see [11] for a new proof using theory of root systems).
Comparing the lists of homogeneous $3$-Sasakian manifolds and of quotients
$G/H=T_{r}G/K,$ from Table I, only
${\mathrm{S}}{\mathrm{p}}(1)\cong{\mathbb{S}}^{3}$ and
${\mathrm{S}}{\mathrm{U}}(n+1)/{\mathrm{S}}({\mathrm{U}}(1)\times{\mathrm{U}}(n-1)),$
$n\geq 2,$ are homogeneous $3$-Sasakian manifolds. Hence $T_{r}G/K$ admits a
$G$-invariant $3$-Sasakian structure if and only if $G/K={\mathbb{C}}{\bf
P}^{n},$ for some $n\geq 1.$ In Theorem 6.2, by a direct procedure, the same
result is reached, also obtaining that there is a unique orthogonal
$3$-Sasakian metric ${\bf g}$ on $T_{r}{\mathbb{C}}{\bf P}^{n}.$ Its
expression, as well as a triple of characteristic vector fields, are
explicitly determined there. Moreover, ${\bf g}$ is the unique orthogonal
metric on $T_{r}{\mathbb{C}}{\bf P}^{n}$ which is Sasakian-Einstein (Theorem
6.4).

On the other hand, Back and Hsiang in [1] proved that
$V_{2}({\mathbb{R}}^{n+1})$ admits invariant Einstein metrics and that any two
homogeneous Einstein metrics are homothetic. The same result was obtained by
M. Kerr in [15]. In this paper, apart from explicitly obtaining such metrics
for $T_{r}{\mathbb{S}}^{n}$ and $T_{r}{\mathbb{R}}{\bf P}^{n}$ (Theorem 7.5),
it is proven that there exists a unique invariant contact metric which is
Einstein and that it is in fact Sasakian-Einstein (Theorem 7.9).

Another relevant fact that we raise is that, because these classes of contact
metric structures can be characterized by using holonomy reduction of its
metric cone (see [4], [6] and Proposition 2.1), each $G$-invariant contact
metric, Sasakian, Sasakian-Einstein or $3$-Sasakian structure on the unit
tangent sphere $T_{1}(G/K),$ can be extended respectively to an $G$-invariant
almost Kähler, Kähler, Kähler Ricci-flat or hyperKähler structure on the
punctured tangent bundle $D(G/K)$ (Theorem 4.4). However, we must point out
that such structures in $D(G/K)$ do not admit differentiable extensions to all
$T(G/K)$ (see Remark 4.5).

To achieve our objectives, we consider the system $\Sigma$ of restricted roots
of $({\mathfrak{g}},{\mathfrak{k}},{\mathfrak{a}}),$ which is either
$\\{\pm\varepsilon\\}$ for spheres and real projective spaces or
$\Sigma=\\{\pm\varepsilon,\pm\varepsilon/2\\}$ for the rest of compact rank-
one symmetric spaces, $\varepsilon$ being a linear function on the
complexification ${\mathfrak{a}}^{\mathbb{C}}$ of ${\mathfrak{a}},$ such that
$\varepsilon({\mathfrak{a}})\subset\sqrt{-1}{\mathbb{R}}$ (see [12] for more
details). The multiplicities $m_{\varepsilon}$,
$m_{\varepsilon/{\scriptscriptstyle 2}}$ of the corresponding restricted roots
appear in Table I.

##  2\. Preliminaries

We briefly recall some basic concepts about almost contact metric structures
and their most significant classes. For general theory on almost contact
metric structures we refer the reader to [3] and for their classification to
[9].

An odd-dimensional $C^{\infty}$ manifold $M$ is said to be 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 moreover $M$ is equipped with
a Riemannian metric $g$ such that

$g(\varphi U,\varphi V)=g(U,V)-\eta(U)\eta(V),$

for all $U,V\in{\mathfrak{X}}(M),$ where ${\mathfrak{X}}(M)$ is the Lie
algebra of the vector fields on $M,$ $(M,\varphi,\xi,\eta,g)$ is called an
almost contact metric manifold. Two almost contact metric manifolds
$(M_{1},\varphi_{1},\xi_{1},$ $\eta_{1},g_{1})$ and
$(M_{2},\varphi_{2},\xi_{2},\eta_{2},g_{2})$ is said to be isomorphic if there
exists an isometry $f\colon(M_{1},g_{1})\to(M_{2},g_{2})$ preserving the
corresponding structures, i.e. $f_{*}\xi_{1}=\xi_{2}$ and
$f_{*}\circ\varphi_{1}=\varphi_{2}\circ f_{*}.$ (Then,
$f^{*}\eta_{2}=\eta_{1}.)$

Let $\nabla$ be the Levi-Civita connection of $(M,g).$ Then, we have

$(\nabla_{U}\Phi)(V,W)=g(V,(\nabla_{U}\varphi)W),\quad(\nabla_{U}\eta)V=g(V,\nabla_{U}\xi)=(\nabla_{U}\Phi)(\xi,\varphi
V),$ (2.1)

where $\Phi$ denotes the fundamental $2$-form, defined by
$\Phi(U,V)=g(U,\varphi V),$ and the exterior derivatives of $\eta$ and $\Phi$
are given by

$2d\eta(U,V)=(\nabla_{U}\eta)V-(\nabla_{V}\eta)U,\quad
3d\Phi(U,V,W)=\mathop{\text{\large$\mathfrak{S}$}\vrule
width=0.0pt,depth=2.0pt}_{U,V,W}(\nabla_{U}\Phi)(V,W).$ (2.2)

We say that $(\varphi,\xi,\eta,g)$ is a contact metric structure if
$d\eta=\Phi.$ Then it is determined by the pair $(\xi,g)$ (or $(\eta,g)).$ A
Riemannian metric $g$ is said to be contact if there exists a vector field
$\xi$ such that $(\xi,g)$ is a contact metric structure and to each of these
vector fields $\xi,$ a characteristic vector field associated with $g.$ If, in
addition, $\xi$ is Killing, the manifold is called $K$-contact and one gets

$\nabla_{U}\xi=-\varphi U.$ (2.3)

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

$[\varphi,\varphi](U,V)=\varphi^{2}[U,V]+[\varphi U,\varphi V]-\varphi[\varphi
U,V]-\varphi[U,\varphi V].$ (2.4)

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

$(\nabla_{U}\varphi)V=g(U,V)\xi-\eta(V)U,\quad U,V\in{\mathfrak{X}}(M).$ (2.5)

Any Sasakian manifold is $K$-contact but the converse may be not true for
dimensions greater than three. A Sasakian manifold $(M,\xi,g)$ is Sasakian-
Einstein if its Riemannian metric $g$ is Einstein. Then on a
$(2n+1)$-dimensional Sasakian-Einstein manifold, the Ricci tensor ${\rm Ric}$
satisfies ${\rm Ric}(U,\xi)=2n\eta(U)$ and the scalar curvature is positive
and equals to $2n(2n+1).$

A Riemannian manifold $(M,g)$ is called $3$-Sasakian if it admits three vector
fields $\\{\xi_{1},\xi_{2},\xi_{3}\\}$ such that

$g(\xi_{i},\xi_{j})=\delta_{ij},\quad[\xi_{i},\xi_{j}]=2\varepsilon_{ijk}\xi_{k},\quad
i,j,k\in\\{1,2,3\\},$

and $(M,g,\xi_{i})$ is a Sasakian manifold, for each $i=1,2,3.$ Because
$(M,g,\xi)$ is a Sasakian manifold, for any
$\xi=\lambda_{1}\xi_{1}+\lambda_{2}\xi_{2}+\lambda_{3}\xi_{3}$ where
$(\lambda_{1},\lambda_{2},\lambda_{3})\in{\mathbb{S}}^{2}(1),$ a $3$-Sasakian
manifold has in fact an ${\mathbb{S}}^{2}(1)$ worth of Sasakian structures.

A more recent characterization for Sasakian, Sasakian-Einstein and
$3$-Sasakian manifold by using holonomy reduction of the associated metric
cone appears in [4]. The reader is also referred to [5], [6], [20] and the
monograph [7] for more information about this topic.

Given an $m$-dimensional Riemannian manifold $(M,g),$ the metric cone
$({\mathcal{C}}(M),\overline{g})$ is the warped product $M\times_{{\rm
Id}_{{\mathbb{R}}^{+}}}{\mathbb{R}}^{+},$ that is the product manifold
${\mathcal{C}}(M)=M\times{\mathbb{R}}^{+}$ equipped with the Riemannian metric
$\overline{g}=r^{2}g+dr^{2}.$ Then $(M,g)$ is Sasakian if the holonomy group
of $({\mathcal{C}}(M),\overline{g})$ reduces to a subgroup of
${\mathrm{U}}(\frac{m+1}{2}).$ In particular, $m=2n+1,$ $n\geq 1,$ and
$({\mathcal{C}}(M),\overline{g})$ is Kähler. For the proof, it is considered
the almost Hermitian structure $(J,\overline{g})$ on ${\mathcal{C}}(M),$
associated to an almost contact metric structure $(\varphi,\xi,\eta)$ on
$(M,g),$ where $J$ is given by

$J(U,\lambda\frac{\partial}{\partial r})=(\varphi
U-\frac{\lambda}{r}\xi,r\eta(U)\frac{\partial}{\partial r}),\quad\mbox{\rm for
all}\;\lambda\colon{\mathbb{R}}^{+}\to{\mathbb{R}}\;\mbox{\rm smooth
function.}$ (2.6)

######  Proposition 2.1.

A Riemannian manifold $(M,g)$ is contact metric if and only if the metric cone
$({\mathcal{C}}(M),\overline{g})$ is almost Kähler. Moreover, the symplectic
$2$-form $F$ in ${\mathcal{C}}(M)$ is the exact form $F=d(r^{2}\eta),$ where
$\eta$ is the contact form of $M.$

###### Proof.

From (2.6), the corresponding fundamental $2$-forms $F$ and $\Phi$ of
$({\mathcal{C}}(M),J,\overline{g})$ and of the contact metric manifold
$(M,\xi,g)$ satisfy $F=r^{2}\Phi+2rdr\wedge\eta=d(r^{2}\eta),$ where tensors
of $M$ (resp., ${\mathbb{R}}^{+})$ and their pull-backs on ${\mathcal{C}}(M)$
via the projection $\pi_{1}\colon{\mathcal{C}}(M)\to M$ (resp.,
$\pi_{2}\colon{\mathcal{C}}(M)\to{\mathbb{R}}^{+}),$ are denote with the same
letter. Then $({\mathcal{C}}(M),\overline{g})$ is almost Kähler.

Conversely, let $J$ be an almost complex structure on ${\mathcal{C}}(M)$
compatible with $\overline{g}$ such that $dF=0,$ $F$ being the Kähler form of
$({\mathcal{C}}(M),J,\overline{g}).$ Identifying $M$ with
$M\times\\{1\\}\subset{\mathcal{C}}(M),$ we consider the induced almost
contact metric structure $(\varphi,\xi,\eta,g)$ on $M$ as hypersurface of
${\mathcal{C}}(M),$ where

$\xi=-J\frac{\partial}{\partial r},\quad\eta=g(\xi,\cdot),\quad
J(U,0)=(\varphi U,\eta(U)\frac{\partial}{\partial r}).$

Then $J$ on $M\cong M\times\\{1\\}$ coincides with the almost complex
structure defined in (2.6). Moreover, for the Levi-Civita connection
$\overline{\nabla}$ of $({\mathcal{C}}(M),\overline{g})$ we have

$\begin{array}[]{lcl}(\overline{\nabla}_{(0,\frac{\partial}{\partial
r})}J)(0,\frac{\partial}{\partial
r})&=&(\overline{\nabla}_{(0,\frac{\partial}{\partial
r})}J)(U,0)=(0,0),\\\\[4.79993pt]
(\overline{\nabla}_{(U,0)}J))(0,\frac{\partial}{\partial
r})&=&-\frac{1}{r}(\nabla_{U}\xi+\varphi U,0),\\\\[4.79993pt]
(\overline{\nabla}_{(U,0)}J)(V,0)&=&((\nabla_{U}\varphi)V+\eta(V)U-g(U,V)\xi,rg(\nabla_{U}\xi+\varphi
U,V)\frac{\partial}{\partial r}).\end{array}$

Now, using (2.1) and (2.2), we get at the points $(x,1)\in M\times\\{1\\},$

$\begin{array}[]{lcl}(\iota_{(0,\frac{\partial}{\partial
r})}dF)((U,0),(V,0))&=&\frac{1}{3}\mathop{\text{\large$\mathfrak{S}$}\vrule
width=0.0pt,depth=2.0pt}(\overline{\nabla}_{(0,\frac{\partial}{\partial
r})})((U,0),(V,0))=\frac{1}{3}\mathop{\text{\large$\mathfrak{S}$}\vrule
width=0.0pt,depth=2.0pt}\overline{g}((U,0),(\overline{\nabla}_{(0,\frac{\partial}{\partial
r})}J)(V,0))\\\\[4.79993pt]
&=&\frac{1}{3}(\overline{g}((V,0),-(\nabla_{U}\xi+\varphi
U,0))+g(\nabla_{V}\xi+\varphi V,U))\\\\[4.79993pt]
&=&\frac{2}{3}(\Phi(U,V)-d\eta(U,V)).\end{array}$

This proves the result. ∎

Under the identification $M\cong M\times\\{1\\}\subset{\mathcal{C}}(M),$ we
have also proven the following.

######  Corollary 2.2.

Each contact metric structure $(\xi,g)$ on an odd-dimensional manifold $M$ can
be extended to a unique almost Kähler structure on its metric cone
$({\mathcal{C}}(M),\overline{g}).$

As a direct consequence of the relationship between Ricci curvature of $M$ and
its metric cone $({\mathcal{C}}(M),\overline{g}),$ it follows that $g$ is
Einstein if and only if $\overline{g}$ is Ricci-flat. Then
$({\mathcal{C}}(M),\overline{g})$ is Kähler Ricci-flat (Calabi-Yau) and the
restricted holonomy group reduces to a subgroup of ${\rm SU}(n+1),$ where
$\dim M=2n+1.$ Moreover, an $m$-dimensional Riemannian manifold $(M,g)$ is
$3$-Sasakian if and only if the holonomy group of the metric cone on $M$
reduces to a subgroup of ${\rm Sp}(\frac{m+1}{4}).$ In particular, $m=4n+3,$
$n\geq 1,$ and $({\mathcal{C}}(M),\overline{g})$ is hyperKähler. This implies
that $3$-Sasakian manifolds are Sasakian-Einstein with Einstein constant
$\lambda=2(2n+1).$

##  3\. Homogeneous almost contact metric manifolds

A connected homogeneous manifold $M$ can be expressed as a quotient manifold
$G/K,$ where $G$ is a connected Lie group acting transitively on $M$ and $K$
is the isotropy subgroup of $G$ at some point $o\in M,$ the origin of $G/K.$
Moreover, if $g$ is a $G$-invariant Riemannian metric on $M=G/K,$ then $(M,g)$
is said to be a homogeneous Riemannian manifold and, if $(\varphi,\xi,\eta)$
is an $G$-invariant almost contact structure on $M$ compatible with $g,$
$(M,\varphi,\xi,\eta,g)$ is said to be a homogeneous almost contact metric
manifold.

We can assume that $G/K$ is a reductive homogeneous space, i.e. there exists
an ${\rm Ad}(K)$-invariant subspace ${\mathfrak{m}}$ of the Lie algebra
${\mathfrak{g}}$ of $G$ such that
${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{m}},$ ${\mathfrak{k}}$ being
the Lie algebra of $K.$ Then the differential map $(\pi_{K})_{*e}$ at the
identity element $e\in G$ of the projection $\pi_{K}\colon G\to G/K,$
$\pi_{K}(a)=aK,$ gives an isomorphism of ${\mathfrak{m}}$ onto $T_{o}(G/K)$
and $g$ is determined by an ${\rm Ad}(K)$-invariant inner product
$\langle\cdot,\cdot\rangle$ on ${\mathfrak{m}}.$

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 $o.$ Hence, for each $\mu\in{\mathfrak{m}},$ a vector
field $\mu^{\tau}$ on ${\mathcal{U}}_{o}$ is defined as

$\mu^{\tau}_{(\exp x)K}=(\tau_{\exp x})_{*o}\mu,\quad\mbox{\rm for all $x\in
O_{\mathfrak{m}},$}$ (3.1)

where $\tau_{b},$ for each $b\in G,$ is the translation $\tau_{b}\colon G/K\to
G/K,$ $\tau_{b}(aK)=baK.$ Then
$[\mu_{1}^{\tau},\mu^{\tau}_{2}]_{o}=[\mu_{1},\mu_{2}]_{\mathfrak{m}}$ (see
[19]), where $[\cdot,\cdot]_{\mathfrak{m}}$ denotes the
${\mathfrak{m}}$-component of $[\cdot,\cdot].$

There exists a one-to-one correspondence between the set of all invariant
affine connections $\nabla$ on $G/K$ and the set of all ${\rm
Ad}(K)$-invariant bilinear functions
$\alpha:{\mathfrak{m}}\times{\mathfrak{m}}\to{\mathfrak{m}}$ [19, Theorem
8.1]. The correspondence is given by

$\alpha(\mu_{1},\mu_{2})=\nabla_{\mu_{1}}\mu_{2}^{\tau},\quad\mu_{1},\mu_{2}\in{\mathfrak{m}}.$
(3.2)

Then the curvature tensor $R$ of $\nabla$ at the origin $o,$ where $R$ is
defined by the sign convention $R(U,V)=\nabla_{[U,V]}-[\nabla_{U},\nabla_{V}]$
for all $U,V\in{\mathfrak{X}}(M),$ is expressed in terms of $\alpha$ by

$\begin{array}[]{lcl}R_{o}(\mu_{1},\mu_{2})\mu_{3}&=&[[\mu_{1},\mu_{2}]_{\mathfrak{k}},\mu_{3}]+\alpha([\mu_{1},\mu_{2}]_{\mathfrak{m}},\mu_{3})\\\\[4.79993pt]
&&+\alpha(\mu_{2},\alpha(\mu_{1},\mu_{3}))-\alpha(\mu_{1},\alpha(\mu_{2},\mu_{3})),\end{array}$
(3.3)

for all $\mu_{1},\mu_{2},\mu_{3}\in{\mathfrak{m}}.$ Using the Koszul formula
in (3.2), the Levi Civita connection of $(M,g)$ is determined for $\alpha$
given by

$\alpha(\mu_{1},\mu_{2})=\frac{1}{2}[\mu_{1},\mu_{2}]_{\mathfrak{m}}+{\mathfrak{U}}(\mu_{1},\mu_{2}),$
(3.4)

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

$2\langle{\mathfrak{U}}(\mu_{1},\mu_{2}),\mu_{3}\rangle=\langle[\mu_{3},\mu_{1}]_{\mathfrak{m}},\mu_{2}\rangle+\langle[\mu_{3},\mu_{2}]_{\mathfrak{m}},\mu_{1}\rangle.$
(3.5)

From (3.2), a $G$-invariant vector field $U$ on $G/K$ is Killing if and only
if
$\langle\alpha(\mu_{1},\mu),\mu_{2}\rangle+\langle\alpha({\mu_{2}},\mu),\mu_{1}\rangle=0,$
for all $\mu_{1},\mu_{2}\in{\mathfrak{m}},$ where $\mu$ is the ${\rm
Ad}(K)$-invariant vector $\mu=U_{o}\in{\mathfrak{m}}.$ Since from (3.4) and
(3.5) we get
$\langle\alpha(\mu_{1},\mu),\mu_{2}\rangle+\langle\alpha({\mu_{2}},\mu),\mu_{1}\rangle=-\langle{\mathfrak{U}}(\mu_{1},\mu_{2}),\mu\rangle,$
$U$ is a Killing vector field if and only if

${\mathfrak{U}}(\mu_{1},\mu_{2})\in\mu^{\bot},\quad\mu_{1},\mu_{2}\in{\mathfrak{m}}.$
(3.6)

The $G$-invariant connection $\widetilde{\nabla}$ corresponding with
$\alpha=0$ is known as the canonical connection [16]. Its torsion
$\widetilde{T}$ and curvature $\widetilde{R}$ are given at the origin by

$\widetilde{T}_{o}(\mu_{1},\mu_{2})=-[\mu_{1},\mu_{2}]_{\mathfrak{m}},\quad\widetilde{R}_{o}(\mu_{1},\mu_{2})={\rm
ad}_{[\mu_{1},\mu_{2}]_{\mathfrak{k}}}.$ (3.7)

Since any $G$-invariant tensor field on $M$ is $\widetilde{\nabla}$-parallel
[17, Proposition I.11], it follows that
$\widetilde{\nabla}g=\widetilde{\nabla}\widetilde{T}=\widetilde{\nabla}\widetilde{R}=0.$

Let ${\mathfrak{g}}_{0}\subset{\mathfrak{k}}$ be the subalgebra generated by
all projections $[\mu_{1},\mu_{2}]_{\mathfrak{k}},$
$\mu_{1},\mu_{2}\in{\mathfrak{m}},$ and let $\widetilde{\mathfrak{g}}$ be the
subalgebra $\widetilde{\mathfrak{g}}:={\mathfrak{g}}_{0}\oplus{\mathfrak{m}}$
of ${\mathfrak{g}},$ known as the transvection algebra. Denote by $G_{0}$ and
$\widetilde{G}$ the connected subgroups of $G$ with Lie algebra
${\mathfrak{g}}_{0}$ and $\widetilde{\mathfrak{g}},$ respectively. We say that
$\widetilde{G}$ is the transvection group of the affinely manifold
$(M,\widetilde{\nabla})$ [17, Ch. I]. Then, because ${\mathfrak{g}}_{0}$
coincides precisely with the algebra generated by all curvature transformation
$\widetilde{R}(\mu_{1},\mu_{2})$ on $T_{o}M\cong{\mathfrak{m}}$ [17,
Proposition I.2], $G_{o}$ is isomorphic to the restricted holonomy group of
$(M,\widetilde{\nabla}).$ Therefore, if $G_{o}$ is assumed to be closed in
$K,$ we obtain a new representation for $M$ as reductive space
$M=\widetilde{G}/G_{o}$ with the same canonical connection.

