DOI: 10.1515/auom-2015-0002
An. Şt. Univ. Ovidius Constanţa
Vol. 23(1),2015, 25–36
Geometry of product complex Cartan
manifolds
Nicoleta Aldea and Gheorghe Munteanu
Abstract
In this paper we consider the product of two complex Cartan manifolds, the outcome being a class of product complex Cartan spaces.
Then, we study the relationships between the geometric objects of a
product complex Cartan space and its components, (e.g. Chern-Cartan
complex nonlinear connection, Cartan tensors). By means of these, we
establish the necessary and sufficient conditions under which a product
complex Cartan space is Landsberg-Cartan or Berwald-Cartan or it has
some other properties.
1
Introduction
The geometry of the holomorphic cotangent bundle, endowed with a complex
homogeneous Hamiltonian on the fibre, is known as complex Cartan geometry,
[10, 9, 14, 5, 15]. It is a natural development, into the complex context, of the
remarkable results from real Hamilton and Cartan geometries, mainly due to
R. Miron, [13, 12, 7]. Of course, there are some formal similarities with the
real approach, but the differences between real and complex cases have the
upper hand.
The study of the complex Cartan spaces has been justified by the existence
of a pseudo-distance on the holomorphic cotangent bundle, pointed out by S.
Kobayashi in [10].
Key Words: complex Cartan metric, product manifold, Landsberg- and Berwald-Cartan
metrics.
2010 Mathematics Subject Classification: Primary 53C60; Secondary 53B40.
Received: 2 May, 2014.
Accepted: 28 June, 2014.
25
GEOMETRY OF PRODUCT COMPLEX CARTAN MANIFOLDS
26
In complex Cartan geometry there are not so many known examples of
complex Cartan metrics. Besides the significant dual Kobayashi metric characterized by J. Faran in [9], there are two rather trivial classes of complex
Cartan metrics: the complex Cartan metrics which come from Hermitian metrics on the base manifold (so called the purely Hermitian metrics in [14]) and
the locally Minkowski complex metrics, [14]. Recently, in the papers, [5, 6],
we initiated the study of some classes of complex Cartan spaces: LandsbergCartan, Kähler-Cartan, Berwald-Cartan, etc, which are exemplified by complex Cartan-Randers metrics. Nevertheless, any other examples are welcome.
Twisted product structures are widely used in geometry to build examples
of Finsler manifolds, [11]. The product of two complex Finsler manifolds is
approached by W. Zhicheng and C. Zhong in [18]. Therefore, it is natural to
extend this construction for complex Cartan spaces. Using some ideas from
the complex Finsler case [18], our aim in the present paper is to introduce and
study product complex Cartan spaces.
The paper is organized as follows. After we make a short survey of complex Cartan spaces (Section 2), in Section 3 we construct a class of complex
Cartan spaces, on the product M := M1 × M2 , where (M1 , C1 ) and (M2 , C2 )
are two complex Cartan manifolds, (Theorem 3.1). We find that the study of a
such product complex Cartan space is reduced to the study of its components
(M1 , C1 ) and (M2 , C2 ). More exactly, the non-vanishing components of ChernCartan complex nonlinear connection associated to the product complex Cartan space are the Chern-Cartan complex nonlinear connection of (M1 , C1 ) and
(M2 , C2 ), respectively.
Based on this, in section 4 we obtain the conditions under which the product complex Cartan space is Kähler-Cartan or weakly Kähler-Cartan, (Theorem 4.1). Also, we prove that the product complex Cartan space is LandsbergCartan or Berwald-Cartan if and only if its components (M1 , C1 ) and (M2 , C2 )
have similar properties, (Theorem 4.2). Finally, in Theorem 4.3 are given the
necessary and sufficient conditions that the product complex Cartan space to
be purely Hermitian.
2
Preliminaries
Geometry of real and complex Finsler spaces is already one classic today,
([1, 8, 16, 2, 14, 17, 3, 4], etc.). Also, the study of Cartan spaces (real and
complex) is enthralling, ([12, 13, 7, 14, 6], etc.). In this section we will give
some preliminaries about complex Cartan geometry with Chern-Cartan and
Berwald complex linear connections. We will set the basic notions (for more
see [14, 6, 5]).
