Full text of DOI:10.1017/S0305004199003825

Math. Proc. Camb. Phil. Soc. (2000), 128, 87
87
Printed in the United Kingdom # 2000 Cambridge Philosophical Society
Characterizing the complex hyperbolic space by Kahler
homogeneous structures
B P. M. GADEA, A. MONTESINOS AMILIBIA  J. MUNOZ MASQUE†
IMAFF, CSIC, Serrano 123, 28006 – Madrid, Spain
Department of Geometry and Topology, Faculty of Mathematics, 46100 – Burjasot,
Valencia, Spain
IFA, CSIC, Serrano 144, 28006 – Madrid, Spain
(Received 9 April 1998; revised 23 October 1998)
1. Introduction
The Kahler case of Riemannian homogeneous structures [3, 15, 18] has been
studied in [1, 2, 6, 7, 13, 16], among other papers. Abbena and Garbiero [1] gave a
classification of Kahler homogeneous structures, which has four primitive classes
ꢀ, …, ꢀ (see [6, theorem 5n1] for another proof and Section 2 below for the result).
"
%
The purpose of the present paper is to prove the following result:
T 1n1. A simply connected irreducible homogeneous Kahler manifold admits
a nonvanishing Kahler homogeneous structure in Abbena–Garbiero’s class ꢀ & ꢀ if
and only if it is the complex hyperbolic space equipped with the Bergman metric of
#
%
negative constant holomorphic sectional curvature.
We thus have a situation similar to the Riemannian case, where a connected,
simply connected and complete Riemannian manifold admits a non-vanishing
homogeneous structure of first class if and only if it is isometric to the hyperbolic
space ([18, theorem 5n2]), and similarly we obtain a vector field ξ (see (4n4) in Section
4) which is the complex analogue of the vector field in the Riemannian case satisfying
] ξ l g(X, ξ) ξkg(ξ, ξ) X, ([8, 18]), for Riemannian homogeneous structures of first
X
class and models of negative constant (ordinary) sectional curvature. Moreover, this
suggests the possibility of a quaternionic analogue to Theorem 1n1 for models of
negative constant quaternionic sectional curvature and also of a Cayleyan analogue.
On the other hand, similarly to the Riemannian case [18, p. 55], one has a solvable
group acting simply transitively on the relevant domain (see Remark 4n2(i)), thus
explaining why positive (holomorphic sectional) curvature is not detected.
2. Preliminaries and notations
As is well known, Ambrose and Singer proved in [3] that a connected, simply
connected and complete Riemannian manifold (M, g) is homogeneous if and only if it
admits a Riemannian homogeneous structure, i.e. a (1, 2) tensor field S satisfying
]g l 0, ]R l 0, ]S l 0,
(2n1)
† Partially supported by the DGES (Spain): P.M.G. and J.M.M. under Project PB-95-0124 and
A.M.A. under Project PB-94-0972.

88
P. M. G, A. M  J. M. M
where ] l ]kS, ] denotes the Levi–Civita connection and R the curvature tensor
of ]. We write S
]g l 0 is equivalent to S
l g(S Y, Z). Then, from ]g l 0 it follows that the condition
XYZ
X
lkS
. We set
XZY
XYZ
R
Z l ]Q Q Zk] ] Zj] ] Z,
R
l g(R Z, W),
XY
X,Y
X
Y
Y
X
XYZW
XY
R
(Z, W) l R
,
X, Y, Z, W ? ꢁ(M).
XY
XYZW
In the sequel X, Y, Z, W, U, ξ, ζ, will stand for vector fields on a C_ manifold M.
We denote by F the Kahler form, defined by F(X, Y) l g(X, JY), by r the Ricci
tensor and by s the scalar curvature.
Sekigawa [17] proved that a connected, simply connected and complete almost
Hermitian manifold (M, g, J) is homogeneous if and only if it admits an almost
Hermitian homogeneous structure, i.e. a (1, 2) tensor field S satisfying the conditions
(2n1) and ]J l 0.
We further suppose that (M, g, J) is Kahler, so ]J l 0. As ]J l 0, the condition
]J l 0 is equivalent to S
l S
homogeneous structure satisfying that invariance condition, then the manifold is
. Hence, if S is an almost Hermitian
XJYJZ
XYZ
Kahler ([1, theorem 5n8]). This property is equivalent to saying that S belongs to the
vector space
$
ꢂ(V) l oS ? " V*:S
lkS
l S
q.