We finish this section with the following result, which will be used later.

######  Lemma 3.1.

Let $(M_{1}=G_{1}/K_{1},\varphi_{1},\xi_{1},\eta_{1},g_{1})$ and
$(M_{2}=G_{2}/K_{2},\varphi_{2},\xi_{2},\eta_{2},g_{2})$ be simply connected
homogeneous almost contact metric manifolds with adapted reductive
decompositions ${\mathfrak{g}}_{1}={\mathfrak{k}}_{1}\oplus{\mathfrak{m}}_{1}$
and ${\mathfrak{g}}_{2}={\mathfrak{k}}_{2}\oplus{\mathfrak{m}}_{2},$
respectively, and $o_{1}$ and $o_{2}$ the corresponding origin points. If
$L\colon({\mathfrak{m}}_{1},\langle\cdot,\cdot\rangle_{1})\to({\mathfrak{m}}_{2},\langle\cdot,\cdot\rangle_{2})$
is a linear isometry satisfying

  1. (i)

$L[\mu,\nu]_{{\mathfrak{m}}_{1}}=[L\mu,L\nu]_{{\mathfrak{m}}_{2}},$

  2. (ii)

$L\circ{\rm ad}_{[\mu,\nu]_{{\mathfrak{k}}_{1}}}={\rm
ad}_{[L\mu,L\nu]_{{\mathfrak{k}}_{2}}}\circ L,$

  3. (iii)

$L{\xi_{1}}_{o_{1}}={\xi_{2}}_{o_{2}}$ and
$L\circ\varphi_{1\,o_{1}}=\varphi_{2\,o_{2}}\circ L,$

for all $\mu,\nu\in{\mathfrak{m}}_{1},$ then there exits a unique isometry $f$
of $(M_{1},g_{1})$ onto $(M_{2},g_{2})$ such that $f(o_{1})=o_{2}$ and the
differential of $f$ at $o_{1}$ coincides with $L.$ Moreover, the almost
contact metric structures on $M_{1}$ and $M_{2}$ are isomorphic via $f.$

###### Proof.

Let $\widetilde{T}_{i}$ be the torsion and let $\widetilde{R}_{i}$ be the
curvature of the canonical connection $\widetilde{\nabla}_{i}$ on $M_{i},$
$i=1,2.$ From (i) and (ii), using (3.7), $L$ maps $\widetilde{T}_{1}$ and
$\widetilde{R}_{1}$ at $o_{1}$ into $\widetilde{T}_{2}$ and
$\widetilde{R}_{2}$ at $o_{2},$ respectively. Since
$\widetilde{\nabla}_{i}\widetilde{T}_{i}=\widetilde{\nabla}_{i}\widetilde{R}_{i}=0,$
it follows from [16, Theorem 7.8], that there exists a unique affine
isomorphism $f$ of $M_{1}$ onto $M_{2}$ such that $f(o_{1})=o_{2}$ and
$f_{*o_{1}}=L.$ Moreover, taking into account that $\widetilde{\nabla}_{1}$
and $\widetilde{\nabla}_{2}$ are metric connections and $L$ is a linear
isometry, $f$ is an isometry.

Next, we show the last part of the lemma. Because $\xi_{1}$ and $\varphi_{1}$
are $G_{1}$-invariant, they are $\widetilde{\nabla}_{1}$-parallel. Hence,
using that $f$ is an affine isomorphism, $f_{*}\xi_{1}$ and
$f_{*}\circ\varphi_{1}\circ f_{*}^{-1}$ are $\widetilde{\nabla}_{2}$-parallel
tensor fields on $M_{2}.$ According to [17, Proposition I.38], they are then
$\widetilde{G}_{2}$-invariant, where $\widetilde{G}_{2}$ denotes the
transvection group of $(M_{2}=G_{2}/K_{2},\widetilde{\nabla}_{2}).$ Since
$\xi_{2}$ and $\varphi_{2}$ are also $\widetilde{G}_{2}$-invariant and, from
(iii), $f_{*\,o_{1}}\xi_{1\,o_{1}}=\xi_{2\,o_{2}}$ and
$(f_{*}\circ\varphi_{1}\circ f_{*}^{-1})_{*\,o_{2}}=\varphi_{2\,o_{2}},$ it
follows, using the transitivity of $\widetilde{G}_{2}$ on $M_{2}$ as subgroup
of $G_{2},$ that $f_{*}\xi_{1}=\xi_{2}$ and $f_{*}\circ\varphi_{1}\circ
f_{*}^{-1}=\varphi_{2}.$ This proves the result. ∎

######  Remark 3.2.

The quaterna
$({\mathfrak{m}}_{i},\widetilde{T}_{i},\widetilde{R}_{i},\langle\cdot,\cdot\rangle_{i}=g_{i\,o_{i}}),$
for each $i=1,2,$ is an infinitesimal model associated to the reductive
homogeneous manifold $(M_{i}=G_{i}/K_{i},g_{i}),$ known as the canonical
infinitesimal model, and the linear isometry $L$ satisfying (i) and (ii) is
said to be an isomorphism of infinitesimal models (see [10] and [22]). Then
Lemma 3.1 says that two simply connected homogeneous almost contact metric
manifolds are isomorphic if there exists an isomorphism between the
corresponding canonical infinitesimal models that preserves the almost contact
metric structures.

##  4\. The tangent sphere bundle of a compact rank-one symmetric space

Let $G/K$ be a compact rank-one symmetric space and let
${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{m}}$ be the associated Cartan
decomposition. Then, because $G$ is compact and semisimple, $G/K$ is equipped
with an $G$-invariant metric determined by the restriction to ${\mathfrak{m}}$
of an ${\rm Ad}(G)$-invariant inner product $\langle\cdot,\cdot\rangle$ of
${\mathfrak{g}}.$

Fixed an (one-dimensional) Cartan subspace ${\mathfrak{a}}$ of
${\mathfrak{m}},$ it follows from [12, Lemma 3.2] that the subset
$\Sigma^{+}\subset\Sigma$ of positive restricted roots of the triple
$({\mathfrak{g}},{\mathfrak{k}},{\mathfrak{a}})$ is either
$\\{\varepsilon\\},$ if $G/K\in\\{{\mathbb{S}}^{n},{\mathbb{R}}{\bf
P}^{n}\;(n\geq 2)\\},$ or $\\{\varepsilon,\frac{1}{2}\varepsilon\\}$ for the
rest of compact rank-one symmetric spaces, $\varepsilon$ being a linear
function $\varepsilon\in({\mathfrak{a}}^{\mathbb{C}})^{*}.$ (For the general
theory of restricted roots, see [14, Ch. III and Ch. VII] and [18, Section
V.2].) For each $\lambda\in\Sigma^{+},$ put

${\mathfrak{m}}_{\lambda}=\\{\mu\in{\mathfrak{m}}\colon{\rm
ad}^{2}_{X}\mu=\lambda^{2}(X)\mu\\},\quad{\mathfrak{k}}_{\lambda}=\\{\nu\in{\mathfrak{k}}\colon{\rm
ad}^{2}_{X}\nu=\lambda^{2}(X)\nu\\},$ (4.1)

$X$ being a generator of ${\mathfrak{a}}.$ Then, using [14, Ch. VII, Lemma
11.3], we have the direct and $\langle\cdot,\cdot\rangle$-orthogonal
decompositions

${\mathfrak{m}}={\mathfrak{a}}\oplus{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{m}}_{\varepsilon/2},\quad{\mathfrak{k}}={\mathfrak{h}}\oplus{\mathfrak{k}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon/2},$

where ${\mathfrak{h}}$ is the centralizer of ${\mathfrak{a}}$ in
${\mathfrak{k}},$ that is the Lie algebra
${\mathfrak{h}}=\\{\nu\in{\mathfrak{k}}\colon[{\mathfrak{a}},\nu]=0\\}$ of the
subgroup $H$ of $K.$ Moreover,
$\dim{\mathfrak{m}}_{\varepsilon}=\dim{\mathfrak{k}}_{\varepsilon}=m_{\varepsilon}$
and
$\dim{\mathfrak{m}}_{\varepsilon/2}=\dim{\mathfrak{k}}_{\varepsilon/2}=m_{\varepsilon/2},$
where $m_{\varepsilon}$ and $m_{\varepsilon/2}$ are the multiplicities of the
corresponding restricted roots $\varepsilon$ and $\varepsilon/2$ (see Table
I). Here and in the sequels, we put
${\mathfrak{m}}_{\varepsilon/2}={\mathfrak{k}}_{\varepsilon/2}=0$ if
$\varepsilon/2$ is not in $\Sigma.$

Applying [14, Ch. VII, Lemma 11.4], it is immediate the following.

######  Lemma 4.1.

We have:

$\begin{array}[]{l}[{\mathfrak{h}},{\mathfrak{m}}_{\lambda}]\subset{\mathfrak{m}}_{\lambda},\quad[{\mathfrak{h}},{\mathfrak{k}}_{\lambda}]\subset{\mathfrak{k}}_{\lambda},\quad[{\mathfrak{a}},{\mathfrak{m}}_{\lambda}]\subset{\mathfrak{k}}_{\lambda},\quad[{\mathfrak{a}},{\mathfrak{k}}_{\lambda}]\subset{\mathfrak{m}}_{\lambda},\;\;\lambda=\varepsilon,\varepsilon/2,\\\\[4.79993pt]
[{\mathfrak{m}}_{\varepsilon},{\mathfrak{m}}_{\varepsilon}]\subset{\mathfrak{h}},\quad[{\mathfrak{m}}_{\varepsilon},{\mathfrak{m}}_{\varepsilon/2}]\subset{\mathfrak{k}}_{\varepsilon/2},\quad[{\mathfrak{m}}_{\varepsilon},{\mathfrak{k}}_{\varepsilon}]\subset{\mathfrak{a}},\quad[{\mathfrak{m}}_{\varepsilon},{\mathfrak{k}}_{\varepsilon/2}]\subset{\mathfrak{m}}_{\varepsilon/2},\\\\[4.79993pt]
[{\mathfrak{m}}_{\varepsilon/2},{\mathfrak{m}}_{\varepsilon/2}]\subset{\mathfrak{h}}\oplus{\mathfrak{k}}_{\varepsilon},\quad[{\mathfrak{m}}_{\varepsilon/2},{\mathfrak{k}}_{\varepsilon}]\subset{\mathfrak{m}}_{\varepsilon/2},\quad[{\mathfrak{m}}_{\varepsilon/2},{\mathfrak{k}}_{\varepsilon/2}]\subset{\mathfrak{a}}\oplus{\mathfrak{m}}_{\varepsilon},\\\\[4.79993pt]
[{\mathfrak{k}}_{\varepsilon},{\mathfrak{k}}_{\varepsilon}]\subset{\mathfrak{h}},\quad[{\mathfrak{k}}_{\varepsilon},{\mathfrak{k}}_{\varepsilon/2}]\subset{\mathfrak{k}}_{\varepsilon/2},\quad[{\mathfrak{k}}_{\varepsilon/2},{\mathfrak{k}}_{\varepsilon/2}]\subset{\mathfrak{h}}\oplus{\mathfrak{k}}_{\varepsilon}.\end{array}$
(4.2)

From [14, Ch. VII, Lemma 2.3], for any
$\mu_{\lambda}\in{\mathfrak{m}}_{\lambda},$ $\lambda\in\Sigma^{+}$, there
exists a unique vector $\nu_{\lambda}\in{\mathfrak{k}}_{\lambda}$ such that
$[u,\mu_{\lambda}]=\sqrt{-1}\lambda(u)\nu_{\lambda}$ and
$[u,\nu_{\lambda}]=-\sqrt{-1}\lambda(u)\mu_{\lambda},$ for all
$u\in{\mathfrak{a}}.$ Then,

$[u,\mu_{\lambda}]=-\lambda_{\mathbb{R}}(u)\nu_{\lambda},\quad[u,\nu_{\lambda}]=\lambda_{\mathbb{R}}(u)\mu_{\lambda},$
(4.3)

where $\lambda_{\mathbb{R}}$ is the linear function
$\lambda_{\mathbb{R}}\colon{\mathfrak{a}}\to{\mathbb{R}}$,
$\lambda\in\Sigma^{+}$, defined by the relation
$\sqrt{-1}\lambda_{\mathbb{R}}=\lambda$. Then fixing in
${\mathfrak{m}}_{\varepsilon}$ and ${\mathfrak{m}}_{\varepsilon/2}$ some
$\langle\cdot,\cdot\rangle$-orthonormal basis $\\{\mu^{j}_{\varepsilon}\\}$
and $\\{\mu^{p}_{\varepsilon/2}\\},$ $j=1,\dots,m_{\varepsilon},$
$p=1,\dots,m_{\varepsilon/2},$ we take the unique bases
$\\{\nu^{j}_{\varepsilon}\\}$ and $\\{\nu^{p}_{\varepsilon/2}\\}$ of
${\mathfrak{k}}_{\varepsilon}$ and ${\mathfrak{k}}_{\varepsilon/2}$ satisfying
(4.3) for $\lambda=\varepsilon$ and $\lambda=\varepsilon/2.$ Clearly these
last two basis are also $\langle\cdot,\cdot\rangle$-orthonormal.

Consider $X$ the unique (basis) vector in ${\mathfrak{a}}$ such that
$\varepsilon_{\mathbb{R}}(X)=1.$ Then, multiplying the inner product
$\langle\cdot,\cdot\rangle$ by a positive constant, we can assume that
$\langle X,X\rangle=1.$

######  Lemma 4.2.

The following identities hold:

$\begin{array}[]{l}[\mu^{j}_{\varepsilon},\mu^{k}_{\varepsilon}]=[\nu^{j}_{\varepsilon},\nu^{k}_{\varepsilon}],\quad[\mu^{j}_{\varepsilon},\nu^{k}_{\varepsilon}]=-[\nu^{j}_{\varepsilon},\mu^{k}_{\varepsilon}]=-\delta_{jk}X,\\\\[4.79993pt]
[\mu^{j}_{\varepsilon},\mu^{p}_{\varepsilon/2}]=[\nu^{j}_{\varepsilon},\nu^{p}_{\varepsilon/2}],\quad[\nu^{j}_{\varepsilon},\mu^{p}_{\varepsilon/2}]=-[\mu^{j}_{\varepsilon},\nu^{p}_{\varepsilon/2}],\\\\[4.79993pt]
[\mu^{p}_{\varepsilon/2},\mu^{q}_{\varepsilon/2}]_{{\mathfrak{k}}_{\varepsilon}}=-[\nu^{p}_{\varepsilon/2},\nu^{q}_{\varepsilon/2}]_{{\mathfrak{k}}_{\varepsilon}}=\frac{1}{2}([\mu^{p}_{\varepsilon/2},\mu^{q}_{\varepsilon/2}]-[\nu^{p}_{\varepsilon/2},\nu^{q}_{\varepsilon/2}]),\\\\[4.79993pt]
[\mu^{p}_{\varepsilon/2},\nu^{q}_{\varepsilon/2}]_{{\mathfrak{m}}_{\varepsilon}}=[\nu^{p}_{\varepsilon/2},\mu^{q}_{\varepsilon/2}]_{{\mathfrak{m}}_{\varepsilon}}=\frac{1}{2}([\mu^{p}_{\varepsilon/2},\nu^{q}_{\varepsilon/2}]+[\nu^{p}_{\varepsilon/2},\mu^{q}_{\varepsilon/2}]),\\\\[4.79993pt]
[\mu^{p}_{\varepsilon/2},\nu^{q}_{\varepsilon/2}]_{\mathfrak{a}}=-\frac{\delta_{pq}}{2}X,\end{array}$

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

###### Proof.

Using that ${\rm ad}_{X}\colon{\mathfrak{g}}\to{\mathfrak{g}}$ is a Lie
derivation, one gets

${\rm ad}^{2}_{X}[U,V]=[{\rm ad}^{2}_{X}U,V]+[U,{\rm ad}_{X}^{2}V]+2[{\rm
ad}_{X}U,{\rm ad}_{X}V],\quad U,V\in{\mathfrak{g}}.$

From here, applying (4.1), (4.2) and (4.3) these equalities are proved. ∎

Next we analyze the tangent bundle $T(G/K)$ and the tangent sphere bundle
$T_{r}(G/K),$ of radius $r>0,$ of $G/K.$

Let us first observe that on the trivial vector bundle $G\times{\mathfrak{m}}$
there are two Lie group actions that commute with each other: 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 the mapping $\phi$ given by

$\phi\colon G\times_{K}{\mathfrak{m}}\to
T(G/K),\quad[(a,x)]\mapsto(\tau_{a})_{*o}x,$ (4.4)

is a $G$-equivariant diffeomorphism which allows us to identify $T(G/K)$ with
$G\times_{K}{\mathfrak{m}}.$ Then, taking into account that each translation
$\tau_{a}$ is an isometry of $(G/K,g),$ it follows from (4.4) that

$T_{r}(G/K)=\phi(G\times_{K}{\mathcal{S}}_{\mathfrak{m}}(r)),$ (4.5)

where ${\mathcal{S}}_{\mathfrak{m}}(r)=\\{x\in{\mathfrak{m}}\colon\langle
x,x\rangle=r^{2}\\}$ is the sphere of radius $r>0$ in the Euclidean space
$({\mathfrak{m}},\langle\cdot,\cdot\rangle).$

On the other hand, let $W$ be the Weyl chamber in ${\mathfrak{a}}$ containing
$X,$

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

which is naturally identified with ${\mathbb{R}}^{+}.$ Since each nonzero
${\rm Ad}(K)$-orbit in ${\mathfrak{m}}$ intersects ${\mathfrak{a}}$ and also
$W,$ it follows that ${\mathfrak{m}}^{R}={\mathfrak{m}}\setminus\\{0\\},$
where ${\mathfrak{m}}^{R}$ denotes the (open dense) subset of regular points
${\mathfrak{m}}^{R}:={\rm Ad}(K)(W)$ of ${\mathfrak{m}}.$ Hence, we have:

  1. (i)

the ${\rm Ad}(K)$-action on the unit sphere ${\mathcal{S}}_{\mathfrak{m}}(1)$
of ${\mathfrak{m}}$ is transitive, or equivalently,

${\mathcal{S}}_{\mathfrak{m}}(r)={\rm Ad}(K)rX,\quad r>0.$ (4.6)

  2. (ii)

the mapping

$\chi\colon G/H\times{\mathbb{R}}^{+}\to
G\times_{K}({\mathfrak{m}}\setminus\\{0\\}),\quad(aH,t)\mapsto[(a,tX)],$ (4.7)

is a well-defined $G$-equivariant diffeomorphism.

The property (i) is a well-known characterization for the class of compact
rank-one symmetric spaces and this implies that for the simply connected case
all their geodesics are closed (see [2]). From (4.4) and (4.7),
$G/H\times{\mathbb{R}}^{+}$ is identified with the punctured tangent bundle
$D(G/K)=T(G/K)\setminus\\{\mbox{\rm zero section}\\}$ of $T(G/K)$ via the
$G$-equivariant diffeomorphism $\phi\circ\chi$ and, using (4.5) and (4.6),
$T_{r}(G/K)$ with $G/H,$ i.e.

$T_{r}(G/K)=(\phi\circ\chi)(G/H\times\\{r\\})\cong G/H.$

According with [12], the decomposition
$\overline{\mathfrak{m}}={\mathfrak{a}}\oplus{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{m}}_{\varepsilon/2}\oplus{\mathfrak{k}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon/2}$
is ${\rm Ad}(H)$-irreducible and moreover, for $G/K\in\\{{\mathbb{C}}{\bf
P}^{n}\;(n\geq 1)\\},$ because ${\mathfrak{a}}$,
${\mathfrak{m}}_{\varepsilon}$ and ${\mathfrak{k}}_{\varepsilon}$ are one-
dimensional, ${\rm Ad}(H)$ acts trivially on
$\overline{\mathfrak{n}}:={\mathfrak{a}}\oplus{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}.$
Then we have the following properties:

  1. (i)

${\mathfrak{g}}={\mathfrak{h}}\oplus\overline{\mathfrak{m}}$ is a reductive
decomposition associated to $G/H,$

  2. (ii)

the tangent space $T_{o_{H}}(G/H)$ at the origin $o_{H}=\\{H\\}$ is identified
with $\overline{\mathfrak{m}},$ and

  3. (iii)

any $G$-invariant Riemannian metric ${\mathbf{g}}={\bf
g}^{a_{0},a_{\lambda},b_{\lambda}},$ $\lambda\in\Sigma^{+},$ on $G/H$ is
determined by

${\mathbf{g}}_{o_{H}}=a_{0}\langle\cdot,\cdot\rangle_{\mathfrak{a}}+a_{\varepsilon}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon}}+b_{\varepsilon}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon}}+a_{\varepsilon/2}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon/2}}+b_{\varepsilon/2}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon/2}},$
(4.8)

where $a_{0},a_{\lambda}$ and $b_{\lambda},$ for $\lambda\in\Sigma^{+},$ are
positive constants, except for $G/K\in\\{{\mathbb{C}}{\bf P}^{n}\;(n\geq
1)\\},$ where

${\bf
g}_{o_{H}}=\prec\cdot,\cdot\succ+a_{\varepsilon/2}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon/2}}+b_{\varepsilon/2}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon/2}},$
(4.9)

$\prec\cdot,\cdot\succ$ being an arbitrary inner product on
$\overline{\mathfrak{n}}.$

######  Definition 4.3.

A $G$-invariant metric ${\bf g}$ on $G/H$ such that at the origin is expressed
as in (4.8) is said to be an orthogonal metric.

