ON HOLOMORPHICALLY PROJECTIVE FLAT
PARABOLICALLY-KÄHLERIAN SPACES ∗
MOHSEN SHIHA
Department of Mathematics, P.O. Box: 249, Teachers Coll., Abha,
Kingdom of Saudi Arabia
E-mail: mohsen sheha@yahoo.com
JOSEF MIKEŠ †
Department of Algebra and Geometry, Fac. Sci., Palacky Univ., Tomkova 40,
779 00 Olomouc, Czech Republic
E-mail: josef.mikes@upol.cz
We consider holomorphically projective mappings of parabolically-Kählerian
spaces and define holomorphically projective flat parabolically-Kählerian spaces.
We found the tensor characteristic of these spaces and obtained their metric tensors.
1. Introduction
Many authors studied holomorphically projective mappings of Kählerian
spaces and their generalizations [1, 17]. Some facts from the theory of holoo(m)
morphically projective mappings of parabolically-Kählerian spaces Kn
were published in [2, 9]−[15].
o(m)
is said to be parabolically-Kählerian
A (pseudo-) Riemannian space Kn
space if together with a metric tensor gij (x) it possesses an affinor structure
Fih (x) of rank m ≥ 2 satisfying the following relations
a) Fαh Fiα = 0,
b)
giα Fjα + gjα Fiα = 0,
∗
c)
h
Fi,j
= 0,
(1)
MSC 2000: 53B20, 53B30.
Keywords: holomorphically projective flat space, holomorphically projective mapping,
parabolically Kählerian space.
This paper is dedicated to Professors Ivan Kolár̆ and Oldr̆ich Kowalski in occasion of
their 70–ties.
† Work supported by the Grant No 201/05/2707 of The Czech Science Foundation and
by the Council of Czech Government MSM No 6198959214.
467
468
where the comma denotes the covariant derivation.
2. Holomorphically projective mappings of
parabolically-Kählerian spaces
The following criteria from the papers [10, 13] hold for holomorphically
o(m)
projective mappings from a parabolically-Kählerian space Kn
onto a
o(m)
parabolically-Kählerian space K̄ n .
o(m)
An analytically planar curve of the parabolically-Kählerian space Kn
is a curve defined by the equations xh = xh (t) which tangent vector
λh = dxh /dt, being translated, remains in the area element formed by
the tangent vector λh and its conjugate λα Fαh , i.e., the conditions
dλh
+ Γhαβ λα λβ = ρ1 (t)λh + ρ2 (t)λα Fαh ,
dt
are fulfilled. Here Γhij is the Christoffel symbol and ρ1 , ρ2 are functions of
the argument t.
o(m)
o(m)
is a holomorphically projeconto K̄ n
The diffeomorphism f of Kn
o(m)
into
tive mapping, if it transform all analytically planar curves of Kn
o(m)
analytically planar curves of K̄ n .
o(m)
o(m)
−→ K̄ n , both spaces being reConsider a concrete mapping f : Kn
ferred to the general coordinate system x with respect to this mapping.
o(m)
This is a coordinate system where two corresponding points M ∈ Kn
o(m)
have equal coordinates x = (x1 , x2 , . . . , xn ); the corand f (M ) ∈ K̄ n
o(m)
will be marked with a bar. For
responding geometric objects in K̄ n
o(m)
example, Γhij and Γ̄hij are components of the Christoffel symbols on Kn
o(m)
and K̄ n
, respectively.
o(m)
o(m)
and K̄ n
are preserved under f , i.e. F̄ih (x) =
Structures of Kn
h
Fi (x). Among others, the structure Fih is covariantly constant, and
ḡiα Fjα + ḡjα Fiα = 0 holds.
o(m)
It is proved in [10, 13] that a parabolically-Kählerian space Kn
admits a
holomorphically projective mapping f onto a parabolically-Kählerian space
o(m)
K̄ n
if and only if the following conditions (in the common coordinate
system x) hold:
Γ̄hij (x) = Γhij (x) + ψi δjh + ψj δih + ϕi Fjh + ϕj Fih ,
(2)
where ϕi is a covector, ψi = ϕα Fiα , and ψi (x) is a gradient, i.e. there is a
function ψ(x), such that ψi (x) = ∂ψ(x)/∂xi .
469
If ϕi 6≡ 0 then a holomorphically projective mapping is called nontrivial;
otherwise it is said to be trivial or affine.