Let M be a n − dimensional complex manifold and z = (z k )k=1,n be
27
GEOMETRY OF PRODUCT COMPLEX CARTAN MANIFOLDS
complex coordinates in a local chart. The complexified of the real tangent
bundle TC M splits into the sum of holomorphic tangent bundle T 0 M and its
conjugate T 00 M . The dual of T 0 M is denoted by T 0∗ M . On the manifold T 0∗ M ,
a point u∗ is characterized by the coordinates u∗ = (z, ζ) = (z k , ζk )k=1,n , and
a change of these has the form z 0k = z 0k (z) and ζk0 =
∂ ∗ zj
ζ ,
∂z 0k j
∗ 0k
rank( ∂∂zzl ) = n.
Definition 2.1. A complex Cartan space is a pair (M, C), where C : T 0∗ M →
R+ , (R+ := [0, ∞)), is a continuous function satisfying the conditions:
0∗ M := T 0∗ M \{0};
^
i) H := C2 is smooth on T
ii) C(z, ζ) ≥ 0, the equality holds if and only if ζ = 0;
iii) C(z, λζ) = |λ|C(z, ζ) for ∀λ ∈ C;
iv) the Hermitian matrix hj̄i (z, ζ) is positive definite, where hj̄i :=
is the fundamental metric tensor.
∂2H
∂ζi ∂ ζ̄j
Equivalently, the condition iv) means that the indicatrix is strongly pseudoconvex.
j̄i
∂H
∂hj̄i
Consequently, from iii ) we have ∂ζ
ζk = ∂∂H
ζ̄ = H, ∂h
∂ζk ζk = ∂ ζ̄ ζ̄k = 0
ζ̄ k
k
k
k
and H = hj̄i ζ̄j ζi . An usual example of complex Cartan space is so called purely
Hermitian complex Cartan space, this means that hj̄i = hj̄i (z).
The geometry of a complex Cartan space consists in the study of the geometric objects of the complex manifold T 0∗ M endowed with the Hermitian
metric structure defined by hj̄i . So, the first step is the study of sections in
the complexified tangent bundle of T 0∗ M , which is decomposed in the sum
TC (T 0∗ M ) = T 0 (T 0∗ M ) ⊕ T 00 (T 0∗ M ).
Let V T 0∗ M ⊂ T 0 (T 0∗ M ) be the vertical bundle, which has the vertical
distribution Vu∗ (T 0∗ M ), locally spanned by { ∂ζ∂k }. A complex nonlinear connection, briefly (c.n.c.), on T 0∗ M is a supplementary subbundle in T 0 (T 0∗ M ) of
V (T 0∗ M ) , i.e., T 0 (T 0∗ M ) = H(T 0∗ M ) ⊕ V (T 0∗ M ). The horizontal distribuδ∗
∂∗
∂
δ∗
and
tion Hu∗ (T 0∗ M ) is locally spanned by { δz
j }, where δz k = ∂z k + Njk ∂ζ
j
0∗
∗
functions Njk are the coefficients of the (c.n.c.) on T M. The pair {δk :=
δ∗
, ∂˙ k := ∂ζ∂k } will be called the adapted frame of the (c.n.c.), which obey
δz k
∗ 0j
∗ 0k
the change rules δk∗ = ∂∂zzk δj∗0 and ∂˙ k = ∂∂zzj ∂˙ 0j . By conjugation everywhere
we have obtained an adapted frame {δ ∗ , ∂˙ k̄ } on T 00∗ (T 0∗ M ). The dual adapted
k̄
u
frames are {d∗ z k , δζk = dζk − Nkj dz j } and {d∗ z̄ k , δ ζ̄k }.