+
XYZ
XZY
XJYJZ
Definition 2n1. A Kahler homogeneous structure on a Kahler manifold (M, g, J) is an
almost Hermitian homogeneous structure S on M such that S ? ꢂ(T M) for all
x
x
+
x ? M.
The classification of Kahler homogeneous structures was obtained by Abbena and
Garbiero in [1, theorem 2n1]. We recall their result here: let V be a 2n-dimensional
real vector space (which is the model for the tangent space at any point of a manifold
equipped with a Kahler homogeneous structure) endowed with a complex structure
#
J and a Hermitian inner product f,g, that is, J lkI, fJX, JYg l fX, Yg, X, Y ? V,
where I denotes the identity isomorphism of V.
If dim V & 6, ꢂ(V) decomposes into the direct sum of the following subspaces
invariant and irreducible under the action of the group U(n):
+
"
ꢀ l oS ? ꢂ(V) :S
l (S
jS
jS
jS
), c (S) l 0q;
+
XYZ
XYZ
YZX
ZXY
JYJZX
JZXJY
"
#
"#
ꢀ l oS ? ꢂ(V) :S
l fX, Yg θ (Z)kfX, Zg θ (Y)jfX, JYg θ (JZ)
+
#
"
"
"
kfX, JZg θ (JY)k2fJY, Zg θ (JX), θ ? V*q;
"
"
"
"
ꢀ l oS ? ꢂ(V) :S
lk (S
jS
jS
jS
), c (S) l 0q;
+
XYZ
XYZ
YZX
ZXY
JYJZX
JZXJY
$
#
"#
ꢀ l oS ? ꢂ(V) :S
l fX, Yg θ (Z)kfX, Zg θ (Y)jfX, JYg θ (JZ)
+
%
#
#
#
kfX, JZg θ (JY)j2fJY, Zg θ (JX), θ ? V*q;
#
#
#
n
#
X, Y, Z ? V, where c is defined by c (S) (X) l Σ
denotes an arbitrary orthonormal basis of V and
S
, for any X ? V, oe , …, e q
i= e e ,X
n
#
"#
"#
"
"
i i
1
1
θ (X) l
c (S) (X), θ (X) l
c (S) (X), X ? V.
"
"#
#
"#
2(nk1)
2(nj1)
If dim V l 4, then ꢂ(V) l ꢀ & ꢀ & ꢀ. If dim V l 2, then ꢂ(V) l ꢀ.
+
+
#
$
%
%

Characterizing the complex hyperbolic space
89
3. The class ꢀ & ꢀ
#
%
We can write the class ꢀ & ꢀ as
#
%
ꢀ & ꢀ l oS ? ꢂ(V) :S
l fX, Yg (θ jθ ) (Z)
+
XYZ
#
%
"
#
kfX, Zg (θ jθ ) (Y)jfX, JYg (θ jθ ) (JZ)
"
#
"
#
kfX, JZg (θ jθ ) (JY)j2fY, JZg (θ kθ ) (JX)q.
(3n1)
"
#
"
#
Remark 3n1. We recall that ꢀ & ꢀ is the sum of the isotypic components ꢀ, ꢀ
of the smallest dimension in the decomposition of ꢂ(V) above. In fact, one has
#
%
#
%
+
ꢀ % ꢀ % V % V* and the respective dimensions of ꢀ, …, ꢀ are (see [1, p. 382])
#
to spaces of (negative) constant holomorphic sectional curvature, which are scarce in
%
"
%
n(nj1) (nk2), 2n, n(nk1) (nj2), 2n. It is thus reasonable that ꢀ & ꢀ corresponds
#
%
all homogeneous Kahler spaces.
L 3n2. A simply connected complete irreducible Kahler manifold (M l G\H, g, J)
of real dimension 2n & 4 which admits a nonvanishing Kahler homogeneous structure
S ? ꢀ & ꢀ, is Einstein.
#
%
Proof. Let ξ be the vector field dual to the 1-form θ l θ jθ and ζ the vector field
"
#
dual to the 1-form θ kθ , both with respect to the metric. From (3n1) we have
"
#
S Y l g(X, Y) ξkg(Y, ξ) Xkg(X, JY) Jξjg(JY, ξ) JXk2g(JX, ζ) JY. (3n2)
X
Taking the ] derivative of this formula, applying Ambrose–Singer’s equations (2n1)
and Sekigawa’s equation ]J l 0 and contracting with g(W, :), we obtain
Z
g(X, Y) g(] ξ, W)kg(] ξ, Y) g(X, W)kg(X, JY) g(] Jξ, W)
Z
Z
Z
jg(] ξ, JY) g(JX, W)k2g(JX, ] ζ) g(JY, W) l 0.