As we will see in the next section, there are many $G$-invariant contact
metrics in $T_{r}{\mathbb{C}}{\bf P}^{n}$ that are not orthogonal.

Under the previous identification $\phi\circ\chi,$ the metric $\overline{\bf
g}=r^{2}{\bf g}+dr^{2}$ on the metric cone ${\mathcal{C}}(G/H)$ gives an
$G$-invariant Riemannian metric, also denoted by $\overline{\bf g},$ on
$D(G/K).$ Then, taking $T_{1}G/K\cong G/H\times\\{1\\}$ as submanifold of
$G/H\times{\mathbb{R}}^{+}\cong D(G/K),$ using Corollary 2.2 and the
characterizations described in Section 2 for Sasakian, Sasakian-Einstein and
$3$-Sasakian structures, the following result is proved.

######  Theorem 4.4.

Each $G$-invariant contact metric, Sasakian, Sasakian-Einstein or $3$-Sasakian
structure $(\xi,{\bf g})$ on the unit tangent sphere $T_{1}(G/K$) can be
extended, respectively, to a unique $G$-invariant almost Kähler, Kähler,
Kähler Ricci-flat or hyperKähler structure on the metric punctured tangent
bundle $(D(G/K),\overline{\bf g}).$

######  Remark 4.5.

From the definition of $\chi$ in (4.7), one gets

$\chi_{*(o_{H},t)}(X,0)=\pi_{*(e,tX)}(X,0),\quad t\in{\mathbb{R}}^{+}.$

Then, $\overline{\bf g}$ on $G\times_{K}({\mathfrak{m}}\setminus\\{0\\})$
satisfies

$\overline{\bf
g}_{[(e,tX)]}(\pi_{*(e,tX)}(X,0),\pi_{*(e,tX)}(X,0))=\overline{\bf
g}_{(o_{H},t)}((X,0),(X,0))=t^{2}a_{0}.$

Hence, $\lim_{t\to 0^{+}}\overline{\bf
g}_{[(e,tX)]}(\pi_{*(e,tX)}(X,0),\pi_{*(e,tX)}(X,0))=0$ and so $\overline{\bf
g}$ cannot be extended to a Riemannian metric on all
$G\times_{K}{\mathfrak{m}}\cong T(G/K).$

##  5\. Invariant contact metric structures

In order to obtain all $G$-invariant contact metric structures on
$T_{r}(G/K),$ we analyze the space of $G$-invariant vector fields on
$T_{r}(G/K)=G/H$ or, equivalently, the subspace ${\rm
Inv}(\overline{\mathfrak{m}})\subset\overline{\mathfrak{m}}$ of ${\rm
Ad}(H)$-invariant vectors of $\overline{\mathfrak{m}},$ as candidates to be
characteristic vector fields. This space contains the standard vector field
$\xi^{S}$ on $T_{r}(G/K),$ for any compact rank-one symmetric space $G/K.$ In
fact $\xi^{S}$ is the $G$-invariant vector field on
$G/H\times{\mathbb{R}}^{+}$ such that $\xi^{S}_{(o_{H},t)}=(X,0),$ for all
$t\in{\mathbb{R}}^{+},$ [12, Proposition 4.8].

Two cases for $G/K$ are distinguished depending on whether $\dim{\rm
Inv}(\overline{\mathfrak{m}})$ is one or greater than one.

######  Theorem 5.1.

If $G/K\in\\{{\mathbb{S}}^{n},{\mathbb{R}}{\bf P}^{n}(n\geq
3),{\mathbb{H}}{\bf P}^{n}(n\geq 1),{\mathbb{C}}a{\bf P}^{2}\\},$ any
$G$-invariant contact metric structure on the tangent sphere bundle
$T_{r}(G/K),$ $r>0,$ is given by the pairs $(\xi,{\bf g}),$ where
$\xi=\pm\frac{1}{\kappa}\xi^{S},$ for some $\kappa>0,$ i.e.
$\xi_{o_{H}}=\pm\frac{1}{\kappa}X,$ and the metric ${\bf g}$ is determined by

${\bf
g}_{o_{H}}=\kappa^{2}\langle\cdot,\cdot\rangle_{\mathfrak{a}}+\frac{\kappa}{2q_{\varepsilon}}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon}}+\frac{\kappa
q_{\varepsilon}}{2}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon}}+\frac{\kappa}{4q_{\varepsilon/2}}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon/2}}+\frac{\kappa
q_{\varepsilon/2}}{4}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon/2}},$
(5.1)

for some positive constants $q_{\lambda},$ $\lambda\in\Sigma^{+}.$ Moreover,
$(\xi,{\bf g})$ is K-contact, indeed Sasakian, if and only if $q_{\lambda}=1.$

###### Proof.

Let ${\bf g}$ be any $G$-invariant Riemannian metric ${\mathbf{g}}={\bf
g}^{a_{0},a_{\lambda},b_{\lambda}},$ $\lambda\in\Sigma^{+},$ on $G/H$ defined
in (4.8) and let $\xi$ be any unit $G$-invariant vector field on $(G/H,{\bf
g}).$ Then $\xi_{o_{H}}$ in $\overline{\mathfrak{m}}$ is ${\rm
Ad}(H)$-invariant and, therefore, the subspace ${\mathbb{R}}\xi_{o_{H}}$ must
be ${\rm Ad}(H)$-irreducible. According with Table I, taking into account that
the decomposition
$\overline{\mathfrak{m}}={\mathfrak{a}}\oplus{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}\oplus{\mathfrak{m}}_{\varepsilon/2}\oplus{\mathfrak{k}}_{\varepsilon/2}$
is ${\rm Ad}(H)$-irreducible and the dimensions of the subspaces
${\mathfrak{m}}_{\varepsilon},$ ${\mathfrak{k}}_{\varepsilon},$
${\mathfrak{m}}_{\varepsilon/2}$ and ${\mathfrak{k}}_{\varepsilon/2}$ are
different from one, we have that ${\rm
Inv}(\overline{\mathfrak{m}})={\mathbb{R}}X.$ Then
$\xi=\pm\frac{1}{\kappa}\xi^{S}$ and the result has already been proven in
[13, Theorem 6.1]. ∎

If $G/K\in\\{{\mathbb{C}}{\bf P}^{n}\,(n\geq 1)\\}$ then ${\rm
Inv}(\overline{\mathfrak{m}})=\overline{\mathfrak{n}}$ and any $G$-invariant
Riemannian metric ${\bf g}$ on $T_{r}{\mathbb{C}}{\bf P}^{n}=G/H$ is expressed
at the origin as in (4.9). Put

$\begin{array}[]{l}a_{0}=\prec X,X\succ,\quad
a_{\varepsilon}=\prec\mu_{\varepsilon},\mu_{\varepsilon}\succ,\quad
b_{\varepsilon}=\prec\nu_{\varepsilon},\nu_{\varepsilon}\succ,\\\\[4.79993pt]
a_{0\varepsilon}=\prec X,\mu_{\varepsilon}\succ,\quad b_{0\varepsilon}=\prec
X,\nu_{\varepsilon}\succ,\quad
c_{\varepsilon}=\prec\mu_{\varepsilon},\nu_{\varepsilon}\succ,\end{array}$

where $\\{X,\mu_{\varepsilon},\nu_{\varepsilon}\\}$ is an orthonormal basis of
$(\overline{\mathfrak{n}},\langle\cdot,\cdot\rangle)$ satisfying (4.3). Then
${\bf g}$ is orthogonal if and only if
$a_{0\varepsilon}=b_{0\varepsilon}=c_{\varepsilon}=0.$ Next, consider the
(unique) orthonormal basis $\\{\xi_{1},\xi_{2},\xi_{3}\\}$ of
$(\overline{\mathfrak{n}},\prec\cdot,\cdot\succ)$ satisfying the following
conditions:

  1. (i)

$\xi_{1}=\frac{1}{\rho}X,$ for some $\rho>0,$ and
$\xi_{2}\in{\mathbb{R}}\\{X,\mu_{\varepsilon}\\},$

  2. (ii)

the bases $\\{\xi_{1},\xi_{2}\\}$ and $\\{\xi_{1},\xi_{2},\xi_{3}\\}$ have,
respectively, the same orientation than $\\{X,\mu_{\varepsilon}\\}$ and
$\\{X,\mu_{\varepsilon},\nu_{\varepsilon}\\}.$

Therefore the matrix $A=(a_{ij})$ such that
$\xi_{j}=a_{j1}X+a_{j2}\mu_{\varepsilon}+a_{j3}\nu_{\varepsilon},$ $j=1,2,3,$
determines the inner product $\prec\cdot,\cdot\succ$ and it is expressed as

$A=\left(\begin{array}[]{ccc}\frac{1}{\rho}&0&0\\\\[4.79993pt]
a_{21}&a_{22}&0\\\\[4.79993pt] a_{31}&a_{32}&a_{33}\end{array}\right),$ (5.2)

where $a_{22}$ and $a_{33}$ are positive numbers. Note that ${\bf g}$ is
orthogonal if and only if $A$ is diagonal.

######  Remark 5.2.

From [12, Proposition 4.8], the induced metric $\widetilde{g}^{S}$ of the
Sasaki metric $g^{S}$ on $T_{r}(G/K)$ is $G$-invariant, determined by

$\widetilde{g}^{S}_{o_{H}}=\left\\{\begin{array}[]{lcl}\langle\cdot,\cdot\rangle_{\mathfrak{m}}+r^{2}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon}},&\mbox{if}&G/K={\mathbb{S}}^{n}\;\mbox{or}\;\;{\mathbb{R}}{\mathbf{P}}^{n},\\\\[4.79993pt]
\langle\cdot,\cdot\rangle_{\mathfrak{m}}+r^{2}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon}}+\frac{r^{2}}{4}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon/2}},&\mbox{if}&G/K={\mathbb{C}}{\mathbf{P}}^{n},\;{\mathbb{H}}{\mathbf{P}}^{n}\;\mbox{or}\;\;{\mathbb{C}}a{\mathbf{P}}^{n}.\end{array}\right.$
(5.3)

Hence $\widetilde{g}^{S}$ is an orthogonal metric, even in
$T_{r}{\mathbb{C}}{\bf P}^{n}.$ Here $a_{0}=a_{\varepsilon}=1$ and
$b_{\varepsilon}=r^{2},$ and $A$ is the diagonal matrix $(1,1,\frac{1}{r}).$

If moreover ${\bf g}$ is contact metric then its characteristic vector field
$\xi$ is a $G$-invariant field with $\xi_{o_{H}}\in\overline{\mathfrak{n}}.$
We say that a $G$-invariant contact metric structure $(\xi,{\bf g})$ on
$T_{r}{\mathbb{C}}{\bf P}^{n}$ is of

  1. $\bullet$

Type AI, AII or AIII, if $\xi_{o_{H}}=\pm\xi_{1},$ $\pm\xi_{2}$ or
$\pm\xi_{3},$

  2. $\bullet$

Type BI, BII or BIII, if $\xi_{o_{H}}=\cos\theta\xi_{i}+\sin\theta\xi_{j},$
for $(i,j)=(1,2),$ $(1,3)$ or $(2,3),$ where
$\theta\in]0,2\pi[\setminus\\{\pi/2,\pi,3\pi/2\\},$

  3. $\bullet$

Type C, if
$\xi_{o_{H}}=\cos\theta\cos\phi\xi_{1}+\sin\theta\cos\phi\xi_{2}+\sin\phi\xi_{3},$
where $\theta\in]0,2\pi[\setminus\\{\pi/2,\pi,3\pi/2\\}$ and
$\phi\in]-\pi/2,\pi/2[\setminus\\{0\\}.$

######  Theorem 5.3.

A $G$-invariant contact metric structure $(\xi,{\bf g})$ on
$T_{r}{\mathbb{C}}{\bf P}^{n}\,(n\geq 1)$ is of

  1. $\bullet$

Type AI if and only if $\xi_{o_{H}}=\pm\frac{1}{\kappa}X$ and

$\begin{array}[]{l}a_{0}=\kappa^{2},\quad
a_{\varepsilon}=\frac{\kappa}{2q_{\varepsilon}},\quad
b_{\varepsilon}=\frac{q_{\varepsilon}}{2\kappa}(4\alpha^{2}+\kappa^{2}),\\\
c_{\varepsilon}=\alpha,\quad a_{0\varepsilon}=b_{0\varepsilon}=0,\quad
a_{\varepsilon/2}=\frac{\kappa}{4q_{\varepsilon/2}},\quad
b_{\varepsilon/2}=\frac{\kappa q_{\varepsilon/2}}{4},\end{array}$

for all $\kappa,q_{\varepsilon},q_{\varepsilon/2}>0$ and
$\alpha\in{\mathbb{R}};$

  2. $\bullet$

Type AII if and only if $\xi_{o_{H}}=\pm\frac{1}{\kappa}\mu_{\varepsilon}$ and

$\begin{array}[]{l}a_{0}=\frac{\kappa}{2q_{\varepsilon}},\quad
a_{\varepsilon}=\kappa^{2},\quad
b_{\varepsilon}=\frac{q_{\varepsilon}}{2\kappa}(4\alpha^{2}+\kappa^{2}),\\\\[4.79993pt]
a_{0\varepsilon}=c_{\varepsilon}=0,\quad b_{0\varepsilon}=\alpha,\quad
a_{\varepsilon/2}=\frac{\kappa}{4q_{\varepsilon/2}},\quad
b_{\varepsilon/2}=\frac{\kappa q_{\varepsilon/2}}{4},\end{array}$

for all $\kappa,q_{\varepsilon},q_{\varepsilon/2}>0$ and
$\alpha\in{\mathbb{R}};$

  3. $\bullet$

Type AIII if and only if $\xi=\pm\frac{1}{\kappa}\nu_{\varepsilon}$ and

$\begin{array}[]{l}a_{0}=\frac{\kappa}{2q_{\varepsilon}},\quad
a_{\varepsilon}=\frac{q_{\varepsilon}}{2\kappa}(4\alpha^{2}+\kappa^{2}),\quad
b_{\varepsilon}=\kappa^{2},\\\\[4.79993pt] a_{0\varepsilon}=\alpha,\quad
b_{0\varepsilon}=c_{\varepsilon}=0,\quad
a_{\varepsilon/2}=b_{\varepsilon/2}=\frac{\kappa}{4},\end{array}$

for all $\kappa,q_{\varepsilon}>0$ and $\alpha\in{\mathbb{R}};$

  4. $\bullet$

Type BI if and only if
$\xi_{o_{H}}=\frac{1}{\kappa}(\epsilon_{\cos\theta}\frac{\varrho}{q_{\varepsilon}}X+\sin\theta\mu_{\varepsilon}),$
where $\epsilon_{\cos\theta}$ denotes the sing of $\cos\theta$ and
$\varrho=\sqrt{q_{\varepsilon}(1-q_{\varepsilon}\sin^{2}\theta)},$ and

$\begin{array}[]{l}a_{0}=\frac{\kappa^{2}\varrho^{2}}{\cos^{2}\theta},\quad
a_{\varepsilon}=\frac{\kappa^{2}}{\cos^{2}\theta}(\cos^{2}\theta+(1-q_{\varepsilon})^{2}\sin^{2}\theta),\quad
b_{\varepsilon}=\frac{q_{\varepsilon}(\kappa^{4}\alpha^{2}\varrho^{2}+\sin^{2}\theta\cos^{2}\theta)}{4\varrho^{2}\sin^{2}\theta},\\\\[4.79993pt]
a_{0\varepsilon}=-\epsilon_{\cos\theta}\frac{\kappa^{2}\varrho}{\cos^{2}\theta}(1-q_{\varepsilon})\sin\theta,\quad
b_{0\varepsilon}=\epsilon_{\cos\theta}\frac{\alpha\kappa^{3}q_{\varepsilon}\varrho}{2\cos\theta},\quad
c_{\varepsilon}=-\frac{\alpha\kappa^{3}\varrho^{2}}{2\sin\theta\cos\theta},\\\\[4.79993pt]
a_{\varepsilon/2}=\frac{\kappa\sqrt{q_{\varepsilon}}}{4q_{\varepsilon/2}},\quad
b_{\varepsilon/2}=\frac{\kappa\sqrt{q_{\varepsilon}}q_{\varepsilon/2}}{4},\end{array}$

for all $\kappa,q_{\varepsilon/2}>0,$
$q_{\varepsilon}\in]0,\frac{1}{\sin^{2}\theta}[$ and $\alpha\in{\mathbb{R}};$

  5. $\bullet$

Type BII if and only if $\xi_{o_{H}}=\frac{q_{\varepsilon}}{2\kappa}\cos\theta
X+\sin\theta(\alpha\mu_{\varepsilon}+\frac{1}{\kappa}\nu_{\varepsilon}),$ and

$\begin{array}[]{l}a_{0}=\frac{4\kappa^{2}q_{\varepsilon}^{2}}{\varrho^{2}},\quad
a_{\varepsilon}=\frac{1}{q_{\varepsilon}^{2}\varrho^{2}}(\varrho^{2}+16\alpha^{2}q_{\varepsilon}^{2}\kappa^{4}\tan^{2}\theta),\quad
b_{\varepsilon}=\kappa^{2}(1+\frac{(4-\varrho)^{2}}{\varrho^{2}}\tan^{2}\theta+\frac{\alpha^{2}}{q_{\varepsilon}^{2}}),\\\\[4.79993pt]
a_{0\varepsilon}=\frac{8\alpha\kappa^{3}q_{\varepsilon}}{\varrho^{2}}\tan\theta,\quad
b_{0\varepsilon}=\frac{2\kappa^{2}q_{\varepsilon}(4-\varrho)}{\varrho^{2}}\tan\theta,\quad
c_{\varepsilon}=\frac{\kappa\alpha}{q_{\varepsilon}^{2}}(4\kappa^{2}q_{\varepsilon}^{2}(4-\varrho)\tan^{2}\theta-\varrho^{2}),\\\\[4.79993pt]
a_{\varepsilon/2}=b_{\varepsilon/2}=\frac{\kappa\sqrt{\varrho}}{2\varrho},\end{array}$

where
$\varrho=q_{\varepsilon}^{2}\cos^{2}\theta+4(\alpha^{2}\kappa^{2}+1)\sin^{2}\theta,$
for all $\kappa,q_{\varepsilon}>0$ and $\alpha\in{\mathbb{R}}.$

  6. $\bullet$

Type BIII if and only if $\xi_{o_{H}}=\frac{1}{\kappa
q_{\varepsilon}}(\cos\theta\mu_{\varepsilon}+q_{\varepsilon}\sin\theta\nu_{\varepsilon}),$
and

$\begin{array}[]{l}a_{0}=\frac{q_{\varepsilon}^{2}}{4},\quad
a_{\varepsilon}=\frac{\kappa^{2}q_{\varepsilon}^{2}}{\varrho^{2}}(1+\alpha^{2}\varrho^{2}\tan^{2}\theta),\quad
b_{\varepsilon}=\frac{\kappa^{2}}{\varrho^{2}}(\varrho^{2}\alpha^{2}+(\varrho^{2}+(1-q_{\varepsilon}^{2})^{2}\sin^{2}\theta\cos^{2}\theta)),\\\\[4.79993pt]
a_{0\varepsilon}=-\frac{\alpha\kappa q_{\varepsilon}^{2}}{2}\tan\theta,\quad
b_{0\varepsilon}=\frac{\alpha\kappa q_{\varepsilon}}{2},\quad
c_{\varepsilon}=-\frac{q_{\varepsilon}\kappa^{2}}{\varrho^{2}}(\alpha^{2}\varrho^{2}\tan\theta+(1-q_{\varepsilon}^{2})\sin\theta\cos\theta),\\\\[4.79993pt]
a_{\varepsilon/2}=b_{\varepsilon/2}=\frac{\kappa
q_{\varepsilon}\sqrt{\varrho}}{4\varrho},\end{array}$

where $\varrho=\cos^{2}\theta+q_{\varepsilon}^{2}\sin^{2}\theta,$ for all
$\kappa,q_{\varepsilon}>0$ and $\alpha\in{\mathbb{R}};$

  7. $\bullet$

Type C if and only if

$\begin{array}[]{lcl}\xi_{o_{H}}&=&\frac{1}{q_{\varepsilon}\delta}(\delta\cos\phi(\cos\theta+q_{\varepsilon}\alpha\sin\theta)+q_{\varepsilon}(\alpha\gamma_{1}-\gamma_{2})\sin\phi)X\\\\[4.79993pt]
&&+(\delta\sin\theta\cos\phi+\gamma_{1})\mu_{\varepsilon}+\frac{1}{\kappa}\nu_{\varepsilon},\;\mbox{and}\end{array}$
$\begin{array}[]{l}a_{0}=q_{\varepsilon}^{2},\quad
a_{\varepsilon}=\frac{1+\alpha^{2}q_{\varepsilon}^{2}}{\delta^{2}},\quad
b_{\varepsilon}=\frac{\kappa^{2}}{\delta^{2}}(\gamma_{1}^{2}+q_{\varepsilon}^{2}\gamma_{2}^{2}+\delta^{2}),\\\\[4.79993pt]
a_{0\varepsilon}=-\frac{\alpha q_{\varepsilon}^{2}}{\delta},\quad
b_{0\varepsilon}=\frac{\kappa q_{\varepsilon}^{2}\gamma_{2}}{\delta},\quad
c_{\varepsilon}=-\frac{\kappa}{\delta^{2}}(\gamma_{1}+\alpha
q_{\varepsilon}^{2}\gamma_{2}),\\\\[4.79993pt]
a_{\varepsilon/2}^{2}=b_{\varepsilon/2}^{2}=\frac{1}{16\delta^{2}}(q_{\varepsilon}^{2}\delta^{2}\cos\theta^{2}\cos^{2}\phi+\varrho),\end{array}$

for all $\kappa,q_{\varepsilon}>0$ and $\alpha\in{\mathbb{R}}$ such that
$\varrho\cos^{2}\theta\cos^{2}\phi\leq\kappa^{2},$ where
$\varrho=\cos^{2}\phi(\sin\theta-
q_{\varepsilon}\alpha\cos\theta)^{2}+4q_{\varepsilon}^{2}\sin^{2}\phi,$

$\begin{array}[]{lcl}\gamma_{1}&=&\frac{\cot\phi}{\kappa^{2}\beta}((1-\kappa^{2}\beta\delta)\sin\theta-
q_{\varepsilon}\alpha\cos\theta),\\\\[4.79993pt]
\gamma_{2}&=&\frac{\cot\phi}{\kappa^{2}\beta
q_{\varepsilon}}(q_{\varepsilon}\alpha\sin\theta+(\kappa^{2}\beta\delta-
q_{\varepsilon}^{2}(\alpha^{2}+\delta^{2}))\cos\theta)\end{array}$

and $\delta$ is positive number such that
$(q_{\varepsilon}\cos\theta\cos\phi)^{2}\delta^{2}-2\kappa^{2}\beta\delta+\varrho=0,$
where $\beta^{2}=\frac{4q_{\varepsilon}^{2}}{\kappa^{2}}.$

When ${\bf g}$ is an orthogonal metric, $(\xi,{\bf g})$ is said to be an
orthogonal contact metric structure.