A Hermitian connection DΓ, of (1, 0)− type is so called Chern-Cartan
connection (cf. [14]), and it is locally given by the following coefficients
Nji = −hj k̄
∂ ∗ hk̄l
i
ζl ; Hjk
:= hm̄i (δk∗ hj m̄ ) ; Vjik := −hj m̄ (∂˙ k hm̄i ),
∂z i
(2.1)
GEOMETRY OF PRODUCT COMPLEX CARTAN MANIFOLDS
28
ı̄
and Hj̄k
= Vj̄ı̄k = 0, where here and hereinafter δk∗ is the adapted frame
i
of the Chern-Cartan (c.n.c.), hj k̄ hk̄l = δjl and Dδk∗ δj∗ = Hjk
δi , Dδk∗ ∂˙ i =
−H i ∂˙ j , D ˙ k δ ∗ = V ik δ ∗ , D ˙ k ∂˙ i = −V ik ∂˙ j , etc. Moreover, we have
jk
∂
j
j
i
∂
j
i
i
Hjk
= ∂˙ i Njk ; Hjk
ζi = Njk ,
(2.2)
i
and Chern-Cartan connection DΓ := (Nji , Hjk
, 0, Vjik , 0) is h∗ − and v ∗ − metrical, (for more details see [14]).
Further on, in order to simplify the writing, we use a bar over indices to
denote the complex conjugation of the variables or of the frames, e.g., ζk̄ := ζ̄k
or ∂˙ k̄ := ∂˙ k , in some places.
In [5] we investigated some classes of complex Cartan spaces. A complex
∗i
Cartan space (M, C) is called Kähler-Cartan iff Tjk
ζi = 0 and weakly Kähler∗i
∗i
i
i
Cartan iff Tjk
ζi ζ j = 0, where Tjk
:= Hjk
− Hkj
is the h∗ −torsion and ζ j :=
hm̄j ζm̄ . A necessary and sufficient condition for Kähler-Cartan property is
Njk = Nkj , [6]. The space (M, C) is called Berwald-Cartan iff the coefficients
i
Hjk
depend only on the position z. This condition is equivalent with the
holomorphicity in ζ of the local coefficients Nji of Chern-Cartan (c.n.c.), (i.e.
∂˙ ı̄ Njk = 0).
Another complex linear connection associated to the Chern-Cartan (c.n.c.)
i
ı̄
is one of Berwald type BΓ := (Nji , Bjk
:= ∂˙ i Njk , Bj̄k
:= (∂˙ ı̄ Nks )ζ s ζj̄ , 0, 0),
i
i
which is neither h∗ − nor v ∗ − metrical, but it has Bjk
= Hjk
. When BΓ is
∗
h − metrical, (M, C) is called complex Landsberg-Cartan space in [5]. This is
equivalent with either BΓ is of (1, 0) - type or the functions Lı̄k := (∂˙ ı̄ Nks )ζ s
vanish identically. Note that any complex Berwald-Cartan space is a complex
Landsberg-Cartan space and any Landsberg-Cartan space with weakly KählerCartan property is a Kähler-Cartan space, (Theorem 3.3 from [5]). Also, any
complex Kähler-Cartan space is a complex Landsberg-Cartan space.
Moreover, on a complex Cartan space, the complex Cartan tensors have
the expressions: V m̄ik = ∂˙ k hm̄i and its complex conjugate V m̄ı̄k = ∂˙ ı̄ hm̄k .
The homogeneity of C leads to V m̄ik ζi = 0 and V m̄ı̄k ζı̄ = 0. Moreover, it is
know that V m̄ik = 0 iff C is purely Hermitian.
3
Product complex Cartan spaces
In this section, we turn to account some ideas from [18], to construct the complex Cartan product metrics. We consider two complex Cartan spaces (M1 , C1 )
and (M2 , C2 ) of complex dimensions m and n, respectively. The complex coordinate in the local charts on M1 and on M2 are denoted by (z α )α=1,m and
0
(z α )α0 =m+1,m+n , respectively. On the manifolds T 0∗ M1 and T 0∗ M2 the points
GEOMETRY OF PRODUCT COMPLEX CARTAN MANIFOLDS
29
0
are characterized by the coordinates (z α , ζα )α=1,m and (z α , ζα0 )α0 =m+1,m+n .