Z
Z
Since dim M & 4, we deduce
]ξ l 0, ]ζ l 0.
(3n3)
Thus, from ]g l 0 we have g(ξ, ξ) l const, g(ζ, ζ) l const. If ξ l 0, ζ ꢀ 0, we also
have
R
ζ l 4(g(X, Jζ) g(Y, ζ)kg(X, ζ) g(Y, Jζ)) Jζ.
(3n4)
XY
The second Ambrose–Singer condition in (2n1) can be written as
(] R)
lkR
kR
kR
kR
.
(3n5)
X
YZWU
S YZWU
X
YS ZWU
YZS WU
YZWS U
X
X
X
Suppose ξ ꢀ 0. Applying Bianchi’s second identity to (3n5) and then substituting
(3n2), we obtain
3
o2g(X, ξ) R
jg(X, W) R
jg(X, U) R
ξk2g(X, JY) R
ξ
ξ
XYZ
YZWU
YZ U
YZW
J ZWU
kg(X, JW) R
kg(X, JU) R
ξq l 0. (3n6)
ξ
YZJ U
YZWJ
Contracting (3n6) with respect to X and W and applying Bianchi’s first identity, we
deduce
(2nj2) R
lk2g(Y, ξ) r(Z, U)j2g(Z, ξ) r(Y, U)
ξ
ZY U
j2g(Y, JZ) r(Jξ, U)kg(Y, U) r(Z, ξ)jg(Y, JU) r(Z, Jξ)
jg(Z, U) r(Y, ξ)kg(Z, JU) r(Y, Jξ).
(3n7)

90
P. M. G, A. M  J. M. M
Contracting (3n7) with regard to Y and U, we obtain r(Z, ξ) l (s\2n) g(Z, ξ). Putting
a l 1\(2nj2), b l s\2n, we can write (3n7) as
1
Rξ l 2θFr(U)j2bθ(JU) FjbU]Fθjb(JU)]F(θ @ J).
(3n8)
U
a
On the other hand, from Bianchi’s first identity one has R (Jξ, :) l
WU
Rξ (U, :)kRξ (W, :). Thus we can write (3n6) as
JW
JU
2θFR jW]FRξ kU]FRξ j2FF(Rξ (W, :)kRξ (U, :))
WU
U
W
JU
JW
j(JW)]FRξ k(JU)]FRξ l 0. (3n9)
JU
JW
Substituting (3n8) in (3n9) and denoting by Ξ(U) the right hand side of (3n8), we deduce
2
θFR jW]FΞ(U)kU]FΞ(W)j2FF(i (Ξ(JU))ki (Ξ(JW)))
WU
W
U
α
j(JW)]FΞ(JU)k(JU)]FΞ(JW) l 0. (3n10)
Taking W l ξ in this formula we obtain
(2g(ξ, ξ) FjθF(θ @ J))F(r(JU)kb(JU)]) l 0.
Contracting this formula with ξ, in account of r(ξ, :) l bg(ξ, :) we deduce
g(ξ, ξ) (θ @ J)F(r(JU)kb(JU)]) l 0.
]
#
Contracting with Jξ we have g(ξ, ξ) (r(JU)kb(JU) ) l 0. Since ξ ꢀ 0 we conclude
that r l bg, that is (M, g, J) is Einstein.
Now suppose ξ l 0, ζ ꢀ 0. It is immediate to see that Qζ, JζQ lk2g(ζ, ζ) Jζ, so that
M has a 2-dimensional involutive distribution, which is also parallel as follows from
(3n2) and ]ζ l 0 in (3n3). Thus ([5, proposition 10n21]) the holonomy representation
Hol (g) leaves invariant a subspace of dimension 2. So, by the De Rham theorem ([5,
theorem 10n41]), we conclude that M is holomorphically isometric to a Kahler
product, but then M would not be irreducible.
T 3n3. A simply connected irreducible homogeneous Kahler manifold of real
dimension 2n & 4 admitting a nonvanishing Kahler homogeneous structure S ? ꢀ & ꢀ
is holomorphically isometric to a bounded symmetric domain of negative constant
#
%
holomorphic sectional curvature.