######  Corollary 5.4.

Any $G$-invariant $K$-contact structure $(\xi,{\bf g})$ on
$T_{r}{\mathbb{C}}{\bf P}^{n}$ $(n\geq 1)$ of Type A is orthogonal.

###### Proof.

Because $\xi$ is Killing, it follows from (3.5) and (3.6) that

${\bf g}([\xi,\mu_{1}]_{\overline{\mathfrak{m}}},\mu_{2})+{\bf
g}([\xi,\mu_{2}]_{\overline{\mathfrak{m}}},\mu_{1})=0,\quad\mu_{1},\mu_{2}\in\overline{\mathfrak{m}}.$
(5.4)

Now, using (4.3) and Theorem 5.3 and putting in (5.4)
$\mu_{1}=\mu_{2}=\mu_{\varepsilon},$ if $(\xi,{\bf g})$ is of Type AI, and
$\mu_{1}=\mu_{2}=X,$ if $(\xi,{\bf g})$ is of Type AII or AIII, we get
$\alpha=0.$ Then, from Theorem 5.3, $(\xi,{\bf g})$ is orthogonal. ∎

######  Theorem 5.5.

All orthogonal contact metric structures on $T_{r}{\mathbb{C}}{\bf P}^{n}$ and
those that are also $K$-contact are described in Table II. Moreover, each
orthogonal contact metric structure of Type AI, determined by a triple of
positive constants $(\kappa,q_{\varepsilon},q_{\varepsilon/2}),$ is isomorphic
to the corresponding of Type AII.

Table II. Orthogonal contact metric structures $(\xi,{\bf
g}^{a_{0},a_{\varepsilon},b_{\varepsilon},a_{\varepsilon/2},b_{\varepsilon/2}})$
on $T_{r}{\mathbb{C}}{\bf P}^{n}$
---
Type | $\xi_{o_{H}}$ | $a_{0}$ | $a_{\varepsilon}$ | $b_{\varepsilon}$ | $a_{\varepsilon/{\scriptscriptstyle 2}}$ | $b_{\varepsilon/{\scriptscriptstyle 2}}$ |  $K$-contact
AI | ${\scriptstyle\pm\frac{1}{\kappa}X}$ | $\kappa^{2}$ | $\frac{\kappa}{2q_{\varepsilon}}$ | $\frac{\kappa q_{\varepsilon}}{2}$ | $\frac{\kappa}{4q_{\varepsilon/2}}$ | $\frac{\kappa q_{\varepsilon/2}}{4}$ | $\scriptstyle{q_{\varepsilon}=q_{\varepsilon/2}=1}$
AII | ${\scriptstyle\pm\frac{1}{\kappa}\mu_{\varepsilon}}$ | $\frac{\kappa}{2q_{\varepsilon}}$ | $\kappa^{2}$ | $\frac{\kappa q_{\varepsilon}}{2}$ | $\frac{\kappa}{4q_{\varepsilon/2}}$ | $\frac{\kappa q_{\varepsilon/2}}{4}$ | $\scriptstyle{q_{\varepsilon}=q_{\varepsilon/2}=1}$
AIII | ${\scriptstyle\pm\frac{1}{\kappa}\nu_{\varepsilon}}$ | $\frac{\kappa}{2q_{\varepsilon}}$ | $\frac{\kappa q_{\varepsilon}}{2}$ | $\kappa^{2}$ | $\frac{\kappa}{4}$ | $\frac{\kappa}{4}$ | $\scriptstyle{q_{\varepsilon}=1}$
BI | ${\scriptstyle\frac{1}{\kappa}(\cos\theta X+\sin\theta\mu_{\varepsilon})}$ | $\kappa^{2}$ | $\kappa^{2}$ | $\frac{1}{4}$ | $\frac{\kappa}{4q_{\varepsilon/2}}$ | $\frac{\kappa q_{\varepsilon/2}}{4}$ | $\scriptstyle{\kappa=\frac{1}{2},\;q_{\varepsilon/2}=1}$
BII | ${\scriptstyle\frac{1}{\kappa}(\cos\theta X+\sin\theta\nu_{\varepsilon})}$ | $\kappa^{2}$ | $\frac{1}{4}$ | $\kappa^{2}$ | $\frac{\kappa}{4}$ | $\frac{\kappa}{4}$ | $\scriptstyle{\kappa=\frac{1}{2}}$
BIII | ${\scriptstyle\frac{1}{\kappa}(\cos\theta\mu_{\varepsilon}+\sin\theta\nu_{\varepsilon})}$ | $\frac{1}{4}$ | $\kappa^{2}$ | $\kappa^{2}$ | $\frac{\kappa}{4}$ | $\frac{\kappa}{4}$ | $\scriptstyle{\kappa=\frac{1}{2}}$
C | ${\scriptstyle 2(\cos\theta\cos\phi X+\sin\theta\cos\phi\mu_{\varepsilon}+\sin\phi\nu_{\varepsilon})}$ | $\frac{1}{4}$ | $\frac{1}{4}$ | $\frac{1}{4}$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\scriptstyle{\rm Yes}$

${\scriptstyle\theta\in]0,2\pi[\setminus\\{\pi/2,\pi,3\pi/2\\},\;\phi\in]-\pi/2,\pi/2[\setminus\\{0\\}}$

######  Remark 5.6.

The tangent sphere bundles $T_{r}{\mathbb{S}}^{2}$ and $T_{r}{\mathbb{R}}{\bf
P}^{2}$ are Lie groups naturally isomorphic to ${\rm
SO}(3)\cong{\mathbb{R}}{\bf P}^{3}$ and ${\rm SO}(3)/\pm{\rm I}_{2},$
respectively. Since ${\mathbb{S}}^{2}\cong{\mathbb{C}}{\bf P}^{1}={\rm
SU}(2)/{\rm S}({\rm U}(1)\times{\rm U}(1)),$ invariant vector fields of
$T_{r}{\mathbb{S}}^{2}$ and $T_{r}{\mathbb{R}}{\bf P}^{2}$ can be viewed as
vectors of the Lie algebra
${\mathfrak{s}}{\mathfrak{u}}(2)\cong{\mathfrak{s}}{\mathfrak{o}}(3)$ and
therefore, the set of their invariant contact metric structures are described
in Theorem 5.3 for $n=1,$ where
${\mathfrak{m}}_{\varepsilon/2}={\mathfrak{k}}_{\varepsilon/2}=0.$

The $G$-invariant contact metric ${\bf g}$ on $T_{r}{\mathbb{C}}{\bf
P}^{n}=G/H,$ $n\geq 1,$ such that ${\bf
g}_{o_{H}}=\frac{1}{4}\langle\cdot,\cdot\rangle_{\overline{\mathfrak{n}}}+\frac{1}{8}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon/2}\oplus{\mathfrak{k}}_{\varepsilon/2}}$
defines $K$-contact metric structures of all types in Table II. Therefore the
set of its associated (Killing) characteristic vector fields consists of all
$G$-invariant vector fields $\xi$ such that $\xi_{o_{H}}$ belongs to the
$2$-sphere ${\mathcal{S}}_{\overline{\mathfrak{n}}}(2)$ of radius $2$ in the
Euclidean space $(\overline{\mathfrak{n}},\langle\cdot,\cdot\rangle).$ In
Theorem 6.2 we prove that ${\bf g}$ is indeed a $3$-Sasakian metric.

From Theorems 5.3 and 5.5 and Corollary 5.4, we have the following.

######  Corollary 5.7.

Any $G$-invariant Riemannian metric ${\bf g}$ on the tangent sphere bundle
$T_{r}{\mathbb{C}}{\bf P}^{n}$ $(n\geq 1)$ such that the pair
$(\xi=\frac{1}{\kappa}\xi^{S},{\bf g})$ is a contact metric structure for some
constant $\kappa>0,$ is determined by

$\begin{array}[]{l}a_{0}=\kappa^{2},\quad
a_{\varepsilon}=\frac{\kappa}{2q_{\varepsilon}},\quad
b_{\varepsilon}=\frac{q_{\varepsilon}}{2\kappa}(4\alpha^{2}+\kappa^{2}),\\\
c_{\varepsilon}=\alpha,\quad a_{0\varepsilon}=b_{0\varepsilon}=0,\quad
a_{\varepsilon/2}=\frac{\kappa}{4q_{\varepsilon/2}},\quad
b_{\varepsilon/2}=\frac{\kappa q_{\varepsilon/2}}{4},\end{array}$ (5.5)

for some positive constants $q_{\varepsilon},$ $q_{\varepsilon/2}$ and
$\alpha\in{\mathbb{R}}.$ Moreover, $(\xi,{\bf g})$ is $K$-contact, indeed
Sasakian, if and only if $q_{\varepsilon}=q_{\varepsilon/2}=1$ and $\alpha=0,$
or equivalently

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

######  Remark 5.8.

Theorem 6.1 in [13] must include the non-orthogonal contact metrics in
$T_{r}{\mathbb{C}}{\bf P}^{n},$ described in (5.5) for $\alpha\neq 0.$

Before proceeding to the proof of Theorems 5.3 and 5.5, we first analyze, as a
consequence of these theorems, the contact character of the induced Sasaki
metric $\widetilde{g}^{S}$ on $T_{r}(G/K).$

For any Riemannian manifold $(M,g),$ the pair
$(\frac{1}{2}\xi^{S},\frac{1}{4}\widetilde{g}^{S})$ is a contact metric
structure on $T_{1}M$ (see for example [3, Ch. 9]) and Tashiro [21] proved
that it is $K$-contact, indeed Sasakian, if and only if $(M,g)$ has constant
sectional curvature $1.$ Now we have the following version of Tashiro’s
Theorem for tangent sphere bundles $T_{r}(G/K)$ of any radius $r>0.$ The proof
follows comparing expressions (5.1) and (5.3) and using Theorems 5.1 and 5.5.

######  Corollary 5.9.

The induced Sasaki metric $\widetilde{g}^{S}$ on $T_{r}(G/K)$ is contact if
and only if $r=\frac{1}{2},$ but it is not $K$-contact. On $T_{{\scriptstyle
1/2}}{\mathbb{C}}{\bf P}^{n}$ $(n\geq 1),$ $\widetilde{g}^{S}$ defines contact
metric structures of Type AI and BI.

For all $r>0,$ $\frac{1}{4r^{2}}\widetilde{g}^{S}$ is a contact metric on
$T_{r}(G/K)$ and it is $K$-contact if and only if $r=1$ and
$G/K={\mathbb{S}}^{n}$ or ${\mathbb{R}}{\bf P}^{n}.$ Then,
$\frac{1}{4}\widetilde{g}^{S}$ is a Sasakian metric. On $T_{r}{\mathbb{C}}{\bf
P}^{n},$ $\frac{1}{4r^{2}}\widetilde{g}^{S}$ defines contact metric structures
of Type AI, AII and BI.

For the proof of Theorems 5.3 and 5.5, we will need to carry out a somewhat
more detailed study of ${\mathbb{C}}{\bf P}^{n}$ as the symmetric space
$G/K={\rm SU}(n+1)/{\rm S}({\rm U}(1)\times{\rm U}(n))$ (see [14, Ch. III and
X]).

Consider the Lie algebra ${\mathfrak{g}}={\mathfrak{s}}{\mathfrak{u}}(n+1)$ of
${\rm SU}(n+1)$ as compact real form of the complex simple Lie algebra
${\mathfrak{a}}_{n}={\mathfrak{s}}{\mathfrak{l}}(n+1,{\mathbb{C}})$ of the
complex $(n+1)\times(n+1)$ matrices of trace $0$ and denote by $E_{jk}$ the
$(n+1)\times(n+1)$-matrix with $1$ in the entry of the $j$th row and the $k$th
column and $0$ elsewhere. A Cartan subalgebra ${\mathfrak{t}}_{\mathbb{C}}$ of
${\mathfrak{a}}_{n}$ consists of the complex diagonal matrices of trace $0,$
generated by the set $\\{T_{i}=E_{ii}-E_{i+1\,i+1},\;(1\leq i\leq n)\\}.$

Let $e_{i},$ $i=1,\dots,n+1,$ be the linear function on
${\mathfrak{t}}_{\mathbb{C}}$ such that $e_{i}(T)=T_{ii},$ for all
$T\in{\mathfrak{t}}_{\mathbb{C}}.$ Then
$\Pi=\\{\alpha_{1},\dots,\alpha_{n}\\}$ is a system of simple roots of
${\mathfrak{a}}_{n},$ where $\alpha_{i}=e_{i}-e_{i+1},$ and we can take as set
of positive roots $\Delta^{+}$ to
$\Delta^{+}=\\{\alpha_{pq}=\alpha_{p}+\dots+\alpha_{q}\;\colon 1\leq p\leq
q\leq n\\}.$ A root vector $E_{\alpha_{pq}}$ of $\alpha_{pq}$ is the matrix
$E_{p\,q+1}.$ Hence, putting

$A_{jk}=\sqrt{-1}(E_{jj}-E_{kk}),\;\;\;B^{0}_{jk}=E_{jk}-E_{kj},\;\;\;B^{1}_{jk}=\sqrt{-1}(E_{jk}+E_{kj}),$
(5.6)

the set of matrices $\\{A_{i\,i+1}\;(1\leq i\leq
n),\;B^{a}_{jk}\;(a=0,1;\;1\leq j<k\leq n+1)\\}$ is a basis of
${\mathfrak{s}}{\mathfrak{u}}(n+1).$ In what follows we shall need the Lie
multiplication table:

$\begin{array}[]{lcl}[A_{rj},A_{kl}]&=&0,\\\\[4.79993pt]
[A_{rj},B^{a}_{kl}]&=&(-1)^{a}\delta_{rk}B^{a+1}_{rl}-\delta_{rl}B^{a+1}_{rk}+(-1)^{a+1}\delta_{jk}B^{a+1}_{jl}+\delta_{jl}B^{a+1}_{jk},\\\\[4.79993pt]
[B^{a}_{rj},B^{b}_{kl}]&=&-\delta_{rk}B^{a+b}_{jl}+(-1)^{b}\delta_{rl}B^{a+b}_{jk}+(-1)^{a}\delta_{jk}B^{a+b}_{rl}+(-1)^{a+b+1}\delta_{jl}B^{a+b}_{rk},\end{array}$
(5.7)

for all $a,b\in\\{0,1\\},$ $a\leq b.$

If $n=1,$ the Lie algebra ${\mathfrak{k}}$ of $K$ is one-dimensional, where
${\mathfrak{k}}={\mathbb{R}}A_{12},$ and
${\mathfrak{m}}={\mathbb{R}}\\{B^{0}_{12},B^{1}_{12}\\}.$ If $n\geq 2,$ a
simple root system of ${\mathfrak{k}}$ is given by
$\overline{\pi}=\\{\alpha_{2},\dots,\alpha_{n}\\}$ and so
$\overline{\Delta}^{+}(t)=\\{\alpha_{pq}\colon 2\leq p\leq q\leq n\\}$ is a
positive root system. Hence,

$\begin{array}[]{lcl}{\mathfrak{k}}&=&{\mathbb{R}}A_{12}\oplus(\sum_{j=2}^{n}{\mathbb{R}}A_{j\,j+1}\oplus\sum_{2\leq
j<k\leq
n+1}({\mathbb{R}}B^{0}_{jk}\oplus{\mathbb{R}}B^{1}_{jk}))\cong{\mathbb{R}}\oplus{\mathfrak{s}}{\mathfrak{u}}(n),\\\\[4.79993pt]
{\mathfrak{m}}&=&\sum_{i=1}^{n}({\mathbb{R}}B^{0}_{1\,i+1}\oplus{\mathbb{R}}B^{1}_{1\,i+1}).\end{array}$

Note that the canonical $G$-invariant complex structure on ${\mathbb{C}}{\bf
P}^{n}$ is determined by the ${\rm Ad}(K)$-invariant endomorphism $I={\rm
ad}_{A_{12}\mid{\mathfrak{m}}}$ of ${\mathfrak{m}}.$ We choose as Cartan
subspace to ${\mathfrak{a}}={\mathbb{R}}X,$ where $X=\frac{1}{2}B^{0}_{12}.$
From (5.7), the centralizer ${\mathfrak{h}}$ of ${\mathfrak{a}}$ in
${\mathfrak{k}}$ is trivial if $n=1.$ If $n=2,$
${\mathfrak{h}}={\mathbb{R}}(A_{12}+2A_{23})$ and, if $n\geq 3,$ it is given
by

${\mathfrak{h}}={\mathbb{R}}(A_{12}+2A_{23})\oplus\Big{(}\sum_{i=3}^{n}{\mathbb{R}}A_{i\,i+1}\oplus\sum_{3\leq
j<k\leq
n+1}({\mathbb{R}}B^{0}_{jk}\oplus{\mathbb{R}}B^{1}_{jk})\Big{)}\cong{\mathbb{R}}\oplus{\mathfrak{s}}{\mathfrak{u}}(n-1).$
(5.8)

Moreover, we have

${\rm ad}_{X}^{2}B^{1}_{12}=-B^{1}_{12},\quad{\rm
ad}_{X}^{2}B^{a}_{1\,j+2}=-\frac{1}{4}B^{a}_{1\,j+2},\quad
a=0,1;\;\;j=1,\dots,n-1.$

Hence, ${\mathfrak{m}}_{\varepsilon}={\mathbb{R}}B^{1}_{12}$ and
${\mathfrak{m}}_{\varepsilon/2}=\sum_{j=1}^{n-1}({\mathbb{R}}B^{0}_{1\,j+2}\oplus{\mathbb{R}}B^{1}_{1\,j+2}).$
Now, taking the trace form $\langle U,V\rangle=-2\,{\rm trace}\,UV,$ the set
$\\{X,\mu_{\varepsilon},\mu_{\varepsilon}^{j,a}\;(a=0,1;\,j=1,\dots,n-1)\\}$
is an orthonormal basis of
$({\mathfrak{m}},\langle\cdot,\cdot\rangle_{\mid{\mathfrak{m}}}),$ where

$\mu_{\varepsilon}=\frac{1}{2}B^{1}_{12},\quad\mu^{j,a}_{\varepsilon/2}=\frac{1}{2}B^{a}_{1\,j+2}.$

Applying (4.3) and (5.7), one gets that
${\mathfrak{k}}_{\varepsilon}={\mathbb{R}}\nu_{\varepsilon}$ and
${\mathfrak{k}}_{\varepsilon/2}={\mathbb{R}}\\{\nu^{j,a}_{\varepsilon/2}\;(a=0,1;\,j=1,\dots,n-1)\\},$
where

$\nu_{\varepsilon}=-\frac{1}{2}A_{12},\quad\nu^{j,a}_{\varepsilon/2}=\frac{1}{2}B^{a}_{2\,j+2}.$

Moreover, the nonzero brackets of these basic elements of
$\overline{\mathfrak{m}}$ satisfy the following equalities:

$\begin{array}[]{l}[X,\mu_{\varepsilon}]=-\nu_{\varepsilon},\quad[X,\nu_{\varepsilon}]=\mu_{\varepsilon},\quad[X,\mu^{j,a}_{\varepsilon/2}]=-\frac{1}{2}\nu^{j,a}_{\varepsilon/2},\quad[X,\nu^{j,a}_{\varepsilon/2}]=\frac{1}{2}\mu^{j,a}_{\varepsilon/2},\\\\[6.0pt]
[\mu_{\varepsilon},\mu^{j,a}_{\varepsilon/2}]=[\nu_{\varepsilon},\nu^{j,a}_{\varepsilon/2}]=\frac{(-1)^{a}}{2}\nu^{j,a+1}_{\varepsilon/2},\quad[\mu_{\varepsilon},\nu^{j,a}_{\varepsilon/2}]=-[\nu_{\varepsilon},\mu^{j,a}_{\varepsilon/2}]=\frac{(-1)^{a}}{2}\mu^{j,a+1}_{\varepsilon/2},\\\\[6.0pt]
[\mu^{j,0}_{\varepsilon/2},\nu^{j,0}_{\varepsilon/2}]=[\mu^{j,1}_{\varepsilon/2},\nu^{j,1}_{\varepsilon/2}]=-\frac{1}{2}X,\quad[\mu^{j,0}_{\varepsilon/2},\nu^{j,1}_{\varepsilon/2}]=-[\mu^{j,1}_{\varepsilon/2},\nu^{j,0}_{\varepsilon/2}]=\frac{1}{2}\mu_{\varepsilon},\\\\[6.0pt]
[\mu_{\varepsilon},\nu_{\varepsilon}]=-X,\quad[\mu^{j,0}_{\varepsilon/2},\mu^{j,1}_{\varepsilon/2}]=-\frac{1}{2}\nu_{\varepsilon}+\frac{1}{2}(\frac{1}{2}(A_{12}+2A_{23})+A_{3\,j+2})\in{\mathfrak{k}}_{\varepsilon}\oplus{\mathfrak{h}},\\\\[4.79993pt]
[\nu^{j,0}_{\varepsilon/2},\nu^{j,1}_{\varepsilon/2}]=\frac{1}{2}\nu_{\varepsilon}+\frac{1}{2}(\frac{1}{2}(A_{12}+2A_{23})+A_{3\,j+2})\in{\mathfrak{k}}_{\varepsilon}\oplus{\mathfrak{h}},\\\\[4.79993pt]
[\mu^{j,a}_{\varepsilon/2},\mu^{k,a}_{\varepsilon/2}]=[\nu^{j,a}_{\varepsilon/2},\nu^{k,a}_{\varepsilon/2}]=-\frac{1}{4}B^{0}_{j+2\,k+2}\in{\mathfrak{h}}\quad(j\neq
k),\\\\[6.0pt]
[\mu^{j,0}_{\varepsilon/2},\mu^{k,1}_{\varepsilon/2}]=[\nu^{j,0}_{\varepsilon/2},\nu^{k,1}_{\varepsilon/2}]=-\frac{1}{4}B^{1}_{j+2\,k+2}\in{\mathfrak{h}}\quad(j\neq
k).\end{array}$ (5.9)

######  Lemma 5.10.

We have

${\mathfrak{h}}={\mathbb{R}}\\{[u,v]_{\mathfrak{h}}\colon
u,v\in\overline{\mathfrak{m}}\\}.$

###### Proof.

From (5.9), we have

$\begin{array}[]{lcl}A_{12}+2A_{23}=4[\mu^{1,0}_{\varepsilon/2},\mu^{1,1}_{\varepsilon/2}]_{\mathfrak{h}},&&A_{i\,i+1}=2([\mu^{i-1,0}_{\varepsilon/2},\mu^{i-1,1}_{\varepsilon/2}]_{\mathfrak{h}}-[\mu^{i-2,0}_{\varepsilon/2},\mu^{i-2,1}_{\varepsilon/2}]_{\mathfrak{h}}),\\\\[4.79993pt]
B^{0}_{jk}=-4[\mu^{j-2,0}_{\varepsilon/2},\mu^{k-2,0}_{\varepsilon/2}]_{\mathfrak{h}},&&B^{1}_{jk}=-4[\mu^{j-2,0}_{\varepsilon/2},\mu^{k-2,1}_{\varepsilon/2}]_{\mathfrak{h}},\end{array}$

for all $i=3,\dots,n$ and $3\leq j<k\leq n+1.$ Then, this lemma follows from
(5.8). ∎

Finally, applying (5.7) and (5.8), we determine the mapping ${\rm
ad}_{\mathfrak{h}}\colon\overline{\mathfrak{m}}\to\overline{\mathfrak{m}}.$

######  Lemma 5.11.

We have

$\begin{array}[]{l}[{\mathfrak{h}},\overline{\mathfrak{n}}]=0,\quad[A_{12}+2A_{23},\mu^{j,a}_{\varepsilon/2}]=(-1)^{a}(\mu^{j,a+1}_{\varepsilon/2}+2\delta_{1j}\mu^{1,a+1}_{\varepsilon/2}),\\\\[4.79993pt]
[A_{12}+2A_{23},\nu^{j,a}_{\varepsilon/2}]=(-1)^{a}(\nu^{j,a+1}_{\varepsilon/2}+2\delta_{1j}\nu^{1,a+1}_{\varepsilon/2}),\\\\[4.79993pt]
[A_{i\,i+1},\mu^{j,a}_{\varepsilon/2}]=(-1)^{a}(\delta_{i\,j+1}\mu^{i-1,a+1}_{\varepsilon/2}-\delta_{i\,j+2}\mu^{i-2,a+1}_{\varepsilon/2}),\\\\[4.79993pt]
[A_{i\,i+1},\nu^{j,a}_{\varepsilon/2}]=(-1)^{a}(\delta_{i\,j+1}\nu^{i-1,a+1}_{\varepsilon/2}-\delta_{i\,j+2}\nu^{i-2,a+1}_{\varepsilon/2}),\\\\[4.79993pt]
[B^{a}_{pq},\mu^{j,b}_{\varepsilon/2}]=(-1)^{a(b+1)}(\delta_{q\,j+2}\mu^{p-2,a+b}_{\varepsilon/2}+(-1)^{a+1}\delta_{p\,j+2}\mu^{q-2,a}_{\varepsilon/2}),\\\\[4.79993pt]
[B^{a}_{pq},\nu^{j,b}_{\varepsilon/2}]=(-1)^{a(b+1)}(\delta_{q\,j+2}\nu^{p-2,a}_{\varepsilon/2}+(-1)^{a+1}\delta_{p\,j+2}\nu^{q-2,a}_{\varepsilon/2}),\end{array}$

for all $j=1,\dots,n-1,$ $i=3,\dots,n$ and $3\leq p<q\leq n+1.$

Proof of Theorem 5.3. From (5.2), we get

$\begin{array}[]{l}a_{0}=\rho^{2},\quad
a_{\varepsilon}=\frac{1}{a_{22}^{2}}(1+a_{21}^{2}\rho^{2}),\quad
b_{\varepsilon}=\frac{1}{a_{22}^{2}a_{33}^{2}}(\rho^{2}A_{13}^{2}+a_{22}^{2}+a_{32}^{2}),\\\\[4.79993pt]
a_{0\varepsilon}=-\frac{\rho^{2}a_{21}}{a_{22}},\quad
b_{0\varepsilon}=\frac{\rho^{2}}{a_{22}a_{33}}A_{13},\quad
c_{\varepsilon}=-\frac{\rho^{2}}{a_{22}^{2}a_{33}}(a_{21}A_{13}+\frac{a_{32}}{\rho^{2}}),\end{array}$
(5.10)

where $A_{13}=a_{21}a_{32}-a_{31}a_{22}.$ Here $A_{ij}$ denotes the adjoint of
the element $a_{ij}$ of $A.$ Put
$\xi_{o_{H}}=\sum_{j=1}^{3}\lambda_{j}\xi_{j},$ where
$\sum_{j=1}^{3}\lambda_{j}^{2}=1.$ Then the $G$-invariant one-form $\eta:={\bf
g}(\xi,\cdot)$ is given at the origin by
$\eta=\langle\kappa_{1}X+\kappa_{2}\mu_{\varepsilon}+\kappa_{3}\nu_{\varepsilon},\cdot\rangle,$
where

$\kappa_{1}=\rho\lambda_{1},\quad\kappa_{2}=\frac{1}{a_{22}}(\lambda_{2}-\rho
a_{21}\lambda_{1}),\quad\kappa_{3}=\frac{1}{a_{22}a_{33}}(\rho
A_{13}\lambda_{1}-a_{32}\lambda_{2}+a_{22}\lambda_{3}).$ (5.11)

Because
$d\eta(u,v)=-\frac{1}{2}\eta([u^{\tau_{H}},v^{\tau_{H}}])=-\frac{1}{2}\eta([u,v]_{\overline{\mathfrak{m}}}),$
for all $u,v\in\overline{\mathfrak{m}},$ where $u^{\tau_{H}}$ and
$v^{\tau_{H}}$ denote the local vector fields on $G/H$ around $o_{H},$ defined
in (3.1), with $u^{\tau_{H}}_{o_{H}}=u$ and $v^{\tau_{H}}_{o_{H}}=v,$ it
follows, using (4.3) and (5.9), that

$\begin{array}[]{lcl}d\eta(X,\mu_{\varepsilon})=\frac{\kappa_{3}}{2},\quad
d\eta(X,\nu_{\varepsilon})=-\frac{\kappa_{2}}{2},&&d\eta(\mu_{\varepsilon},\nu_{\varepsilon})=\frac{\kappa_{1}}{2},\\\\[4.79993pt]
d\eta(\mu^{j,0}_{\varepsilon/2},\mu^{j,1}_{\varepsilon/2})=-d\eta(\nu^{j,0}_{\varepsilon/2},\nu^{j,1}_{\varepsilon/2})=\frac{\kappa_{3}}{4},&&d\eta(\mu^{j,0}_{\varepsilon/2},\nu^{j,0}_{\varepsilon/2})=d\eta(\mu^{j,1}_{\varepsilon/2},\nu^{j,1}_{\varepsilon/2})=\frac{\kappa_{1}}{4},\\\\[4.79993pt]
d\eta(\mu^{j,0}_{\varepsilon/2},\nu^{j,1}_{\varepsilon/2})=-d\eta(\mu^{j,1}_{\varepsilon/2},\nu^{j,0}_{\varepsilon/2})=-\frac{\kappa_{2}}{4},&&\end{array}$

the rest of components of $d\eta$ at $o_{H}$ being zero. From here, the
$G$-invariant $(1,1)$-tensor field $\varphi$ such that $d\eta={\bf
g}(\cdot,\varphi\cdot)$ is given by

$\varphi\xi_{i}=\frac{1}{2}\sum_{l=1}^{3}\kappa_{l}\sum_{j,k=1}^{3}\epsilon_{i,j,k}A_{jl}\xi_{k},\quad
i\in\\{1,2,3\\}.$ (5.12)

Putting $\varkappa_{1}=a_{33}\kappa_{2}-a_{32}\kappa_{3}$ and
$\varkappa_{2}=a_{33}(a_{22}\kappa_{1}-a_{21}\kappa_{2})+\kappa_{3}A_{13},$ we
obtain

$\begin{array}[]{l}\varphi\xi_{1}=\frac{1}{2\rho}(-a_{22}\kappa_{3}\xi_{2}+\varkappa_{1}\xi_{3}),\quad\varphi\xi_{2}=\frac{1}{2\rho}({\kappa_{3}}a_{22}\xi_{1}-\rho\varkappa_{2}\xi_{3}),\quad\varphi\xi_{3}=\frac{1}{2\rho}(-\varkappa_{1}\xi_{1}+\rho\varkappa_{2}\xi_{2}),\\\\[4.79993pt]
\varphi\mu^{j,0}_{\varepsilon/2}=-\frac{\kappa_{3}}{4a_{\varepsilon/2}}\mu^{j,1}_{\varepsilon/2}-\frac{\kappa_{1}}{4b_{\varepsilon/2}}\nu^{j,0}_{\varepsilon/2}+\frac{\kappa_{2}}{4b_{\varepsilon/2}}\nu^{j,1}_{\varepsilon/2},\quad\varphi\mu^{j,1}_{\varepsilon/2}=\frac{\kappa_{3}}{4a_{\varepsilon/2}}\mu^{j,0}_{\varepsilon/2}-\frac{\kappa_{2}}{4b_{\varepsilon/2}}\nu^{j,0}_{\varepsilon/2}-\frac{\kappa_{1}}{4b_{\varepsilon/2}}\nu^{j,1}_{\varepsilon/2},\\\\[4.79993pt]
\varphi\nu^{j,0}_{\varepsilon/2}=\frac{\kappa_{1}}{4a_{\varepsilon/2}}\mu^{j,0}_{\varepsilon/2}+\frac{\kappa_{2}}{4a_{\varepsilon/2}}\mu^{j,1}_{\varepsilon/2}+\frac{\kappa_{3}}{4b_{\varepsilon/2}}\nu^{j,1}_{\varepsilon/2},\quad\varphi\nu^{j,1}_{\varepsilon/2}=-\frac{\kappa_{2}}{4a_{\varepsilon/2}}\mu^{j,0}_{\varepsilon/2}+\frac{\kappa_{1}}{4a_{\varepsilon/2}}\mu^{j,1}_{\varepsilon/2}-\frac{\kappa_{3}}{4b_{\varepsilon/2}}\nu^{j,0}_{\varepsilon/2}.\end{array}$

Hence, using a straightforward calculation, $\varphi$ satisfies
$\varphi^{2}=-I+\eta\otimes\xi,$ or equivalently ${\bf g}$ is compatible with
the structure $(\varphi,\xi,\eta),$ and so $(\xi,{\bf g})$ is a contact metric
structure, if and only if the following conditions hold:

$\left\\{\begin{array}[]{l}a_{22}^{2}\kappa_{3}^{2}+\varkappa_{1}^{2}=4\rho^{2}(1-\lambda_{1}^{2}),\quad
a_{22}^{2}\kappa_{3}^{2}+\rho^{2}\varkappa_{2}^{2}=4\rho^{2}(1-\lambda_{2}^{2}),\\\\[4.79993pt]
\varkappa_{1}^{2}+\rho^{2}\varkappa_{2}^{2}=4\rho^{2}(1-\lambda_{3}^{2});\\\\[4.79993pt]
\varkappa_{1}\varkappa_{2}=4\rho\lambda_{1}\lambda_{2},\quad
a_{22}\kappa_{3}\varkappa_{2}=4\rho\lambda_{1}\lambda_{3},\quad
a_{22}\kappa_{3}\varkappa_{1}=4\rho^{2}\lambda_{2}\lambda_{3};\\\\[4.79993pt]
a_{\varepsilon/2}(\kappa_{1}^{2}+\kappa^{2}_{2})+b_{\varepsilon/2}\kappa^{2}_{3}=16a_{\varepsilon/2}^{2}b_{\varepsilon/2},\quad
b_{\varepsilon/2}(\kappa_{1}^{2}+\kappa_{2}^{2})+a_{\varepsilon/2}\kappa^{2}_{3}=16a_{\varepsilon/2}b_{\varepsilon/2}^{2},\\\\[4.79993pt]
\kappa_{1}\kappa_{3}(a_{\varepsilon/2}-b_{\varepsilon/2})=\kappa_{2}\kappa_{3}(a_{\varepsilon/2}-b_{\varepsilon/2})=0.\end{array}\right.$
(5.13)

Moreover, because $\varphi\xi=0,$ we also have

$a_{22}\lambda_{2}\kappa_{3}=\lambda_{3}\varkappa_{1},\quad\rho\lambda_{3}\varkappa_{2}=a_{22}\lambda_{1}\kappa_{3},\quad\lambda_{1}\varkappa_{1}=\rho\lambda_{2}\varkappa_{2}.$
(5.14)

Type AI: Using (5.11) and (5.14), it follows that $\kappa_{3}=0$ and
$\varkappa_{1}=0.$ Hence, $\kappa_{2}=0.$ Then we have that $a_{21}=A_{13}=0$
and also $a_{31}=0.$ Now, (5.13) hold if and only if

$a_{22}a_{33}=\frac{2}{\rho},\quad\rho^{2}=16a_{\varepsilon/2}b_{\varepsilon/2}.$

Then, putting $\kappa=\rho,$ $q_{\varepsilon}=\frac{a_{22}^{2}\kappa}{2},$
$\alpha=-\frac{a_{32}\kappa}{2a_{22}}$ and
$q_{\varepsilon/2}=\frac{\kappa}{4a_{\varepsilon/2}},$ and using (5.10) the
theorem is proved for this case.

Type AII: From (5.11) and (5.13), it follows that $\kappa_{1}=\kappa_{3}=0$
and then $a_{32}=a_{21}=\varkappa_{2}=0.$ Hence, the equations in (5.13)
reduce to

$a_{33}=2\rho
a_{22},\quad\frac{1}{a_{22}^{2}}=16a_{\varepsilon/2}b_{\varepsilon/2}.$

Then the result for this case is obtained from (5.10), putting
$\kappa=\frac{1}{a_{22}},$ $q_{\varepsilon}=\frac{1}{2a_{22}\rho^{2}},$
$\alpha=\frac{\rho}{2a_{22}^{2}}A_{13}$ and
$q_{\varepsilon/2}=\frac{1}{4a_{22}a_{\varepsilon/2}}.$

Type AIII: From (5.11), we get $\kappa_{1}=\kappa_{2}=0$ and
$\kappa_{3}=\pm\frac{1}{a_{33}},$ and from (5.14), it follows that
$\varkappa_{1}=\varkappa_{2}=0.$ Then, $A_{13}=a_{32}=0$ and also $a_{31}=0.$
Hence, (5.13) holds if and only if

$a_{22}=2\rho a_{33},\quad
a_{\varepsilon/2}=b_{\varepsilon/2}=\frac{1}{4a_{33}}.$

Now, putting $\kappa=\frac{1}{a_{33}},$
$q_{\varepsilon}=\frac{1}{2\rho^{2}a_{33}}$ and
$\alpha=-\frac{\rho^{2}a_{21}}{a_{22}}$ and using (5.10) we get the theorem
for this case.

Type BI: From (5.11) and (5.14) it follows that $\kappa_{3}=0$ and
$\lambda_{1}\varkappa_{1}=\rho\lambda_{2}\varkappa_{2}.$ Then (5.13) holds if
moreover $\varkappa_{2}^{2}=4\lambda_{1}^{2}$ and
$\kappa_{1}^{2}+\kappa_{2}^{2}=16a_{\varepsilon/2}b_{\varepsilon/2}.$ Then,
putting $\kappa=\frac{1}{a_{22}},$ $\alpha=A_{13}$ and
$q_{\varepsilon}=\frac{1}{\lambda_{2}}(\lambda_{2}-\rho a_{21}\lambda_{1}),$
we have

$a_{31}=-\kappa\alpha q_{\varepsilon},\quad
a_{21}=\frac{\lambda_{2}(1-q_{\varepsilon})}{\rho\lambda_{1}}.$ (5.15)

On the other hand, the condition
$\lambda_{1}\varkappa_{1}=\rho\lambda_{2}\varkappa_{2}$ is equivalent to
$\rho^{2}\lambda_{1}^{2}=\kappa^{2}q_{\varepsilon}(1-q_{\varepsilon}\lambda_{2}^{2}).$
Hence, using (5.11),
$q_{\varepsilon}=\frac{\kappa_{1}^{2}+\kappa_{2}^{2}}{\kappa^{2}}$ and
consequently,
$a_{\varepsilon/2}b_{\varepsilon/2}=\frac{\kappa^{2}q_{\varepsilon}}{16}.$
Moreover, we have that $q_{\varepsilon}<\frac{1}{\lambda^{2}}.$ We put

$a_{\varepsilon/2}=\frac{\kappa\sqrt{q_{\varepsilon}}}{4q_{\varepsilon/2}},\quad
b_{\varepsilon/2}=\frac{\kappa\sqrt{q_{\varepsilon}}q_{\varepsilon/2}}{4},$
(5.16)

where $q_{\varepsilon/2}$ is a positive constant. Finally, the condition
$\varkappa_{2}^{2}=4\lambda_{1}^{2}$ is equivalent to
$(\rho\lambda_{1}-a_{21}\kappa^{2}\lambda_{2}q_{\varepsilon})^{2}a_{33}^{2}=4\kappa^{2}\lambda_{1}^{2}.$
Then, putting $\lambda_{1}=\cos\theta$ and $\lambda_{2}=\sin\theta$ and
$\varrho=\sqrt{q_{\varepsilon}(1-q_{\varepsilon}\sin^{2}\theta)},$ we have

$\rho=\frac{\kappa\varrho}{|\cos\theta|},\quad
a_{33}=\frac{2\varrho}{q_{\varepsilon}|\cos\theta|}.$ (5.17)

Now, from (5.10), using (5.15), (5.16) and (5.17), the result is proved for
this case.