Here and further, we use the Greek letters α, β, γ, ... and α0 , β 0 , γ 0 , ... for the
ranges of indices 1, m and m + 1, m + n, respectively. Also, for the complex
Cartan spaces (M1 , C1 ) and (M2 , C2 ) the fundamental metric tensors are denoted by
0 0
∂2K
∂2L
k β̄α :=
and lβ̄ α :=
,
∂ζα ∂ ζ̄β
∂ζα0 ∂ ζ̄β 0
respectively, where K := C21 and L := C22 . Moreover, making the notations
0
0
k α := ∂˙ α K and lα := ∂˙ α L, the homogeneity of both K and L gives k α ζα =
0
0 0
K, k β̄α ζβ̄ ζα = K, lα ζα0 = L and lβ̄ α ζβ̄ 0 ζα0 = L.
The product M := M1 × M2 is a m + n − dimensional complex manifold
and z = (z k )k=1,m+n are the local complex coordinates. Since there is a
natural isomorphism between T 0∗ M and T 0∗ M1 ⊕ T 0∗ M2 , (z i , ζi )i=1,m+n are
the local complex coordinate on the complex manifold T 0∗ M. We use the Latin
letters i, j, k, ... for the ranges of indices 1, m + n. We consider a function
0∗ M × T^
0∗ M ⊂ T
0∗ M → R+ given by
^
C : T^
1
2
q
C(z, ζ) = f (K(z α , ζα ), L(z α0 , ζα0 )),
(3.1)
where f is a real valued smooth function, f : R+ × R+ → R+ , which satisfies
the conditions:
a) f (K, L) = 0 if and only if (K, L) = (0, 0);
b) f (λK, λL) = λf (K, L), for ∀λ ≥ 0;
∂f
c) fK 6= 0, fL 6= 0 and fK fL − f fKL 6= 0 for K, L 6= 0, where fK := ∂K
,
fL :=
∂f
∂L ,
fKL :=
∂2f
∂K∂L
= fLK .
The homogeneity condition b) implies:
2
KfK + LfL = f ; KfKK + LfKL = 0 ; KfKL + LfLL = 0 ; fKL
= fKK fLL .
The function C defined in (3.1) can be a complex Cartan metric. Indeed,
the properties of the function f asserts the conditions i) − iii) from Definition
0
2.1. With the notations H := C2 = f (K(z α , ζα ), L(z α , ζα0 )), ζ α := ∂˙ α H,
0
0
ζ α := ∂˙ α H and ζ i := ∂˙ i H we find the explicit form of the Hermitian matrix
2
0
∂ H
(hβ̄α ) (hβ̄ α )
j̄i
=
=
h (z, ζ)
,
0
0 0
(hβ̄α ) (hβ̄ α )
i,j=1,m+n
∂ζi ∂ ζ̄j i,j=1,m+n
where
(hβ̄α ) :=
0
(hβ̄ α )
∂2H
= fKK (k α k β̄ ) +
∂ζα ∂ ζ̄β
2
0
:= ∂ζ∂α ∂Hζ̄ 0 = fKL (k α lβ̄ );
β
fK (k β̄α );
GEOMETRY OF PRODUCT COMPLEX CARTAN MANIFOLDS
0
0
∂2H
= fKL (lα k β̄ );
∂ζα0 ∂ ζ̄β
2
0
0
:= ∂ζ∂ 0 ∂Hζ̄ 0 = fLL (lα lβ̄ )
α
β
30
(hβ̄α ) :=
(hβ̄
0
α0
)
+ fL (lβ̄
0
α0
).
The same technique as in [18] lead us to the following result:
Theorem 3.1. The pair (M := M1 × M2 , C), with C defined by (3.1), is a
complex Cartan space, called product complex Cartan space, if and only if the
function f satisfies the conditions:
fK
fK + KfKK
> 0 ; fL > 0 ; fK fL − f fKL > 0;
> 0 ; fL + LfLL > 0.