Proof. Suppose ξ ꢀ 0. Since by the Lemma the manifold is Einstein (3n10) gives us
1
θF
R
jU]FW]j(JU)]F(JW)]k2F(W, U) F l 0.
(3n11)
0ab
1
WU
Contracting this formula with ξ we deduce, as k l g(ξ, ξ),
k
1
R
kk(W]FU]j(JW)]F(JU)]j2F(W, U) F) l θF
R
(ξ, :)
0ab
WU
WU
ab
jθ(U) W]kθ(W) U]jθ(JU) (JW)]kθ(JW) (JU)]k2F(W, U) (θ @ J) . (3n12)
1
On the other hand, from Bianchi’s first identity we deduce R (ξ, :) l
Rξ (W, :)kRξ (U, :). Substituting these two summands by their expression
WU
U
W

Characterizing the complex hyperbolic space
91
from (3n8) and then substituting the expression for R (ξ, :) in (3n12) we
obtain
WU
1
R
l W]FU]j(JW)]F(JU)]j2F(W, U) F,
WU
ab
from which
s
R
l
og(Y, W) g(Z, U)kg(Y, U) g(Z, W)jg(Y, JW) g(Z, JU)
kg(Y, JU) g(Z, JW)j2g(Y, JZ) g(W, JU)q.
YZWU
4n(nj1)
That is, (M, g, J) is a model of constant holomorphic sectional curvature
c l s\n(nj1). To determine the sign of s, we prove that (for ξ l 0 or not) if
(M, g, J) is a space of constant holomorphic sectional curvature c, then ζ l 0 and
c lkg(ξ, ξ). In fact, comparing the expressions for R
from substitution in the usual expression
ξ respectively obtained
XJX
R
Z l cog(X, Z) Ykg(Y, Z) Xjg(JX, Z) JYkg(JY, Z) JXj2g(X, JY) JZq
XY
and from ] ξ l S ξ, since n & 2 we obtain the system of equations
X
X
g(X, ξ) g(JX, ζ)kg(JX, ξ) g(X, ζ) l 0,
cg(X, ξ) lkg(ξ, ξ) g(X, ξkζ),
(3n13)
(3n14)
0 l (2g(ξ, ζ)kg(ξ, ξ)kc) g(X, X)jg(X, ξ) g(X, ζ)
#
#
jg(JX, ξ) g(JX, ζ)k2g(X, ζ) k2g(JX, ζ) . (3n15)
From (3n14) we have g(ξ, ξ) ζ l (g(ξ, ξ)jc) ξ which, if it is true, implies (3n13). In
(3n15) we can take X orthogonal to ζ and Jζ, so g(ξ, ξ)jc l 2g(ξ, ζ), thus
g(ξ, ξ) (g(ξ, ξ)jc) l 2g(ξ, g(ξ, ξ) ζ) l 2g(ξ, ξ) (g(ξ, ξ)jc), hence
g(ξ, ξ) (g(ξ, ξ)jc) l 0.
(3n16)
Suppose g(ξ, ξ) l 0. As ξ l 0, (3n15) simplifies to
#
#
cg(X, X)j2g(X, ζ) j2g(JX, ζ) l 0.
Since dim M & 4, we can take X orthogonal to ζ and to Jζ, so cg(X, X) l 0, that is,
the manifold is flat. From (3n4) we deduce
g(X, ζ) g(JY, ζ)kg(Y, ζ) g(JX, ζ) l 0.
Taking X l ζ we have g(ζ, ζ) ζ l 0, which implies ζ l 0, but then S vanishes.
On the other hand, if g(ξ, ξ) ꢀ 0, then (3n16) gives g(ξ, ξ)jc l 0; hence ζ l 0.
The case ξ l 0, ζ ꢀ 0 has been already discarded in the previous lemma.
Remark 3n4. In the conditions of Lemma 3n2, for ξ l 0, ζ ꢀ 0, substituting (3n2) in
(3n5), as ]J l 0 we have
(] R)
l 2g(JX, ζ) (R
jR
jR
jR
) l 0.
X
YZWU
JYZWU
YJZWU
YZJWU
YZWJU
Thus (M, g, J) is Hermitian symmetric. As we have seen, it is holomorphically
isometric to a Kahler product, with a 2-dimensional factor, which is a space of
negative constant curvature k4g(ζ, ζ), as it follows from (3n4).