Type BII: Put $\kappa=\frac{1}{a_{33}}$ and $\beta=\frac{1}{\lambda_{3}}(\rho
A_{13}\lambda_{1}+a_{22}\lambda_{3}).$ Then, from (5.11), we get

$\kappa_{3}=\frac{\kappa\beta}{a_{22}}\lambda_{3},\quad A_{13}=\frac{(\beta-
a_{22})\lambda_{3}}{\rho\lambda_{1}},$ (5.18)

and, using (5.14), we have $\varkappa_{1}=0$ and
$\rho\lambda_{3}\varkappa_{2}=a_{22}\lambda_{1}\kappa_{3}.$ Here, the
condition $\varkappa_{1}=0$ is equivalent to
${\rho\lambda_{1}}a_{21}=-a_{32}\kappa^{2}\beta\lambda_{3}.$ Then (5.13)
reduces to

$\left\\{\begin{array}[]{l}\rho\lambda_{3}\varkappa_{2}=a_{22}\lambda_{1}\kappa_{3},\\\\[4.79993pt]
a_{22}\kappa_{3}\varkappa_{2}=4\rho\lambda_{1}\lambda_{3},\\\\[4.79993pt]
a_{\varepsilon/2}^{2}=b_{\varepsilon/2}^{2}=\frac{\kappa^{2}_{1}+\kappa_{2}^{2}+\kappa^{2}_{3}}{16}.\end{array}\right.$
(5.19)

From (5.18), the first two equations in (5.19) can be expressed as
$\rho\varkappa_{2}=\kappa\beta\lambda_{1}$ and
$\kappa\beta\varkappa_{2}=4\rho\lambda_{1}.$ Then
$\beta=\frac{2\rho}{\kappa}>0$ and $\varkappa_{2}=2\lambda_{1}.$ Moreover,
putting $q_{\varepsilon}=a_{22}$ and $\alpha=a_{32},$ we get

$A_{13}=\frac{(2\rho-\kappa
q_{\varepsilon})\lambda_{3}}{\kappa\rho\lambda_{1}},\quad
a_{21}=-\frac{2\alpha\kappa\lambda_{3}}{\lambda_{1}},\quad\varkappa_{2}=\frac{(\rho(q_{\varepsilon}^{2}\lambda_{1}+4(\alpha^{2}\kappa^{2}+1)\lambda_{3}^{2})-2\kappa
q_{\varepsilon}\lambda_{3}^{2})}{\kappa q_{\varepsilon}\lambda_{1}}.$

Now, putting $\lambda_{1}=\cos\theta,$ $\lambda_{2}=\sin\theta$ and
$\varrho=q_{\varepsilon}^{2}\cos^{2}\theta+4(\alpha^{2}\kappa^{2}+1)\sin^{2}\theta,$
we have that $\rho=\frac{2\kappa q_{\varepsilon}}{\varrho}$ and, moreover,

$A_{13}=\frac{4-\varrho}{2\kappa}\tan\theta,\quad
a_{21}=-2\alpha\kappa\tan\theta,\quad
a_{31}=\frac{q_{\varepsilon}^{2}-4(1+\alpha^{2}\kappa^{2})}{2\kappa
q_{\varepsilon}}\sin\theta\cos\theta.$

Then the result for this case follows applying again (5.10).

Type BIII: From (5.11) and (5.14), we get $\kappa_{1}=\varkappa_{2}=0.$
Moreover, putting $\kappa=\frac{1}{a_{33}}$ and
$\beta=\frac{1}{\lambda_{3}}(a_{22}\lambda_{3}-a_{32}\lambda_{2}),$ it follows
that $a_{21}=\frac{\kappa^{2}A_{13}\beta\lambda_{3}}{\lambda_{2}}.$ On the
other hand, the equations in (5.13) reduces to

$\left\\{\begin{array}[]{l}a_{22}^{2}\kappa_{3}^{2}+\varkappa_{1}^{2}=4\rho^{2},\\\\[4.79993pt]
a_{22}^{2}\kappa_{3}^{2}=4\rho^{2}\lambda_{3}^{2},\\\\[4.79993pt]
a_{22}\kappa_{3}\varkappa_{1}=4\rho^{2}\lambda_{2}\lambda_{3},\\\\[4.79993pt]
a_{\varepsilon/2}^{2}=b_{\varepsilon/2}^{2}=\frac{\kappa_{2}^{2}+\kappa_{3}^{2}}{16}.\end{array}\right.$
(5.20)

Taking into account that $\kappa_{3}=\frac{\kappa\beta}{a_{22}}\lambda_{3}$
and $\varkappa_{1}=\frac{1}{\kappa
a_{22}}(\lambda_{2}-a_{32}\kappa^{2}\beta\lambda_{3}),$ we get from the second
equation in (5.20) that $\kappa^{2}\beta^{2}=4\rho^{2}.$ Then, from the third
equation we have $\varkappa_{1}=\kappa\beta\lambda_{2}$ and the first equation
holds. Hence, we obtain that
$a_{22}=\frac{1}{\kappa^{2}\beta}(\lambda_{2}^{2}+4\rho^{2}\lambda_{3}^{2})$
and it implies that $\beta>0,$ that is, $\beta=\frac{2\rho}{\kappa}.$ Now,
putting $\lambda_{2}=\cos\theta,$ $\lambda_{3}=\sin\theta,$
$q_{\varepsilon}=2\rho$ and $\alpha=\frac{2\kappa A_{13}\rho}{\varrho},$ where
$\varrho=\cos^{2}\theta+4\rho^{2}\sin^{2}\theta,$ we have

$\begin{array}[]{l}a_{21}=\frac{2\alpha\varrho}{q_{\varepsilon}}\tan\theta,\quad
a_{22}=\frac{\varrho}{\kappa q_{\varepsilon}},\quad
a_{31}=-\frac{2\alpha\varrho}{q_{\varepsilon}},\\\\[4.79993pt]
a_{32}=\frac{1-q_{\varepsilon}^{2}}{\kappa
q_{\varepsilon}}\sin\theta\cos\theta,\quad
a_{\varepsilon/2}=b_{\varepsilon/2}=\frac{\kappa
q_{\varepsilon}\sqrt{\varrho}}{4\varrho}.\end{array}$

Then, from (5.10), we get the result for this case.

Type C: Using (5.14), we get

$\varkappa_{1}=\frac{a_{22}\kappa_{3}\lambda_{2}}{\lambda_{3}},\quad\varkappa_{2}=\frac{a_{22}\kappa_{3}\lambda_{1}}{\rho\lambda_{3}}.$
(5.21)

Then, from the first equation in (5.13), we have
$a_{22}^{2}\kappa^{2}_{3}=4\rho^{2}\lambda_{3}^{2}.$ Hence, one gets

$\varkappa_{1}^{2}=4\rho^{2}\lambda_{2}^{2},\quad\varkappa_{2}^{2}=4\lambda_{1}^{2}$

and the equations in (5.13) hold if, moreover,
$a_{\varepsilon/2}^{2}=b^{2}_{\varepsilon/2}=\frac{\kappa^{2}_{1}+\kappa^{2}_{2}+\kappa^{2}_{3}}{16}.$
Put $\kappa=\frac{1}{a_{33}},$ $\alpha=a_{21}$ and
$\beta=\frac{1}{\lambda_{3}}(\rho
A_{13}\lambda_{1}-a_{32}\lambda_{2}+a_{22}\lambda_{3}).$ Then
$\kappa_{3}=\frac{\kappa\beta\lambda_{3}}{a_{22}}$ and the equations in (5.21)
are written as $\varkappa_{1}=\kappa\beta\lambda_{2}$ and
$\rho\varkappa_{2}=\kappa\beta\lambda_{1}.$ Hence, we have

$a_{32}=\frac{(1-\kappa^{2}\beta
a_{22})\lambda_{2}-\rho\alpha\lambda_{1}}{\kappa^{2}\beta\lambda_{3}},\quad
A_{13}=\frac{\kappa^{2}\beta
a_{22}\lambda_{1}+\rho(\alpha(\lambda_{2}-\rho\alpha\lambda_{1})-a_{22}^{2}\rho\lambda_{1})}{\kappa^{2}\beta\rho\lambda_{3}}.$

Moreover, putting $q_{\varepsilon}=\rho,$ we obtain
$\kappa^{2}\beta^{2}=4q_{\varepsilon}^{2}.$ Since
$a_{22}\lambda_{3}=\beta\lambda_{3}+a_{32}\lambda_{2}-q_{\varepsilon}A_{13}\lambda_{1},$
it follows that $a_{22}$ is a solution of the second order equation
$q_{\varepsilon}^{2}\lambda_{1}^{2}a_{22}^{2}-\kappa^{2}\beta
a_{22}+\varrho=0,$ where
$\varrho=4q_{\varepsilon}^{2}\lambda_{3}^{2}+\lambda_{2}^{2}+\alpha
q_{\varepsilon}\lambda_{1}(\alpha q_{\varepsilon}\lambda_{1}-2\lambda_{2}).$
Hence, it must happen that $\varrho\lambda_{1}^{2}\leq\kappa^{2}$ and
$\beta>0.$ Therefore, putting $\lambda_{1}=\cos\theta\cos\phi,$
$\lambda_{2}=\sin\theta\cos\phi$ and $\lambda_{3}=\sin\phi,$ we conclude that,
for each pair of positive constants $\kappa$ and $q_{\varepsilon}$ and
$\alpha\in{\mathbb{R}}$ such that
$\varrho\cos^{2}\theta\cos^{2}\phi\leq\kappa^{2},$ there exists a contact
metric structure $(\xi,{\bf g})$ of Type C and the result follows, putting
$\gamma_{1}=a_{32}$ and $\gamma_{2}=A_{31},$ using (5.10).

Proof of Theorem 5.5. From Theorem 5.3, $(\xi,{\bf g})$ is orthogonal, for
each Type AI-AIII, BI-BIII and C, if and only if the following conditions
hold:

  1. $\bullet$

Type A (AI-III): $\alpha=0,$

  2. $\bullet$

Type BI: $q_{\varepsilon}=1$ and $\alpha=0.$ Then, $\varrho=|\cos\theta|,$

  3. $\bullet$

Type BII: $q_{\varepsilon}=2$ and $\alpha=0.$ Then, $\varrho=4,$

  4. $\bullet$

Type BIII: $q_{\varepsilon}=1$ and $\alpha=0.$ Then, $\varrho=1,$

  5. $\bullet$

Type C: $\gamma_{1}=\gamma_{2}=0$ and $\alpha=0.$ Then,
$\kappa=q_{\varepsilon}=\frac{1}{2},$ $\delta=2$ and
$\varrho=1-\cos^{2}\theta\cos^{2}\phi.$

From here, we obtain the coefficients $a_{0},$ $a_{\varepsilon},$
$b_{\varepsilon},$ $a_{\varepsilon/2}$ and $b_{\varepsilon/2}$ of ${\bf g}$,
given in Table II.

Next, let
${\mathfrak{U}}\colon\overline{\mathfrak{m}}\times\overline{\mathfrak{m}}\to\overline{\mathfrak{m}}$
be the symmetric bilinear function defined in (3.5) for the Euclidean space
$(\overline{\mathfrak{m}},{\bf g}_{o_{H}}),$ where ${\bf g}$ is an orthogonal
metric on $T_{r}{\mathbb{C}}{\bf P}^{n}.$ Using (4.3), (4.8) and (5.9),
together with Lemma 4.2, we have

$\begin{array}[]{l}{\mathfrak{U}}(X,\mu_{\varepsilon})=\frac{a_{0}-a_{\varepsilon}}{2b_{\varepsilon}}\nu_{\varepsilon},\quad{\mathfrak{U}}(X,\nu_{\varepsilon})=\frac{b_{\varepsilon}-a_{0}}{2a_{\varepsilon}}\mu_{\varepsilon},\quad{\mathfrak{U}}(X,\mu^{j,a}_{\varepsilon/2})=\frac{a_{0}-a_{\varepsilon/2}}{4b_{\varepsilon/2}}\nu^{j,a}_{\varepsilon/2},\\\\[4.79993pt]
{\mathfrak{U}}(X,\nu^{j,a}_{\varepsilon/2})=\frac{b_{\varepsilon/2}-a_{0}}{4a_{\varepsilon/2}}\mu^{j,a}_{\varepsilon/2},\quad{\mathfrak{U}}(\mu_{\varepsilon},\nu_{\varepsilon})=\frac{a_{\varepsilon}-b_{\varepsilon}}{2a_{0}}X,\quad{\mathfrak{U}}(\mu_{\varepsilon},\mu^{j,a}_{\varepsilon/2})=\frac{(-1)^{a}(a_{\varepsilon/2}-a_{\varepsilon})}{4b_{\varepsilon/2}}\nu^{j,a+1}_{\varepsilon/2},\\\\[4.79993pt]
{\mathfrak{U}}(\mu_{\varepsilon},\nu^{j,a}_{\varepsilon/2})=\frac{(-1)^{a}(b_{\varepsilon/2}-a_{\varepsilon})}{4a_{\varepsilon/2}}\mu^{j,a+1}_{\varepsilon/2},\quad{\mathfrak{U}}(\nu_{\varepsilon},\mu^{j,a}_{\varepsilon/2})=\frac{(-1)^{a}(b_{\varepsilon}-a_{\varepsilon/2})}{4a_{\varepsilon/2}}\mu^{j,a+1}_{\varepsilon/2},\\\\[4.79993pt]
{\mathfrak{U}}(\nu_{\varepsilon},\nu^{j,a}_{\varepsilon/2})=\frac{(-1)^{a}(b_{\varepsilon/2}-b_{\varepsilon})}{4b_{\varepsilon/2}}\nu^{j,a+1}_{\varepsilon/2},\quad{\mathfrak{U}}(\mu^{j,a}_{\varepsilon/2},\nu^{j,a}_{\varepsilon/2})=\frac{(a_{\varepsilon/2}-b_{\varepsilon/2})(a_{0}+a_{\varepsilon})}{4a_{0}a_{\varepsilon}}X,\\\\[4.79993pt]
{\mathfrak{U}}(\mu^{j,0}_{\varepsilon/2},\nu^{j,1}_{\varepsilon/2})=-{\mathfrak{U}}(\mu^{j,1}_{\varepsilon/2},\nu^{j,0}_{\varepsilon/2})=-\frac{(a_{\varepsilon/2}-b_{\varepsilon/2})}{4a_{\varepsilon}}\mu_{\varepsilon},\end{array}$
(5.22)

for all $j=1,\dots,n-1;$ $a=0,1$ and the rest of its components being zero.
Therefore, applying (3.6), we obtain in Table II the set of all $G$-invariant
orthogonal metrics ${\bf g}$ for which $(\xi,{\bf g})$ is $K$-contact.

Finally we prove that the contact metric structures of Type AI and Type AII
are isomorphic for each triple of positive constants
$(\kappa,q_{\varepsilon},q_{\varepsilon/2}).$

Denote by ${\bf g}^{\kappa,q_{\varepsilon},q_{\varepsilon/2}}_{1}$ and ${\bf
g}^{\kappa,q_{\varepsilon},q_{\varepsilon/2}}_{2}$ the $G$-invariant contact
metrics on $T_{r}{\mathbb{C}}{\bf P}^{n}=G/H$ associated to the structures of
Type AI y AII, respectivamente, and consider the linear isometry
$L\colon(\overline{\mathfrak{m}},{\bf
g}^{\kappa,q_{\varepsilon},q_{\varepsilon/2}}_{1o_{H}})\to(\overline{\mathfrak{m}},{\bf
g}^{\kappa,q_{\varepsilon},q_{\varepsilon/2}}_{2o_{H}}),$ given by

$\begin{array}[]{l}LX=\mu_{\varepsilon},\quad L\mu_{\varepsilon}=-X,\quad
L\nu_{\varepsilon}=\nu_{\varepsilon},\quad
L\mu^{j,a}_{\varepsilon/2}=(-1)^{a}\mu^{j,a+1}_{\varepsilon/2},\quad
L\nu^{j,a}_{\varepsilon/2}=\nu^{j,a}_{\varepsilon/2},\end{array}$ (5.23)

for all $j=1,\dots,n-1$ and $a=0,1.$ Let us check that $L$ satisfies the
conditions (i), (ii) and (iii) in Lemma 3.1. By a straightforward calculation,
using (5.9), $L$ verifies (i). For (ii), note that the non-zero
${\mathfrak{h}}$-components of the brackets in (5.9) are of the form
$[\mu^{j,a}_{\varepsilon/2},\mu^{k,b}_{\varepsilon/2}]_{\mathfrak{h}},$ for
$j,k=1,\dots,n-1$ and $a,b=0,1,$ and we get

$[L\mu^{j,a}_{\varepsilon/2},L\mu^{k,b}_{\varepsilon/2}]_{\mathfrak{h}}=[\mu^{j,a}_{\varepsilon/2},\mu^{k,b}_{\varepsilon/2}]_{\mathfrak{h}}.$

Then, from Lemma 5.10, the condition (ii) is equivalent to

$L\circ{\rm ad}_{u}={\rm ad}_{u}\circ L,\quad u\in{\mathfrak{h}},$ (5.24)

which is satisfied using Lemma 5.11. From (5.23), (iii) is verified. Finally,
because ${\rm SU}(n+1)$ is simply connected and ${\rm S}({\rm U}(1)\times{\rm
U}(n-1))$ is connected, diffeomorphic to ${\rm SU}(1)\times{\rm
SU}(n-1)\times{\mathbb{S}}^{1},$ it follows that $T_{r}({\mathbb{C}}{\bf
P}^{n})$ is also simply connected.

##  6\. Invariant Sasakian and 3-Sasakian structures

From Theorem 5.5, there are three families of orthogonal $K$-contact
structures $(\xi^{\kappa}_{i},{\bf g}^{\kappa}_{i}),$ $i=1,2,3,$ on
$T_{r}{\mathbb{C}}{\bf P}^{n},$ depending of a parameter $\kappa>0,$ and
determined at $o_{H}$ by

$\begin{array}[]{lclclcl}\xi^{\kappa}_{1}=\pm\frac{1}{\kappa}X,&&{\bf
g}^{\kappa}_{1}&=&\kappa^{2}\langle\cdot,\cdot\rangle_{\mathfrak{a}}+\frac{\kappa}{2}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}}+\frac{\kappa}{4}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon/2}\oplus{\mathfrak{k}}_{\varepsilon/2}},\\\\[4.79993pt]
\xi^{\kappa}_{2}=\pm\frac{1}{\kappa}\mu_{\varepsilon},&&{\bf
g}^{\kappa}_{2}&=&\kappa^{2}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon}}+\frac{\kappa}{2}\langle\cdot,\cdot\rangle_{{\mathfrak{a}}\oplus{\mathfrak{k}}_{\varepsilon}}+\frac{\kappa}{4}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon/2}\oplus{\mathfrak{k}}_{\varepsilon/2}},\\\\[4.79993pt]
\xi^{\kappa}_{3}=\pm\frac{1}{\kappa}\nu_{\varepsilon},&&{\bf
g}^{\kappa}_{3}&=&\kappa^{2}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon}}+\frac{\kappa}{2}\langle\cdot,\cdot\rangle_{{\mathfrak{a}}\oplus{\mathfrak{m}}_{\varepsilon}}+\frac{\kappa}{4}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon/2}\oplus{\mathfrak{k}}_{\varepsilon/2}}.\end{array}$

They are, respectively, of Type AI, AII and AIII. Since
$(\xi^{\kappa}_{1},{\bf g}^{\kappa}_{1})$ is Sasakian [12, Theorem 1.1], it
follows from the last part of Theorem 5.5 that $(\xi^{\kappa}_{2},{\bf
g}^{\kappa}_{2})$ is also Sasakian. However, as can be seen in the following
proposition, the procedure used in the proof of Theorem 5.5 is not applicable
when the contact metric structure $(\xi^{\kappa}_{3},{\bf g}^{\kappa}_{3})$ is
considered.