Moreover,
0 0
m−1 n−1
i) det(hj̄i ) = fK
fL (fK fL − f fKL ) det(k β̄α ) det(lβ̄ α );
ii) The inverse of (hj̄i ) is
hαβ̄ hαβ̄ 0
,
hij̄ =
hα0 β̄ hα0 β̄ 0
(3.2)
where
hαβ̄ :=
fL fKK
1
ζα ζβ̄ ]; hαβ̄ 0 := − fKL
fK [kαβ̄ −
A
A ζα ζβ̄ 0 ;
fKL
fK fLL
1
hα0 β̄ := − A ζα0 ζβ̄ ; hα0 β̄ 0 := fL [lα0 β̄ 0 − A ζα0 ζβ̄ 0 ];
A := fK fL + LfK fLL + KfL fKK and (kαβ̄ ) and (lα0 β̄ 0 ) are the inverse of
0 0
k and lβ̄ α , respectively.
β̄α
Now, we consider the local coefficients of Chern-Cartan (c.n.c.) associated
to the complex Cartan metrics C1 and C2 :
0
N̂αβ = −kαδ̄
0
∂ ∗ k δ̄γ
∂ ∗ lδ̄ γ
0 β 0 = −l 0 0
ζ
;
Ň
ζγ 0 .
γ
α
α δ̄
∂z β
∂z β 0
Next, by means of these, a technical computation yields the expressions of
the local coefficients of Chern-Cartan (c.n.c.), associated to product complex
Cartan space (M := M1 × M2 , C).
Lemma 3.1. Let (M := M1 × M2 , C) be the product complex Cartan space,
with C defined by (3.1). Then, the local coefficients of Chern-Cartan (c.n.c.)
corresponding to (M := M1 × M2 , C) are given by following
Nαβ Nα0 β
(Nji ) =
,
Nαβ 0 Nα0 β 0
where Nαβ = N̂αβ , Nα0 β 0 = N̆α0 β 0 and Nα0 β = Nαβ 0 = 0.
31
GEOMETRY OF PRODUCT COMPLEX CARTAN MANIFOLDS
Proof. It results by a trivial computation, taking into account (3.2) and
Nji = −hj k̄
∂ ∗ ˙ k̄
∂ ∗ ˙ γ̄
∂ ∗ ˙ γ̄ 0
0
(
∂
(
∂
H)
−
h
(∂ H),
H)
=
−h
j
γ̄
j
γ̄
∂z i
∂z i
∂z i
where
0
0
∂˙ γ̄ H = fK k γ̄ , ∂˙ γ̄ H = fL lγ̄ ;
∗
∗ γ̄δ
∂ ∗ ˙ γ̄
γ̄
(∂ H) = fKK ∂∂zK
+ fK ∂∂zkβ ζδ ;
β k
∂z∗β
∗
0
0
∂
γ̄
(∂˙ γ̄ H) = fKL ∂∂zK
;
β l
∂z∗β
∗
∂
∂
L
γ̄
γ̄
˙
(
∂
H)
=
f
k
KL
β0
β0
∂z
∂∗
∂z β 0
∂z
0δ0
∗ γ̄
0
∂ ∗ L γ̄ 0
(∂˙ γ̄ H) = fLL ∂z
+ fL ∂∂zl β0 ζδ0 .
β0 l
Indeed, we have
∂ ∗ ˙ γ̄ 0
∂ ∗ ˙ γ̄
Nαβ = −hαγ̄ ∂z
H)
β (∂ H) − hαγ̄ 0 ∂z β (∂
= − f1K [kαγ̄ −
∗
fL fKK
γ̄
ζα ζγ̄ ][fKK ∂∂zK
β k
A
∗ γ̄δ
+ fK ∂∂zkβ ζδ ] +
2
fKL
∂∗K
γ̄ 0
0
A ∂z β ζα ζγ̄ l
=
N̂αβ .
Analogous, we obtain the other components of the matrix (Nji ).