92
P. M. G, A. M  J. M. M
Now, there are ([10, pp. 518–520]) four non-flat non-compact Hermitian symmetric
surfaces: SU(1, 1)\S(U(1)iU(1)), SO*(4)\U(2), SO (2, 1)\SO(2)i1 and Sp(1)\U(1).
All of them are isomorphic and can be identified to the real hyperbolic space
!
SO (2, 1)\SO(2)i1. Taking the half-plane Poincare model H l oz l xjiy ? #:y " 0q
!
for it, we have the metric and the complex structure (particular cases of (4n2) and
(4n3) below)
#
#
dx jdy
c
c
g lk
,
J l " dxk " dy.
cy cx
#
cy
It is immediate that the vector field ζ lk(cy\2) c\cy (a particular case of (4n4)
below) satisfies the equation ] ζ l 2g(X, Jζ) Jζ, ] being the Levi–Civita connection
X
of g. We conclude that in the simply connected reducible case one can also consider
spaces holomorphically isometric to products of the 2-dimensional Siegel domain
model of the complex hyperbolic space, that is, the above half-plane Poincare model,
equipped with the Kahler homogeneous structure S given by S Y l 2g(X, Jζ) JY,
and any other Hermitian symmetric space, endowed with its standard vanishing
Kahler homogeneous structure as a Hermitian symmetric space.
X
4. Proof of Theorem 1n1
As for the converse, it suffices to give a non-vanishing Kahler homogeneous
structure on any model of negative constant holomorphic sectional curvature. Thus,
to finish the proof we explicitly give such a structure on the Siegel domain
n
k=
n
n+
k k
"
"
D l (z l xjiy, u , …, u ) ? # :yk u u " 0 ,
ꢀ
(
equipped with the Kahler structure obtained by a convenient Cayley transform ([14,
*
+
"
n+
"
p. 5]) from that of the unit open ball in #
([9, p. 227] or [12, p. 169], changing the
sign in the second summand of the numerator in the last expression of the metric).
A calculation shows that
1
k k
k
k
g lk
odz dzj4(yk u u ) du du
ꢀ
ꢀ
+
k k
#
c(yk u u )
ꢀ
k
k
k
k
k
k
k
k
j2i(dz u du kdz u du )j4( u du ) ( u du )q.
ꢀ
ꢀ
ꢀ
ꢀ
(D , g , J ) is a Kahler manifold, with
+
+
+
n
k=
+
n
k=
c
c
cu
c
cz
c
cu
k
k
J l i
" dzj
" du k " dzk
" du
.
ꢀ
"
ꢀ
"
0cz
The Riemannian manifold (D , g ) is homogeneous, hence complete. Since (D , g , J )
is connected, simply connected and complete, it is a model of negative constant
holomorphic sectional curvature. Furthermore, we seek for a homogeneous structure
S of the type
1
+
k
k
+
+
+
+
S Y l g(X, Y) ξkg(ξ, Y) Xkg(X, JY) Jξjg(ξ, JY) JX.
X
Since ]ξ l 0, we must find a vector field ξ on D satisfying
+
] ξ l g(X, ξ) ξkg(ξ, ξ) Xkg(X, Jξ) Jξ.
(4n1)
X

Characterizing the complex hyperbolic space
93
k
k
k
1
Putting u l v jiw , the metric g and the complex structure J are written as
+
+
j
j
#
#
#
#
g lk
odx jdy j4[yk ((v ) j(w ) )]
ꢀ
+
k
k
#
# #
c[yk ((v ) j(w ) )]
ꢀ
ꢀ
j k
k
k
k l
k
k
k
l
k
k
k
k
k
k
#
#
i
((dv ) j(dw ) )k4[dx (w dv kv dw )kdy (v dv jw dw )]
ꢀ
ꢀ
ꢀ
k l
k
l
l k
k l
k
l
k
l
j8 [(v v jw w ) (dv dv jdw dw )j(v w kv w ) (dv dw kdw dv )]q, (4n2)
ꢀ
k,l
c
c
cx
c
0cw
c
cv
k
k
J l " dxk " dyj
" dv k
" dw
.