######  Proposition 6.1.

There is no isomorphism $L$ between the canonical infinitesimal models
$(\overline{\mathfrak{m}},\widetilde{T}_{o_{H}},\widetilde{R}_{o_{H}},{\bf
g}^{\kappa}_{1\,o_{H}})$ and
$(\overline{\mathfrak{m}},\widetilde{T}_{o_{H}},\widetilde{R}_{o_{H}},{\bf
g}^{\kappa}_{3\,o_{H}})$ of $T_{r}{\mathbb{C}}{\bf P}^{n}=G/H,$ for $n\geq 2,$
such that $L{\mathfrak{a}}={\mathfrak{k}}_{\varepsilon}.$

###### Proof.

From the condition (5.24) and taking into account that $H=S(U(1)\times
U(n-1))$ is connected, it follows that any isomorphism $L$ of infinitesimal
models on $T_{r}{\mathbb{C}}{\bf P}^{n}$ is ${\rm Ad}(H)$-invariant. Hence,
from the irreducibility of the decomposition
$\overline{\mathfrak{m}}={\mathfrak{a}}\oplus{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}\oplus{\mathfrak{m}}_{\varepsilon/2}\oplus{\mathfrak{k}}_{\varepsilon/2},$
either $L{\mathfrak{m}}_{\varepsilon/2}={\mathfrak{m}}_{\varepsilon/2}$ or
$L{\mathfrak{m}}_{\varepsilon/2}={\mathfrak{k}}_{\varepsilon/2}.$ Proceeding
by reductio ad absurdum, suppose that there exists an isomorphism $L$ such
that $L{\mathfrak{a}}={\mathfrak{k}}_{\varepsilon}.$ Then, if
$L{\mathfrak{m}}_{\varepsilon/2}={\mathfrak{k}}_{\varepsilon/2}$ and so
$L{\mathfrak{k}}_{\varepsilon/2}={\mathfrak{m}}_{\varepsilon/2},$ we have from
Lemma 4.1 that

$L[{\mathfrak{m}}_{\varepsilon/2},{\mathfrak{m}}_{\varepsilon/2}]_{\overline{\mathfrak{m}}}\subset
L{\mathfrak{k}}_{\varepsilon},\quad[L{\mathfrak{m}}_{\varepsilon/2},L{\mathfrak{m}}_{\varepsilon/2}]\subset{\mathfrak{k}}_{\varepsilon}.$

Hence, using (i) in Lemma 3.1 and that
$[{\mathfrak{m}}_{\varepsilon/2},{\mathfrak{m}}_{\varepsilon/2}]_{\overline{\mathfrak{m}}}\neq
0$ by (5.9), $L{\mathfrak{k}}_{\varepsilon}={\mathfrak{k}}_{\varepsilon}.$ But
it is absurd since by hypothesis
$L{\mathfrak{a}}={\mathfrak{k}}_{\varepsilon}.$

If $L{\mathfrak{m}}_{\varepsilon/2}={\mathfrak{m}}_{\varepsilon/2}$ and so
$L{\mathfrak{k}}_{\varepsilon/2}={\mathfrak{k}}_{\varepsilon/2},$ then using
again Lemma 4.1,

$L[{\mathfrak{a}},{\mathfrak{m}}_{\varepsilon/2}]_{\overline{\mathfrak{m}}}\subset
L{\mathfrak{k}}_{\varepsilon/2}={\mathfrak{k}}_{\varepsilon/2},\quad[L{\mathfrak{a}},L{\mathfrak{m}}_{\varepsilon/2}]_{\overline{\mathfrak{m}}}=[{\mathfrak{k}}_{\varepsilon},{\mathfrak{m}}_{\varepsilon/2}]_{\overline{\mathfrak{m}}}\subset{\mathfrak{m}}_{\varepsilon/2}.$

Therefore, it must happen that
$[{\mathfrak{a}},{\mathfrak{m}}_{\varepsilon/2}]_{\overline{\mathfrak{m}}}=[{\mathfrak{k}}_{\varepsilon},{\mathfrak{m}}_{\varepsilon/2}]_{\overline{\mathfrak{m}}}=0,$
which is absurd because $n\geq 2.$

∎

######  Theorem 6.2.

The $K$-contact structures $(\xi^{\kappa}_{i},{\bf g}^{\kappa}_{i})$ on
$T_{r}{\mathbb{C}}{\bf P}^{n},$ for $i=1,2,3$ and $\kappa>0,$ are Sasakian.
Moreover, the tangent sphere bundle $T_{r}G/K$ of a compact rank-one symmetric
space $G/K$ admits an $G$-invariant $3$-Sasakian structure if and only if
$G/K={\mathbb{C}}{\bf P}^{n},$ for some $n\geq 1.$ Then there exists a unique
orthogonal $3$-Sasakian metric ${\bf g}$ on $T_{r}{\mathbb{C}}{\bf P}^{n},$
given at $o_{H}$ by

${\bf
g}_{o_{H}}=\frac{1}{4}\langle\cdot,\cdot\rangle_{\overline{\mathfrak{n}}}+\frac{1}{8}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon/2}\oplus{\mathfrak{k}}_{\varepsilon/2}}.$
(6.1)

######  Remark 6.3.

From Remark 5.6, note that $T_{r}{\mathbb{S}}^{2}$ and $T_{r}{\mathbb{R}}{\bf
P}^{2},$ as particular cases of Theorem 6.2, admit an invariant $3$-Sasakian
structure.

Proof of Theorem 6.2. First, by a direct procedure, we will prove that
$(\xi_{3}^{\kappa}=\pm\frac{1}{\kappa}\nu_{\varepsilon},{\bf g}^{\kappa}_{3})$
is normal. From (5.12), the $(1,1)$-tensor field $\varphi_{3}$ of the
corresponding almost contact metric structure is determined by

$\begin{array}[]{l}\varphi_{3}X=-\mu_{\varepsilon},\quad\varphi_{3}\mu_{\varepsilon}=X,\quad\varphi_{3}\nu_{\varepsilon}=0,\\\\[4.79993pt]
\varphi_{3}\mu^{j,a}_{\varepsilon/2}=(-1)^{a+1}\mu^{j,a+1}_{\varepsilon/2},\quad\varphi_{3}\nu^{j,a}_{\varepsilon/2}=(-1)^{a}\nu^{j,a+1}_{\varepsilon/2},\end{array}$
(6.2)

for all $j=1,\dots,n-1;$ $a=0,1,$ and, using (5.9), also as

$\varphi_{{3}_{\mid{\mathfrak{m}}_{\lambda}\oplus{\mathfrak{k}}_{\lambda}}}=\frac{1}{\lambda_{\mathbb{R}}(X)}{\rm
ad}_{\nu_{\varepsilon}},\quad\varphi_{3}X={\rm ad}_{\nu_{\varepsilon}}X,$

where $\lambda\in\Sigma^{+}=\\{\varepsilon,\varepsilon/2\\}.$ Then, since the
subspaces $\overline{\mathfrak{n}}$ and
${\mathfrak{m}}_{\varepsilon/2}\oplus{\mathfrak{k}}_{\varepsilon/2}$ of
$\overline{\mathfrak{m}}$ are ${\rm ad}_{\nu_{\varepsilon}}$-invariant, it
follows that

${\rm ad}_{\nu_{\varepsilon}}\circ\varphi_{3}=\varphi_{3}\circ{\rm
ad}_{\nu_{\varepsilon}}.$ (6.3)

On the other hand, taking into account that
$2d\eta^{\kappa}_{3}(u,v)=-\eta^{\kappa}_{3}([u,v]_{\overline{\mathfrak{m}}}),$
for all $u,v\in\overline{\mathfrak{m}},$ where $\eta^{\kappa}_{3}={\bf
g}^{\kappa}_{3}(\xi^{\kappa}_{3},\cdot),$ we get that the (skew-symmetric)
$G$-invariant tensor field ${\bf
N}_{3}=[\varphi_{3},\varphi_{3}]+2d\eta^{\kappa}_{3}\otimes\xi^{\kappa}_{3}$
is determined by

${\bf
N}_{3}(u,v)=-[u,v]_{\overline{\mathfrak{m}}}+[\varphi_{3}u,\varphi_{3}v]_{\overline{\mathfrak{m}}}-\varphi_{3}[\varphi_{3}u,v]_{\overline{\mathfrak{m}}}-\varphi_{3}[u,\varphi_{3}v]_{\overline{\mathfrak{m}}}.$
(6.4)

From (6.3) and using that the orthogonal complement
$(\xi_{3}^{\kappa})^{\bot}$ in $\overline{\mathfrak{m}}$ of $\xi_{3}^{\kappa}$
is ${\rm ad}_{\nu_{\varepsilon}}$-preserving, it follows that ${\bf
N}_{3}(\xi^{\kappa}_{3},\cdot)=0.$ Moreover, from (6.4),

${\bf N}_{3}(\varphi_{3}u,v)={\bf N}_{3}(u,\varphi_{3}v),\quad
u,v\in(\xi^{\kappa}_{3})^{\bot},$

and so, ${\bf N}_{3}(u,\varphi_{3}u)=0.$ Then, applying (6.2), we have ${\bf
N}_{3}(X,\mu_{\varepsilon})={\bf
N}_{3}(\mu^{j,a}_{\varepsilon/2},\mu^{j,a+1}_{\varepsilon/2})={\bf
N}_{2}(\nu^{j,a}_{\varepsilon/2},\nu^{j,a+1}_{\varepsilon/2})=0$ and

$\begin{array}[]{lcl}{\bf N}_{3}(X,\mu^{j,a}_{\varepsilon/2})&=&(-1)^{a+1}{\bf
N}_{3}(\mu_{\varepsilon},\mu^{j,a+1}_{\varepsilon/2}),\\\\[4.79993pt] {\bf
N}_{3}(X,\nu^{j,a}_{\varepsilon/2})&=&(-1)^{a}{\bf
N}_{3}(\mu_{\varepsilon},\nu^{j,a+1}_{\varepsilon/2}),\\\\[4.79993pt] {\bf
N}_{3}(\mu^{j,a}_{\varepsilon/2},\mu^{k,b}_{\varepsilon/2})&=&(-1)^{a+b+1}{\bf
N}_{3}(\mu^{j,a+1}_{\varepsilon/2},\mu^{k,b+1}_{\varepsilon/2}),\\\\[4.79993pt]
{\bf
N}_{3}(\nu^{j,a}_{\varepsilon/2},\nu^{k,b}_{\varepsilon/2})&=&(-1)^{a+b+1}{\bf
N}_{3}(\nu^{j,a+1}_{\varepsilon/2},\nu^{k,b+1}_{\varepsilon/2}),\\\\[4.79993pt]
{\bf
N}_{3}(\mu^{j,a}_{\varepsilon/2},\nu^{k,b}_{\varepsilon/2})&=&(-1)^{a+b}{\bf
N}_{3}(\mu^{j,a+1}_{\varepsilon/2},\nu^{k,b+1}_{\varepsilon/2}),\end{array}$
(6.5)

for all $j,k=1,\dots,n-1$ and $a,b=0,1.$ Using (5.9), we have for $j\neq k,$

$[\mu^{j,a}_{\varepsilon/2},\nu^{k,b}_{\varepsilon}]=[\mu^{j,a}_{\varepsilon/2},\mu^{k,b}_{\varepsilon/2}]_{\overline{\mathfrak{m}}}=[\nu^{j,a}_{\varepsilon/2},\nu^{k,b}_{\varepsilon/2}]_{\overline{\mathfrak{m}}}=0.$

Hence, from (6.2), (6.4) and (6.5), we get

${\bf N}_{3}(\mu^{j,a}_{\varepsilon/2},\mu^{k,b}_{\varepsilon/2})={\bf
N}_{3}(\nu^{j,a}_{\varepsilon/2},\nu^{k,b}_{\varepsilon/2})=0,$

for all $a,b=0,1,$ and $j,k=1,\dots,n-1.$ Moreover, for $j\neq k,$ we obtain
${\bf N}_{3}(\mu^{j,a}_{\varepsilon/2},\nu^{k,b}_{\varepsilon/2})=0.$ Then we
conclude that ${\bf N}_{3}$ is identically null if and only if

${\bf N}_{3}(X,\mu^{j,a}_{\varepsilon/2})={\bf
N}_{3}(X,\nu^{j,a}_{\varepsilon/2})={\bf
N}_{3}(\mu^{j,a}_{\varepsilon/2},\nu^{j,b}_{\varepsilon/2})=0,\quad a\leq b,$

which is satisfied by direct calculation using (5.9) and (6.2) in (6.4). This
proves that $(\xi_{3}^{\kappa}=\pm\frac{1}{\kappa},{\bf g}^{\kappa}_{3})$ is
also Sasakian.

For $\kappa=\frac{1}{2},$ the three Sasakian metrics ${\bf g}^{\kappa}_{i},$
$i=1,2,3,$ coincide with the metric ${\bf g}$ determined in (6.1) and clearly
$(\\{\xi_{1},\xi_{2},\xi_{3}\\},{\bf g})$ is a $G$-invariant $3$-Sasakian
structure on $T_{r}{\mathbb{C}}{\bf P}^{n},$ where $\xi_{1\,{o_{H}}}=2X,$
$\xi_{2\,{o_{H}}}=2\nu_{\varepsilon},$ and
$\xi_{3\,{o_{H}}}=2\mu_{\varepsilon}.$ From Theorem 5.5, ${\bf g}$ is the
unique $G$-invariant orthogonal $3$-Sasakian metric on tangent sphere bundles
of compact rank-one symmetric spaces, thus proving the result.

######  Theorem 6.4.

The $3$-Sasakian metric ${\bf g}$ is the unique orthogonal Sasakian-Einstein
metric on $T_{r}{\mathbb{C}}{\bf P}^{n}.$

###### Proof.

From Theorem 6.2, all orthogonal Sasakian metrics on $T_{r}{\mathbb{C}}{\bf
P}^{n}$ are of the form ${\bf g}^{\kappa}_{i},$ for $i=1,2,3$ and $\kappa>0.$

Let
$\alpha_{i}\colon\overline{\mathfrak{m}}\times\overline{\mathfrak{m}}\to\overline{\mathfrak{m}}$
be the ${\rm Ad}(H)$-invariant bilinear function that determines the Levi-
Civita connection of ${\bf g}^{\kappa}_{i}.$ Then, applying (5.9) and (5.22)
in (3.4), we have that the non-vanishing components of $\alpha_{1}$ and
$\alpha_{3}$ are given by

$\begin{array}[]{l}\alpha_{1}(X,\mu_{\varepsilon})=(\kappa-1)\nu_{\varepsilon},\quad\alpha_{1}(\mu_{\varepsilon},X)=\kappa\nu_{\varepsilon},\quad\alpha_{1}(X,\nu_{\varepsilon})=-\alpha_{3}(\nu_{\varepsilon},X)=(1-\kappa)\mu_{\varepsilon},\\\\[4.79993pt]
\alpha_{1}(\nu_{\varepsilon},X)=-\alpha_{3}(X,\nu_{\varepsilon})=-\kappa\xi_{\varepsilon},\quad\alpha_{3}(X,\mu_{\varepsilon})=-\alpha_{3}(\mu_{\varepsilon},X)=-\frac{1}{2}\nu_{\varepsilon},\\\\[4.79993pt]
\alpha_{1}(X,\mu^{j,a}_{\varepsilon/2})=\frac{(2\kappa-1)}{2}\mu^{j,a}_{\varepsilon/2},\quad\alpha_{1}(\mu^{j,a}_{\varepsilon/2},X)=\kappa\nu^{j,a}_{\varepsilon/2},\quad\alpha_{1}(X,\nu^{j,a}_{\varepsilon/2})=\frac{1-2\kappa}{2}\mu^{j,a}_{\varepsilon/2},\\\\[4.79993pt]
\alpha_{1}(\nu^{j,a}_{\varepsilon/2},X)=-\kappa\mu^{j,a}_{\varepsilon/2},\quad\alpha_{1}(\mu_{\varepsilon},\nu_{\varepsilon})=-\alpha_{1}(\nu_{\varepsilon},\mu_{\varepsilon})=-\frac{1}{2}X,\\\\[4.79993pt]
\alpha_{1}(\mu^{j,a}_{\varepsilon/2},\mu_{\varepsilon})=\alpha_{1}(\nu^{j,a}_{\varepsilon/2},\nu_{\varepsilon})=\alpha_{3}(\mu^{j,a}_{\varepsilon/2},\mu_{\varepsilon})=\frac{(-1)^{a+1}}{2}\nu_{\varepsilon/2}^{j,a+1},\\\\[4.79993pt]
\alpha_{1}(\mu^{j,a}_{\varepsilon/2},\nu_{\varepsilon})=-\alpha_{1}(\nu^{j,a}_{\varepsilon/2},\mu_{\varepsilon})=-\alpha_{3}(\nu^{j,a}_{\varepsilon/2},\mu_{\varepsilon})=\frac{(-1)^{a}}{2}\mu^{j,a+1}_{\varepsilon/2},\\\\[4.79993pt]
\alpha_{3}(\mu^{j,a}_{\varepsilon/2},X)=\frac{1}{2}\nu^{j,a}_{\varepsilon/2},\quad\alpha_{3}(\nu^{j,a}_{\varepsilon/2},X)=-\frac{1}{2}\mu^{j,a}_{\varepsilon/2},\;\;\alpha_{3}(\mu_{\varepsilon},\nu_{\varepsilon})=-\kappa
X,\quad\alpha_{3}(\nu_{\varepsilon},\mu_{\varepsilon})=(1-\kappa)X,\\\\[4.79993pt]
\alpha_{3}(\nu^{j,a}_{\varepsilon/2},\nu_{\varepsilon})=(-1)^{a+1}\kappa\nu_{\varepsilon/2}^{j,a+1},\quad\alpha_{3}(\mu^{j,a}_{\varepsilon/2},\nu_{\varepsilon})=(-1)^{a}\kappa\mu^{j,a+1}_{\varepsilon/2},\\\\[4.79993pt]
\alpha_{3}(\nu_{\varepsilon},\mu^{j,a}_{\varepsilon/2})=(-1)^{a}(\kappa-1)\mu^{j,a+1}_{\varepsilon/2},\quad\alpha_{3}(\nu_{\varepsilon},\nu^{j,a}_{\varepsilon/2})=(-1)^{a}\frac{1-2\kappa}{2}\nu^{j,a+1}_{\varepsilon/2},\end{array}$

for all $j=1,\dots,n-1,$ together, for $i=1,2,3.$ with

$\begin{array}[]{l}\alpha_{i}(\mu^{j,0}_{\varepsilon/2},\mu^{j,1}_{\varepsilon/2})=-\alpha_{i}(\nu^{j,0}_{\varepsilon/2},\nu^{j,1}_{\varepsilon/2})=-\frac{1}{4}\nu_{\varepsilon},\\\\[4.79993pt]
\alpha_{i}(\mu^{j,0}_{\varepsilon/2},\nu^{j,1}_{\varepsilon/2})=-\alpha_{i}(\mu^{j,1}_{\varepsilon/2},\nu^{j,0}_{\varepsilon/2})=-\alpha_{i}(\nu^{j,0}_{\varepsilon/2},\mu^{j,1}_{\varepsilon/2})=\alpha_{i}(\nu^{j,1}_{\varepsilon/2},\mu^{j,0}_{\varepsilon/2})=\frac{1}{4}\mu_{\varepsilon}.\end{array}$

Denote by $R_{i}$ the Riemannian curvature, expressed as $(0,4)$-tensor field,
and the Ricci curvature ${\rm Ric}_{i}$ of $(T_{r}{\mathbb{C}}{\bf P}^{n},{\bf
g}^{\kappa}_{i}).$ Then, using (3.3) and (5.9), we get

$\begin{array}[]{lclcl}R_{1}(\mu_{\varepsilon},X,\mu_{\varepsilon},X)&=&R_{3}(X,\nu_{\varepsilon},X,\nu_{\varepsilon})&=&\frac{\kappa^{3}}{2},\\\\[4.79993pt]
R_{1}(\mu_{\varepsilon},\nu_{\varepsilon},\mu_{\varepsilon},\nu_{\varepsilon})&=&R_{3}(X,\mu_{\varepsilon},X,\mu_{\varepsilon})&=&\frac{\kappa(2-3\kappa)}{4},\\\\[4.79993pt]
R_{1}(\mu_{\varepsilon},\mu^{j,a}_{\varepsilon/2},\mu_{\varepsilon},\mu^{j,a}_{\varepsilon/2})&=&R_{1}(\mu_{\varepsilon},\nu^{j,a}_{\varepsilon/2},\mu_{\varepsilon},\nu^{j,a}_{\varepsilon/2})&=&\frac{\kappa}{16},\\\\[4.79993pt]
R_{3}(X,\mu^{j,a}_{\varepsilon/2},X,\mu^{j,a}_{\varepsilon/2})&=&R_{3}(X,\nu^{j,a}_{\varepsilon/2},X,\nu^{j,a}_{\varepsilon/2})&=&\frac{\kappa}{16},\end{array}$

for all $j=1,\dots,n-1,$ $a=0,1.$ Hence, ${\rm
Ric}_{1}(\mu_{\varepsilon},\mu_{\varepsilon})={\rm Ric}_{3}(X,X)=n-\kappa.$ On
the other hand, because $(\xi_{1}^{\kappa},{\bf g}_{1}^{\kappa})$ and
$(\xi^{\kappa}_{3},{\bf g}^{\kappa}_{3})$ are Sasakian structures, the
condition to be ${\bf g}_{1}^{\kappa}$ or ${\bf g}_{3}^{\kappa}$ Einstein
implies that the Einstein constant $\lambda$ must be equals to $2(2n-1)$ and
so ${\rm Ric}_{1}(\mu_{\varepsilon},\mu_{\varepsilon})={\rm
Ric}_{3}(X,X)=(2n-1)\kappa.$ Hence, $\kappa=\frac{1}{2}.$ We get the same
result for the families of metrics ${\bf g}^{\kappa}_{2}$ because, according
with Theorem 5.5, the Sasakian structures $(\xi^{\kappa}_{1},{\bf
g}^{\kappa}_{1})$ and $(\xi^{\kappa}_{2},{\bf g}^{\kappa}_{2})$ are isomorphic
for every $\kappa.$ ∎

##  7\. Invariant Einstein metrics on $T_{r}{\mathbb{S}}^{n}$ and
$T_{r}{\mathbb{R}}{\mathbf{P}}^{n}$

Consider the unit sphere ${\mathbb{S}}^{n}$ of ${\mathbb{R}}^{n+1}$ as the
quotient $G/K={\rm SO}(n+1)/{\rm SO}(n)$ and the real projective space
${\mathbb{R}}{\bf P}^{n}$ expressed as $G/K={\rm SO}(n+1)/{\rm S}({\rm
O}(1)\times{\rm O}(n)).$ According with [12, Proposition 4.2], $T_{r}G/K$ is
the quotient manifold $G/H,$ where $H={\rm SO}(n-1)$ for ${\mathbb{S}}^{n}$
and $H={\rm S}({\rm O}(1)\times{\rm O}(n-1))$ for ${\mathbb{R}}{\bf P}^{n}.$

The set of matrices $\\{B^{0}_{jk},\,1\leq j<k\leq n+1\\}$ is an orthonormal
basis of the Lie algebra ${\mathfrak{s}}{\mathfrak{o}}(n+1)$ of $G=SO(n+1),$
with respect to the invariant trace form $\langle A,B\rangle=-\frac{1}{2}{\rm
trace}\,AB,$ for all $A,B\in{\mathfrak{s}}{\mathfrak{o}}(n+1).$ Then the Lie
algebra ${\mathfrak{k}}={\mathfrak{s}}{\mathfrak{o}}(n)$ of $K,$ where $K={\rm
SO}(n)$ or $K={\rm S}({\rm O}(1)\times{\rm O}(n)),$ and its orthogonal
complement ${\mathfrak{m}}$ in ${\mathfrak{s}}{\mathfrak{o}}(n+1)$ are given
by