4
Some classes of product complex Cartan spaces
The expression of the Chern-Cartan (c.n.c.) of a product complex Cartan
space is very convenient, it makes that some properties of both complex Cartan manifolds (M1 , C1 ) and (M2 , C2 ) to be devolved on the product complex
Cartan manifold and, conversely.
Theorem 4.1. Let (M, C) be the product complex Cartan space of the complex
Cartan spaces (M1 , C1 ) and (M2 , C2 ), with M := M1 × M2 and C defined by
(3.1).Then,
i) (M, C) is weakly Kähler-Cartan if and only if both spaces (M1 , C1 ) and
(M2 , C2 ) are weakly Kähler-Cartan.
ii) (M, C) is Kähler-Cartan if and only if both spaces (M1 , C1 ) and (M2 , C2 )
are Kähler-Cartan.
Proof. i) Corresponding to the product complex Cartan space (M, C) we have
∗i
∗i
∗i
j
Tjk
ζi ζ j = (Tjβ
ζi ζ j , Tjβ
0 ζi ζ )
and
0
∗i
Tjk
ζi ζ j = (Njk − Nkj )ζ j = (Nαk − Nkα )ζ α + (Nα0 k − Nkα0 )ζ α .
Using Lemma 3.1, it results
32
GEOMETRY OF PRODUCT COMPLEX CARTAN MANIFOLDS
0
∗i
Tjβ
ζi ζ j = (Nαβ − Nβα )ζ α + (Nα0 β − Nβα0 )ζ α
∗γ
= fK (N̂αβ − N̂βα )k α = fK T̂αβ
ζγ ζ α and
0
∗i
j
Tjβ
= (Nαβ 0 − Nβ 0 α )ζ α + (Nα0 β 0 − Nβ 0 α0 )ζ α
0 ζi ζ
0
0
0
= fL (N̆α0 β 0 − N̆β 0 α0 )lα = fL T̆α∗γ0 β 0 ζγ 0 ζ α ,
0
0
∗γ
where the functions T̂αβ
ζγ ζ α and T̆α∗γ0 β 0 ζγ 0 ζ α contain the horizontal torsion
coefficients associated to the spaces (M1 , C1 ) and (M2 , C2 ), respectively. Due
∗γ
∗i
the above relations, we obtain that Tjk
ζi ζ j = (0, 0) iff T̂αβ
ζγ ζ α = 0 and
0
0
T̆α∗γ0 β 0 ζγ 0 ζ α = 0 which prove i).
∗i
ζi ) =
ii) Using again Lemma 3.1, for the product space (M, C) we have (Tjk
N̂αβ − N̂βα
0
(Njk −Nkj ) =
, which attest that Njk = Nkj
0
N̆α0 β 0 − N̆β 0 α0
iff N̂αβ = N̂βα and N̆α0 β 0 = N̆β 0 α0 .
Theorem 4.2. Let (M, C) be the product complex Cartan space of the complex
Cartan spaces (M1 , C1 ) and (M2 , C2 ), with M := M1 × M2 and C defined by
(3.1).Then,
i) (M, C) is Landsberg-Cartan if and only if both spaces (M1 , C1 ) and
(M2 , C2 ) are Landsberg-Cartan.
ii) (M, C) is Berwald-Cartan if and only if both spaces (M1 , C1 ) and
(M2 , C2 ) are Berwald-Cartan.
Proof. i) The functions Lı̄k associated to the product space (M, C) can be writ!
β̄
β̄
L
L
0
0
γ
γ
.