(4n3)
ꢀ
1
A long but straightforward computation shows that the vector field
+
k
k
cy
c
c
cy
k k
ξ lk (yk u u )
(4n4)
ꢀ
2
satisfies (4n1) for all X. Specifically, if X is given by
c
X l α jβ
cx
c
cy
c
cv
c
j
γ
jδ
,
ꢀ
0 k
cwk1
k
k
we obtain
c
c
cx
c
cy
k
k
k
k
] ξ l 2 (αj (δ v kγ w ))
j
(γ v jδ w )
ꢀ
ꢀ
4 ( 9
:
X
k
k
k
k
c
c
j
γ
jδ
.
ꢀ
0 k
Hence, (D , g , J ) admits the Kahler homogeneous structure S given by
cwk1*
k
k
cv
+
+
+
c
c
cy
c
k k
S Y lk (yk u u ) g(X, Y) kg
, Y X
ꢀ
(
0cy 1
X
2
c
c
jg(X, JY) jg
, JY JX .
(4n5)
0cy 1 *
cx
According to Heintze’s Theorem ([11, theorem 4]), a connected homogeneous
Kahler manifold of negative curvature is holomorphically isometric to the complex
hyperbolic space. Hence
C 4n1. A connected homogeneous Kahler manifold of real dimension 2n & 4
and negative curvature admits a nonvanishing Kahler homogeneous structure
S ? ꢀ & ꢀ.
#
%
Remark 4n2. (i) The Siegel domain D thus admits at least two Kahler homogeneous
+
structures: S l 0, as D is a realization of the noncompact Hermitian symmetric
+
to the fact that the solvable group #H (see [11, p. 33]) acts simply transitively on
D (see [11, p. 32], [5, p. 181], [4, p. 92]) by holomorphic isometries. Hence, we can
space U(1, n)\U(1)iU(n); and the non-vanishing structure S in (4n5), corresponding
n
+
canonical connection on the reductive homogeneous space (D , g , J ) identified to
n
identify D with #H and thus it is immediate to obtain the expression of the
+
the solvable group #H .
+
+
+
n

94
P. M. G, A. M  J. M. M
(ii) For dim M l 2 only the class ꢀ remains. This case has been studied by Abbena
and Garbiero [1, p. 391].
%
REFERENCES
[1] E. A and S. G. Almost Hermitian homogeneous structures. Proc. Edinb.
Math. Soc. (2) 31 (1988), 375–395.
[2] E. A and S. G. Almost Hermitian homogeneous manifolds and Lie groups.
Nihonkai Math. J. 4 (1993), 1–15.
[3] W. A and I. M. S. On homogeneous Riemannian manifolds. Duke Math. J. 25
(1958), 647–669.
[4] J. B, F. T and L. V. Generalized Heisenberg groups and Damek–Ricci
harmonic spaces (Springer-Verlag, 1995).
[5] A. B. Einstein manifolds (Springer-Verlag, 1987).
[6] S. C and A. F. Homogeneous structures on Kahler submanifolds of complex
projective spaces. Proc. Edinb. Math. Soc. (2) 39 (1996), 381–395.
[7] M. F, A. F and S. S. Almost-Hermitian geometry. Diff. Geom.
Appl. 4 (1994), 259–282.
[8] P. M. G and J. A. O. Reductive homogeneous pseudo-Riemannian manifolds.
Monatsh. Math. 124 (1997), 17–34.
[9] S. G. Curvature and homology (Dover, 1982).
[10] S. H. Differential geometry, Lie groups and symmetric spaces (Academic Press, 1978).
[11] E. H. On homogeneous manifolds of negative curvature. Math. Ann. 211 (1974), 23–34.
[12] S. K and K. N. Foundations of Differential Geometry, II (Interscience Publ.,
1969).
[13] A. L and L. V. Naturally reductive s-manifolds. Note Mat. 10 (1990), 363–370.
[14] S. M. Automorphisms of Siegel domains. Lect. Notes in Math., vol. 286 (Springer-
Verlag, 1972).
[15] V. P. Classifying 3- and 4-dimensional homogeneous Riemannian manifolds by
Cartan triples. Pacific J. Math. 173 (1996), 511–532.
[16] S. S. Riemannian geometry and holonomy groups (Longman, Essex, 1989).
[17] K. S. Notes on homogeneous almost Hermitian manifolds. Hokkaido Math. J. 7
(1978), 206–213.
[18] F. T and L. V. Homogeneous structures on Riemannian manifolds. London
Math. Soc. Lect. Notes Ser., vol. 83 (Cambridge Univ. Press, 1983).