Condition (2) is equivalent to
ḡij,k = 2ψk ḡij + ψi ḡjk + ψj ḡik + ϕi ḡjα Fkα + ϕj ḡiα Fkα .
o(m)
Under a holomorphically projective mapping f : Kn
following conditions hold:
(3)
o(m)
−→ K̄ n
h
R̄hijk = Rijk
+ ψij δkh − ψik δjh + ϕij Fkh − ϕik Fjh − (ϕjk − ϕkj )Fih ,
o(m)
h
h
where Rijk
and R̄ijk
are Riemannian tensors of Kn
ϕij = ϕi,j − ψi ϕj − ϕi ψj ,
ψij = ϕαj Fiα
o(m)
and K̄ n
, the
(4)
,
(= ψji = ψi,j − ψi ψj ).
(5)
3. Holomorphically projective flat parabolically-Kählerian
space
o(m)
is said to be holomorphically proA parabolically-Kählerian space Kn
jective flat, if it admits a holomorphically projective mapping onto a flat
space, i.e. the space with the vanishing Riemannian tensor.
We have the following theorem.
o(m)
is holomorphically
Theorem 3.1 The parabolically-Kählerian space Kn
projective flat if and only if the following conditions are true for the Riemannian tensor
Rhijk = c (2 Fhi Fjk + Fhj Fik − Fhk Fij )
(6)
where c = const, Fij = giα Fjα .
o(m)
admit a holomorphically
Proof. Let a parabolically-Kählerian space Kn
h
projective mapping onto a flat space V̄n (R̄ijk
= 0), which should be a pao(m)
same.
rabolically-Kählerian space K̄ n
h
If R̄ijk
= 0 then after omitting the index h (4) takes the form
Rhijk = −ψij gkh + ψik gjh − ϕij Fhk + ϕik Fhj + (ϕjk − ϕkj )Fhi .
(7)
Let us symmetrize (7) at indices h and i. Then, using the properties of the
Riemannian tensor we get:
0 = −ψij gkh + ψik gjh − ϕij Fhk + ϕik Fhj − ψhj gki
+ ψhk gji − ϕhj Fik + ϕhk Fij .
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Analyzing of this formula, we obtain ψij = 0 and
ϕij = c Fij ,
(8)
where c is a certain function. Thus (8) takes the form (6).
On the basis (5), formula (8) takes the form
ϕi,j = ψi ϕj + ϕi ψj + c Fij .
(9)
The condition of integrability takes the form: c,k Fij − c,j Fik = 0. From
foregoing one it is implied, that c,i = 0 and c = const.
So, we have shown that the Riemannian tensor at all holomorphically proo(m)
jective flat parabolically-Kählerian spaces Kn
satisfies (6).
o(m)
It is easy to check that any parabolically-Kählerian space Kn , in which
the Riemannian tensor satisfies (6), admits holomorphically projective mapo(m)
ping onto a flat space K̄ n .
Make sure that the system of equations (3) and (9) is completely integrable
o(m)
and has the solution ḡij (x), ϕi (x) for any initial conditions
in this Kn
o
ḡij (xo ) =ḡij
o
for which detk ḡij k 6≡ 0,
o
o
ḡij =ḡji
o
and ϕi (xo ) =ϕi
and
o
(10)
o
ḡiα Fjα (xo )+ ḡjα Fiα (xo ) = 0.
o(m)
admits a holomorphically projective mapConsequently, the space Kn
o(m)
with the metric tensor ḡij (x) and the structure
ping onto a space K̄ n
o(m)
h
is a flat space.
Fih (x). Using (4) we can see, that R̄ijk
= 0, hence K̄ n
This completes the proof.
The direct analysis of (6) leads us to the following
Lemma 3.1 A holomorphically projective flat parabolically-Kählerian
o(m)
space Kn
is a Ricci flat symmetric space, i.e. a Ricci tensor is vano(m)
ishing and the Riemannian tensor is covariantly constant in this Kn .
4. On isometries between holomorphically projective flat
parabolically-Kählerian spaces
o(m,c)
We denote Kn
a holomorphically projective flat parabolically-Kählerian space, which determined by (6), and prove the following theorem.
Theorem 4.1 Two holomorphically projective flat parabolically-Kählerian
o(m,c)
o(m̄,c̄)
spaces Kn
and K̄ n
are locally isometric if and only if m̄ = m, the
metric signatures are coincident, and the constants c and c̄ have the same
sign.
471
o(m,c)
o(m̄,c̄)
o(m,c)
o(m̄,c̄)
Proof. Let us consider the given spaces Kn
and K̄ n
which are
related to the coordinate systems x and x̄ respectively. It is natural to
consider the case, when the constants c and c̄ are not equal to zero.