ten as Lı̄k = (∂˙ ı̄ Nks )ζ s = (∂˙ ı̄ Nkα )ζ α +(∂˙ ı̄ Nkα0 )ζ α and (Lı̄k ) =
0
0
Lβ̄γ Lβ̄γ 0
Taking into account Lemma 3.1, it get
0
β̄
N̂ )k α = fK L̂β̄γ ;
Lβ̄γ = (∂˙ β̄ Nγα )ζ α + (∂˙ β̄ Nγα0 )ζ α = fK (∂˙γα
0
β̄
β̄
α
β̄
α
Lγ 0 = (∂˙ Nγ 0 α )ζ + (∂˙ Nγ 0 α0 )ζ = 0;
0
0
0
0
Lβ̄ = (∂˙ β̄ Nγα )ζ α + (∂˙ β̄ Nγα0 )ζ α = 0 and
γ
0
0
0
0
0
0
0
Lβ̄γ 0 = (∂˙ β̄ Nγ 0 α )ζ α + (∂˙ β̄ Nγ 0 α0 )ζ α = fL (∂˙ β̄ N̆γ 0 α0 )lα = fL L̆β̄γ 0 , where the
0
0
0
functions L̂β̄ := (∂˙ β̄ N̂γα )ζ α and L̆β̄0 := (∂˙ β̄0 0 N̆ )ζ α correspond to the spaces
γ
γ
γ α
0
(M1 , C1 ) and (M2 , C2 ), respectively. Thus, Lı̄k = 0 iff L̂β̄γ = 0 and L̆β̄γ 0 = 0.
γ̄
∂˙ N̂αβ
0
ii) Using Lemma 3.1 it implies (∂˙ ı̄ Njk ) =
and so,
0
0
∂˙ γ̄ N̆α0 β 0
∂˙ ı̄ Njk = 0 iff N̂αβ and N̆α0 β 0 are holomorphic in ζγ̄ and ζγ̄ 0 , respectively.
Corollary 4.1. Let (M, C) be the product complex Cartan space of the complex
Cartan spaces (M1 , C1 ) and (M2 , C2 ), with M := M1 × M2 and C defined by
33
GEOMETRY OF PRODUCT COMPLEX CARTAN MANIFOLDS
(3.1). If one of the spaces (M1 , C1 ) and (M2 , C2 ) is Kähler-Cartan and the
other is Berwald-Cartan, then (M, C) is Landsberg-Cartan.
Proof. Under our assumptions, the spaces (M1 , C1 ) and (M2 , C2 ) are
Landsberg-Cartan. Thus, applying Theorem 4.2 i), it results our claim.
Further on, we find the explicit form of the complex Cartan tensors V j̄ik
associated to the product complex Cartan space (M := M1 × M2 , C), with C
defined by (3.1). The complex Cartan tensors corresponding to the metrics
C1 and C2 are following
0 0 0
0
0 0
V̂ β̄αγ := ∂˙ γ k β̄α and V̆ β̄ α γ := ∂˙ γ lβ̄ α ,
respectively, and their complex conjugates. A direct computation leads to the
result.
Lemma 4.1. Let (M, C) be the product complex Cartan space of the complex
Cartan spaces (M1 , C1 ) and (M2 , C2 ), with M := M1 × M2 and C defined by
(3.1). Then,
0
V j̄ik = (V β̄αγ , V β̄αγ , V β̄
0
αγ
, V β̄
0
αγ 0
0
0
0
, V β̄α γ , V β̄α γ , V β̄
0
α0 γ
, V β̄
0
α0 γ 0
),
where
V β̄αγ
= ∂˙ γ hβ̄α = fKKK k β̄ k α k γ
β̄
δ̄αγ
V
β̄αγ 0
V β̄
V β̄
0
0
αγ
0
0
V β̄α γ
V β̄
0
0
β̄α γ
+fKK (V̂
ζδ̄ k + k k + k k ) + fK V̂
0
0
γ
β̄α
= ∂˙ h = (fKKL k β̄ k α + fKL k β̄α )lγ ;
0
0
= ∂˙ γ hβ̄ α = (fKKL k α k γ + fKL V̂ δ̄αγ ζ )lβ̄ ;
β̄αγ
;
δ̄
αγ 0
V β̄α γ
V β̄
(4.3)
β̄γ α
0
α0 γ
α0 γ 0
0
0
0
0
0
= ∂˙ γ hβ̄ α = (fKLL lβ̄ lγ + fKL lβ̄γ )k α ;
0
0
= ∂˙ γ hβ̄α = (fKKL k β̄ k γ + fKL k β̄γ )lα ;
0
0
0 0 0
= ∂˙ γ hβ̄α = (fKLL lα lγ + fKL V̆ δ̄ α γ ζδ̄0 )k β̄ ;
0 0
0
0
0 0
= ∂˙ γ hβ̄ α = (fKLL lβ̄ lα + fKL lβ̄ α )k γ ;
0
0 0
0
0
0
= ∂˙ γ hβ̄ α = fLLL lβ̄ lα lγ
0
0
0
0
+fLL (V̆ δ̄ α γ ζδ̄0 lβ̄ + lβ̄
0
γ 0 α0
l
+ lβ̄
0
α0 γ 0
l ) + fL V̆ β̄
0
α0 γ 0
.