${\mathfrak{k}}={\mathbb{R}}\\{B^{0}_{jk}\\}_{2\leq j<k\leq
n+1},\quad{\mathfrak{m}}={\mathbb{R}}\\{B^{0}_{1k}\\}_{2\leq k\leq n+1}.$

Fixed as Cartan subspace to ${\mathfrak{a}}={\mathbb{R}}X,$ where
$X=B^{0}_{12},$ the centralizer ${\mathfrak{h}}$ of ${\mathfrak{a}}$ in
${\mathfrak{k}}$ is, using (5.7), trivial for $n=2$ and, for $n\geq 3,$ it is
given by

${\mathfrak{h}}={\mathbb{R}}\\{B^{0}_{jk}\\}_{3\leq j<k\leq n+1}.$ (7.1)

Since ${\rm ad}^{2}_{X}B^{0}_{1k}=-B^{0}_{1k},$ for $k=3,\dots,n+1,$ it
follows that
${\mathfrak{m}}_{\varepsilon}={\mathbb{R}}\\{B^{0}_{1k}\\}_{k=3,\dots,n+1}$
and, because ${\rm ad}_{X}B^{0}_{1k}=-B^{0}_{2k},$ we put

$\mu^{j}_{\varepsilon}=B^{0}_{1\,2+j},\quad\nu^{j}_{\varepsilon}=B^{0}_{2\,2+j},\quad
j=1,\dots,n-1.$ (7.2)

Then $\\{\mu^{j}_{\varepsilon}\\}$ and $\\{\nu^{j}_{\varepsilon}\\}$ are
orthonormal basis of ${\mathfrak{m}}_{\varepsilon}$ and
${\mathfrak{k}}_{\varepsilon},$ respectively, satisfying (4.3). From (5.7), we
have

$[\mu^{j}_{\varepsilon},\mu^{k}_{\varepsilon}]=[\nu^{j}_{\varepsilon},\nu^{k}_{\varepsilon}]=-B^{0}_{2+j\,2+k},\quad[\mu^{j}_{\varepsilon},\nu^{k}_{\varepsilon}]=-\delta_{jk}X.$
(7.3)

Hence the bilinear function ${\mathfrak{U}}$ for the Euclidean space
$(\overline{\mathfrak{m}},{\bf g}_{o_{H}}),$ where
$\overline{\mathfrak{m}}=\overline{\mathfrak{n}}={\mathfrak{a}}\oplus{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}$
and ${\bf g}_{o_{H}}={\bf
g}_{o_{H}}^{a_{0},a_{\varepsilon},b_{\varepsilon}}=a_{0}\langle\cdot,\cdot\rangle_{\mathfrak{a}}+a_{\varepsilon}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon}}+b_{\varepsilon}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon}},$
$a_{0},a_{\varepsilon},b_{\varepsilon}\in{\mathbb{R}}^{+},$ is determined by

${\mathfrak{U}}(X,\mu^{j}_{\varepsilon})=\frac{a_{0}-a_{\varepsilon}}{2b_{\varepsilon}}\nu^{j}_{\varepsilon},\quad{\mathfrak{U}}(X,\nu^{j}_{\varepsilon})=\frac{b_{\varepsilon}-a_{0}}{2a_{\varepsilon}}\mu^{j}_{\varepsilon},\quad{\mathfrak{U}}(\mu^{j}_{\varepsilon},\nu^{k}_{\varepsilon})=\frac{a_{\varepsilon}-b_{\varepsilon}}{2a_{0}}\delta_{jk}X,$
(7.4)

for all $j,k=1,\dots,n-1,$ the rest of components being zero. Then, using
(5.7) again, the function
$\alpha\colon\overline{\mathfrak{m}}\times\overline{\mathfrak{m}}\to\overline{\mathfrak{m}}$
that determines the Levi-Civita connection $\nabla$ of $(G/H,{\bf g}),$ is
given by

$\begin{array}[]{lclclcl}\alpha(X,\mu^{j}_{\varepsilon})&=&\frac{(a_{0}-a_{\varepsilon}-b_{\varepsilon})}{2b_{\varepsilon}}\nu^{j}_{\varepsilon},&&\alpha(\mu^{j}_{\varepsilon},X)&=&\frac{(a_{0}-a_{\varepsilon}+b_{\varepsilon})}{2b_{\varepsilon}}\nu^{j}_{\varepsilon},\\\\[4.79993pt]
\alpha(X,\nu^{j}_{\varepsilon})&=&\frac{(a_{\varepsilon}+b_{\varepsilon}-a_{0})}{2a_{\varepsilon}}\mu^{j}_{\varepsilon},&&\alpha(\nu^{j}_{\varepsilon},X)&=&\frac{(b_{\varepsilon}-a_{\varepsilon}-a_{0})}{2a_{\varepsilon}}\mu^{j}_{\varepsilon},\\\\[4.79993pt]
\alpha(\mu^{j}_{\varepsilon},\nu^{k}_{\varepsilon})&=&\frac{(a_{\varepsilon}-b_{\varepsilon}-a_{0})}{2a_{0}}\delta_{jk}X,&&\alpha(\nu^{k}_{\varepsilon},\mu^{j}_{\varepsilon})&=&\frac{(a_{\varepsilon}-b_{\varepsilon}+a_{0})}{2a_{0}}\delta_{jk}X,\end{array}$

the rest of components being zero. Hence, applying (4.3), (5.7), (7.2) and
(7.3) in (3.3), we have the following.

######  Proposition 7.1.

The Riemannian curvature tensor $R$ of $(G/H,{\bf g})$ at $o_{H}$ is given by

$\begin{array}[]{lcl}R(X,\mu^{j}_{\varepsilon},X,\mu^{j}_{\varepsilon})&=&\frac{1}{4b_{\varepsilon}}(2b_{\varepsilon}(a_{0}+a_{\varepsilon}-b_{\varepsilon})+(a_{0}-a_{\varepsilon})^{2}-b^{2}_{\varepsilon});\\\\[4.79993pt]
R(X,\nu^{j}_{\varepsilon},X,\nu^{j}_{\varepsilon})&=&\frac{1}{4a_{\varepsilon}}(2a_{\varepsilon}(a_{0}-a_{\varepsilon}+b_{\varepsilon})+(a_{0}-b_{\varepsilon})^{2}-a^{2}_{\varepsilon});\\\\[4.79993pt]
R(\mu^{j}_{\varepsilon},\mu^{k}_{\varepsilon},\mu^{j}_{\varepsilon},\mu^{k}_{\varepsilon})&=&a_{\varepsilon},\quad
R(\nu^{j}_{\varepsilon},\nu^{k}_{\varepsilon},\nu^{j}_{\varepsilon},\nu^{k}_{\varepsilon})=b_{\varepsilon},\;k\neq
j;\\\\[4.79993pt]
R(\mu^{j}_{\varepsilon},\mu^{k}_{\varepsilon},\nu^{j}_{\varepsilon},\nu^{k}_{\varepsilon})&=&\frac{1}{4a_{0}}(4a_{0}b_{\varepsilon}-(a_{0}-a_{\varepsilon}+b_{\varepsilon})^{2}),\;k\neq
j;\\\\[4.79993pt]
R(\mu^{j}_{\varepsilon},\nu^{j}_{\varepsilon},\mu^{k}_{\varepsilon},\nu^{l}_{\varepsilon})&=&-\frac{1}{4a_{0}}(2a_{0}(a_{0}-a_{\varepsilon}-b_{\varepsilon})\delta_{kl}+(a_{0}^{2}-(a_{\varepsilon}-b_{\varepsilon})^{2})\delta_{jk}\delta_{jl});\\\\[4.79993pt]
R(\mu^{j}_{\varepsilon},\nu^{k}_{\varepsilon},\mu^{k}_{\varepsilon},\nu^{j}_{\varepsilon})&=&-\frac{1}{4a_{0}}(a_{0}^{2}-(a_{\varepsilon}-b_{\varepsilon})^{2}),\;k\neq
j,\end{array}$ (7.5)

the rest of its components being zero.

######  Corollary 7.2.

The $\xi^{S}$-sectional curvature on $(T_{r}G/K=G/H,{\bf g})$ is a (positive)
constant $c$ if and only if either $a_{\varepsilon}=b_{\varepsilon},$ then
$c=\frac{a_{0}}{4a_{\varepsilon}^{2}},$ or
$a_{0}=a_{\varepsilon}+b_{\varepsilon},$ then
$c=\frac{1}{a_{\varepsilon}+b_{\varepsilon}}.$

###### Proof.

From (7.5) it follows that the $\xi^{S}$-sectional curvatures
$K(X,\mu^{j}_{\varepsilon})$ and $K(X,\nu_{\varepsilon}^{j})$ at the origin
$o_{H}$ coincide, for some $j=1,\dots,n-1,$ if and only if
$a_{\varepsilon}=b_{\varepsilon}$ or $a_{0}=a_{\varepsilon}+b_{\varepsilon}.$
Put $c=K(X,\mu^{j}_{\varepsilon})=K(X,\nu^{j}_{\varepsilon}).$ Then
$c=\frac{a_{0}}{4a_{\varepsilon}^{2}}$ if $a_{\varepsilon}=b_{\varepsilon}$
and $c=\frac{1}{a_{\varepsilon}+b_{\varepsilon}}$ if
$a_{0}=a_{\varepsilon}+b_{\varepsilon}.$

Next, let $u$ be any vector in $\overline{\mathfrak{m}}$ orthogonal to $X.$
Then,
$u=\sum_{j=1}^{n-1}(u_{j}\mu^{j}_{\varepsilon}+\overline{u}_{j}\nu^{j}_{\varepsilon}),$
for some $u_{j},\overline{u}_{j}\in{\mathbb{R}}.$ From Proposition 7.1, we get

$\begin{array}[]{lcl}R(X,u,X,u)&=&\sum_{j=1}^{n-1}((u_{j})^{2}R(X,\mu^{j}_{\varepsilon},X,\mu^{j}_{\varepsilon})+(\overline{u}_{j})^{2}R(X,\nu^{j}_{\varepsilon},X,\nu^{j}_{\varepsilon}))\\\\[4.79993pt]
&=&c\,a_{0}(a_{\varepsilon}\sum_{j=1}^{n-1}(u_{j})^{2}+b_{\varepsilon}\sum_{j=1}^{n-1}(\overline{u}_{j})^{2})=c\,a_{0}{\bf
g}(u,u).\end{array}$

Hence, $K(X,u)=c.$ Because ${\bf g}$ is a homogeneous Riemannian metric, this
implies that the $\xi^{S}$-sectional curvature on $T_{r}G/K$ is constant
equals to $c.$ ∎

Next, let $Q$ be the Ricci operator of $(G/H,{\bf g}),$ defined by ${\rm
Ric}(\cdot,\cdot)=g(Q\cdot,\cdot).$ Then one gets that the vectors
$\\{X,\mu^{j}_{\varepsilon},\nu^{j}_{\varepsilon};\;j=1,\dots,n-1\\}$ form an
orthogonal basis of eigenvectors of $Q$ at $o_{H}.$ Moreover, we have

######  Lemma 7.3.

The subspaces ${\mathfrak{a}},$ ${\mathfrak{m}}_{\varepsilon}$ and
${\mathfrak{k}}_{\varepsilon}$ of $\overline{\mathfrak{m}}$ are eigenspaces of
the Ricci operator $Q$ at $o_{H}$ and the corresponding eigenvalues
$\rho_{0},$ $\rho_{\varepsilon}$ and $\varrho_{\varepsilon}$ are given by

$\begin{array}[]{lcl}\rho_{0}&=&\frac{n-1}{2a_{0}a_{\varepsilon}b_{\varepsilon}}(a_{0}^{2}-(a_{\varepsilon}-b_{\varepsilon})^{2}),\\\\[4.79993pt]
\rho_{\varepsilon}&=&\frac{1}{2a_{0}a_{\varepsilon}b_{\varepsilon}}(2(n-1)a_{0}b_{\varepsilon}+a^{2}_{\varepsilon}-b_{\varepsilon}^{2}-a_{0}^{2}),\\\\[4.79993pt]
\varrho_{\varepsilon}&=&\frac{1}{2a_{0}a_{\varepsilon}b_{\varepsilon}}(2(n-1)a_{0}a_{\varepsilon}-a^{2}_{\varepsilon}+b_{\varepsilon}^{2}-a_{0}^{2}).\end{array}$
(7.6)

######  Remark 7.4.

From this lemma, the scalar curvature $s$ of $(G/H,{\bf g})$ is given by

$s=\frac{n-1}{2a_{0}a_{\varepsilon}b_{\varepsilon}}(2(n-1)a_{0}(a_{\varepsilon}+b_{\varepsilon})-(a_{\varepsilon}-b_{\varepsilon})^{2}-a_{0}^{2}).$

Now we are in a position to determine all the invariant metrics that are
Einstein on $T_{r}{\mathbb{S}}^{n}$ and $T_{r}{\mathbb{R}}{\bf P}^{n}.$

######  Theorem 7.5.

There exists, except for a homothetic coefficient $\alpha>0,$ a unique
$G$-invariant Einstein Riemannian metric ${\bf g}^{\alpha}$ on $T_{r}(G/K),$
for $G/K={\mathbb{S}}^{n}$ or ${\mathbb{R}}{\bf P}^{n}$ $(n\geq 2),$ and it is
determined by

${\bf
g}^{\alpha}_{o_{H}}=\alpha\Big{(}\langle\cdot,\cdot\rangle_{\mathfrak{a}}+\frac{n}{2(n-1)}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}}\Big{)}.$
(7.7)

###### Proof.

Let ${\bf g}={\bf g}^{a_{0},a_{\varepsilon},b_{\varepsilon}}$ be any
$G$-invariant Riemannian metric on $T_{r}(G/K)=G/H.$ From (7.6), it follows
that $\rho_{\varepsilon}=\varrho_{\varepsilon}$ if and only if

$a_{\varepsilon}=b_{\varepsilon}\quad\mbox{\rm or}\quad
a_{0}=\frac{1}{n-1}(a_{\varepsilon}+b_{\varepsilon}).$ (7.8)

If $a_{\varepsilon}=b_{\varepsilon},$ clearly the equality
$\rho_{0}=\rho_{\varepsilon}=\varrho_{\varepsilon}$ holds if and only if
$a_{\varepsilon}=\frac{n}{2(n-1)}a_{0}$ and, putting $\alpha=a_{0},$ we have
that ${\bf g}={\bf g}^{\alpha}.$ If
$a_{0}=\frac{1}{n-1}(a_{\varepsilon}+b_{\varepsilon}),$ then

$\begin{array}[]{lcl}\rho_{0}&=&\frac{1}{2(a_{\varepsilon}+b_{\varepsilon})a_{\varepsilon}b_{\varepsilon}}((a_{\varepsilon}+b_{\varepsilon})^{2}-(n-1)^{2}(a_{\varepsilon}-b_{\varepsilon})^{2}),\\\\[4.79993pt]
\rho_{\varepsilon}&=&\frac{n(n-2)}{2(n-1)a_{\varepsilon}b_{\varepsilon}}(a_{\varepsilon}+b_{\varepsilon}).\end{array}$

Let us see that $\rho_{0}=\rho_{\varepsilon}$ cannot occur in this case. Here,

$\Big{(}\frac{n(n-2)}{n-1}-1\Big{)}(a_{\varepsilon}+b_{\varepsilon})^{2}+(n-1)^{2}(a_{\varepsilon}-b_{\varepsilon})^{2}=0.$

Then, for $n=2$ we get that $a_{\varepsilon}b_{\varepsilon}=0$ and, for $n>2,$
that $a_{\varepsilon}=b_{\varepsilon}=0,$ but $a_{\varepsilon}$ and
$b_{\varepsilon}$ are positive constants. ∎

From Corollary 7.2, Theorem 7.5 and (7.4), we have directly the following.

######  Corollary 7.6.

The standard vector field $\xi^{S}$ is Killing with constant
$\xi^{S}$-sectional curvature equals to $\frac{(n-1)^{2}}{n^{2}\alpha}$ on the
tangent sphere bundle $T_{r}(G/K),$ for $G/K={\mathbb{S}}^{n}$ or
${\mathbb{R}}{\bf P}^{n},$ equipped with the $G$-invariant Einstein metric
${\bf g}^{\alpha}.$

######  Corollary 7.7.

There exists an invariant Riemannian metric on $T_{r}{\mathbb{S}}^{n}$ or
$T_{r}{\mathbb{R}}{\bf P}^{n}$ with constant sectional curvature if and only
if $n=2.$ On $T_{r}{\mathbb{S}}^{2}\cong SO(3)$ or on $T_{r}{\mathbb{R}}{\bf
P}^{2}\cong SO(3)/\pm I_{2},$ an invariant Riemannian metric ${\bf g}$ has
constant sectional curvature if and only if ${\bf
g}_{o_{H}}=\alpha\langle\cdot,\cdot\rangle,$ for some $\alpha>0.$ Then, the
constant sectional curvature $c$ is $\frac{1}{4\alpha}.$

###### Proof.

Suppose that $(G/H,{\bf g})$ has constant sectional curvature $c.$ From
Corollary 7.2, $a_{\varepsilon}=b_{\varepsilon}$ or
$a_{0}=a_{\varepsilon}+b_{\varepsilon}.$ If $a_{\varepsilon}=b_{\varepsilon},$
then $c=\frac{a_{0}}{4a_{\varepsilon}^{2}}$ and from (7.5),

$c=K(\mu^{1}_{\varepsilon},\nu^{1}_{\varepsilon})=\frac{4a_{\varepsilon}-3a_{0}}{4a^{2}_{\varepsilon}}.$

Hence, $a_{0}=a_{\varepsilon}=b_{\varepsilon}$ and $c=\frac{1}{4\alpha},$
where $\alpha=a_{0}.$ If $a_{0}=a_{\varepsilon}+b_{\varepsilon},$ then
$K(\mu^{1}_{\varepsilon},\nu^{1}_{\varepsilon})=-\frac{1}{2a_{\varepsilon}}<0$
and this case does not occur. If $n\geq 3,$ we have that
$K(\mu^{1}_{\varepsilon},\mu^{2}_{\varepsilon})=\frac{1}{a_{\varepsilon}}=\frac{1}{\alpha}\neq\frac{1}{4\alpha}.$

Then $n=2$ and ${\bf g}_{o_{H}}=\alpha\langle\cdot,\cdot\rangle,$ for some
$\alpha>0.$ Moreover, from Theorem 7.5, ${\bf g}$ is Einstein and hence, ${\bf
g}$ has in fact constant sectional curvature. ∎

######  Corollary 7.8.

The induced Sasaki metric $\tilde{g}^{S}$ on $T_{r}(G/K),$ for
$G/K={\mathbb{S}}^{n}$ or ${\mathbb{R}}{\bf P}^{n},$ is Einstein if and only
if $r=1$ and $n=2.$ Then $(T_{1}{\mathbb{S}}^{2},\tilde{g}^{S})$ and
$(T_{1}{\mathbb{R}}{\bf P}^{2},\tilde{g}^{S})$ have constant sectional
curvature $c=\frac{1}{4}.$

###### Proof.

From (5.3), $\tilde{g}^{S}={\bf g}^{\alpha},$ for some $\alpha>0,$ if and only
if $r=1$ and $n=2.$ This implies that $\alpha=1$ and
$\tilde{g}^{S}_{o_{H}}=\langle\cdot,\cdot\rangle.$ From Corollary 7.7,
$\tilde{g}^{S}$ has constant sectional curvature $c=\frac{1}{4}.$ ∎

######  Theorem 7.9.

There exists a unique $G$-invariant contact metric ${\bf g}$ on $T_{r}(G/K),$
for $G/K={\mathbb{S}}^{n}$ or ${\mathbb{R}}{\bf P}^{n},$ $n\geq 2,$ which is
Einstein. Such a metric is Sasakian and it is determined by

${\bf
g}_{o_{H}}=\frac{(n-1)^{2}}{n^{2}}\Big{(}\langle\cdot,\cdot\rangle_{\mathfrak{a}}+\frac{n}{2(n-1)}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon}\oplus{\mathfrak{k}}_{\varepsilon}}\Big{)}.$

If $n=2,$ ${\bf g}$ is the $3$-Sasakian metric ${\bf
g}_{o_{H}}=\frac{1}{4}\langle\cdot,\cdot\rangle,$ with constant sectional
curvature $c=1.$

###### Proof.

From Theorem 5.5 case (i), if $G/K\in\\{{\mathbb{S}}^{n},{\mathbb{R}}{\bf
P}^{n}\;(n\geq 3)\\},$ any $G$-invariant contact metric structure $(\xi,{\bf
g})$ on $T_{r}(G/K)$ is determined by $\xi_{o_{H}}=\pm\frac{1}{\kappa}X$ and
${\bf
g}_{o_{H}}=\kappa^{2}\langle\cdot,\cdot\rangle_{\mathfrak{a}}+\frac{\kappa}{2q_{\varepsilon}}\langle\cdot,\cdot\rangle_{{\mathfrak{m}}_{\varepsilon}}+\frac{\kappa
q_{\varepsilon}}{2}\langle\cdot,\cdot\rangle_{{\mathfrak{k}}_{\varepsilon}},$
for some $\kappa,q_{\varepsilon}>0.$ Then, using Theorem 7.5, ${\bf g}$ is
Einstein if and only if $q_{\varepsilon}=1,$ or equivalently $(\xi,{\bf g})$
is Sasakian, and $\kappa=\frac{n-1}{n}.$ For $n=2,$ we apply Theorem 5.5 case
(ii) and again Theorem 7.5. ∎

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