Theorem 4.3. Let (M, C) be the product complex Cartan space of the complex
Cartan spaces (M1 , C1 ) and (M2 , C2 ), with M := M1 × M2 and C defined by
(3.1).Then, C is a purely Hermitian metric if and only if C1 and C2 are purely
Hermitian metrics and
f (K, L) = c1 K + c2 L, c1 , c2 > 0.
GEOMETRY OF PRODUCT COMPLEX CARTAN MANIFOLDS
34
Proof. We suppose that C is a purely Hermitian metric. Then, V j̄ik = 0
and contracting in (4.3) with suitable combinations of ζ∗¯ ζ∗ ζ∗ , where ∗ ∈
{α, β, γ, α0 , β 0 , γ 0 }, we obtain the following system of equations
KfKKK + 2fKK
fKKL
=
0 ; LfLLL + 2fLL = 0
= fKLL = fKL = 0.
Now, the homogeneity condition b) implies: fKK = 0 and fLL = 0 which
give f (K, L) = c1 K + c2 L, c1 , c2 > 0. Replacing this in (4.3), it results
0 0 0
fK V̂ β̄αγ = fL V̆ β̄ α γ = 0. Since fK > 0 and fL > 0, the last conditions
0 0 0
leads to V̂ β̄αγ = V̆ β̄ α γ = 0, i.e. the metrics C1 and C2 are purely Hermitian.
Conversely, if C1 and C2 are purely Hermitian metrics and f (K, L) = c1 K +
c2 L, c1 , c2 > 0, then (4.3) is reduced to V j̄ik = 0 which completes the proof.
Since any purely Hermitian complex Cartan metric is itself one BerwaldCartan, by Theorem 4.2 ii) we have justified the following result.
Corollary 4.2. Let (M, C) be the product complex Cartan space of the complex
Cartan spaces (M1 , C1 ) and (M2 , C2 ), with M := M1 × M2 and C defined by
(3.1). If C1 and C2 are purely Hermitian metrics, then (M, C) is a BerwaldCartan space.
If we choose the function f : R+ × R+ → R+ to be
p
f (K, L) = aK 2 + bKL + cL2 ,
(4.4)
with a, b, c ≥ 0, a2 + b2 + c2 6= 0, then
1
C(z, ζ) = (aK 2 + bKL + cL2 ) 4
(4.5)
is a complex Cartan metric on the product manifold M := M1 ×M2 . According
to Theorem 4.3, C is a purely Hermitian metric if and only if b2 = 4ac and the
complex Cartan metrics K and L are purely Hermitian.
So, considering two purely Hermitian complex Cartan metrics K and L,
the relation (4.5) with a, b, c ≥ 0 and b2 6= 4ac, gives us examples of nonpurely Hermitian complex Cartan metrics. Moreover, these obtained metrics
are Berwald-Cartan.
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Nicoleta ALDEA,
Department of Mathematics and Computer Science,
Transilvania University of Brasov,
Iuliu Maniu 50, 500091 Brasov, Romania.
Email: nicoleta.aldea@lycos.com
Gheorghe MUNTEANU,
Department of Mathematics and Computer Science,
Transilvania University of Brasov,
Iuliu Maniu 50, 500091 Brasov, Romania.
Email: gh.munteanu@unitbv.ro