We will search an isometric mapping f : Kn
−→ K̄ n
. As it is
known, the mapping f : x̄h = x̄h (x1 , x2 , . . . , xn ) is an isometric mapping if
and only if
gij (x) = ḡαβ (x̄(x))∂i x̄α ∂j x̄β .
(11)
Denote x̄hi ≡ ∂i x̄h . From (11) it follows that
∂i x̄h = x̄hi ,
β
α h
∂j x̄hi = Γ̄hαβ x̄α
i x̄j − Γij x̄α ,
o(m,c)
where Γhij and Γ̄hij are the Christoffel symbols of Kn
h
(12)
o(m̄,c̄)
and K̄ n
.
x̄hi (x)
The system (12) for the unknown functions x̄ (x),
has a solution
for initial conditions x̄h (xo ) = x̄ho and x̄hi (xo ) = yih , where the following
properties are satisfied
p
ḡαβ (x̄o )yiα yjβ = gij (xo ), Fiα (xo ) yαh = c̄/c F̄αh (x̄o ) yiα ,
(13)
o(m,c)
where Fih and F̄ih are the structures of Kn
o(m,c)
and K̄n
, respectively.
yih
Initial conditions
from (13) exist if only if m̄ = m, the signatures of the
metric g and ḡ are coincident, and the constants c and c̄ have the same
sign. Conditions (13) follow from (11) and from an integrability condition
β γ δ
of system (12): Rhijk = R̄αβγδ x̄α
h x̄i x̄j x̄k .
5. Holomorphically projective mappings of holomorphically
projective flat parabolically-Kählerian spaces
We can prove the next theorem in the similar way as Theorem 3.1.
Theorem 5.1 If the holomorphically projective flat parabolically-Kählerian
o(m,c)
space Kn
admits a holomorphically projective mapping onto some parao(m)
o(m)
bolically-Kählerian space K̄ n , then K̄ n
is a holomorphically projective
o(m,c̄)
flat parabolically-Kählerian space K̄ n
too.
In addition the next theorem holds
Theorem 5.2 Any holomorphically projective flat parabolically-Kählerian
o(m,c)
space Kn
admits a nontrivial holomorphically projective mapping onto
o(m,c̄)
some holomorphically projective flat parabolically-Kählerian space K̄ n
with a given constant c̄ and a given signature of the metric ḡij .
472
Proof. The availability of this theorem follows from the existence of the
solutions ḡij (x) and ϕi (x) of equations (3) and
ϕi,j = ψi ϕj + ϕi ψj + c Fij − c̄ F̄ij ,
o
α
where F̄ij = ḡiα Fj , for any initial conditions (10) for which det k ḡij k
o
o
o
o
o(m,c)
ḡij = ḡji and ḡiα Fjα (xo )+ ḡjα Fiα (xo ) = 0, in the space Kn
.
6= 0,
Theorem 5.3 Between any holomorphically projective flat parabolicallyKählerian spaces it is possible to establish a nontrivial holomorphically projective mapping.
Proof. Let us have two arbitrary holomorphically projective flat paraboo(m,c̄)
o(m,c)
. By Theorem 5.2, there exists
and K̄ n
lically-Kählerian spaces Kn
o(m,c)
o(m,c̄)
o(m,c)
, on which Kn
with a signature of a metric of K̄ n
some space K̃n
admits nontrivial holomorphically projective mapping. By Theorem 4.1,
o(m,c)
o(m,c̄)
are isometric, which prove the theorem.
and K̃n
the spaces K̄ n
6. Metric of holomorphically projective flat
parabolically-Kählerian spaces
In a symmetric space its a metric tensor may be rebuilt in some Riemannian
coordinate system (y 1 , y 2 , . . . y n ) at a point xo by the known formulas [5]
o
gij = gij +
∞
1 X (−1)k 22k+2 σ1 σ2
mi mσ1 · · · mσk−1 j ,
2
(2k + 2)!
(14)
k=1
o
o
o
o
o
where mij =Riαjβ y α y β , mij = miα gαi and gij , gij and Rhijk are the components of the metric, its inverse and Riemannian tensors at the point xo .
Taking into account the representation of Riemannian tensor (6) and properties of structures Fih the formulas (14) take the form:
o
o
o
gij = gij − c Fi Fj ,
(15)
where Fi = F iα y α , F ij are the components of tensor Fij at xo .
Note, that for a given point xo of holomorphically projective flat paraboo(m,c)
lically-Kählerian space Kn
the metric and structure tensors may be
simultaneously reduced to the form:




0 0 bab
0 0 0
o
o


∗
gij =  0 e 0  and F hi =  0 0 0  ,
Em 0 0
bTab 0 0
473
where bTab is a transposed matrix bab , a, b = 1, m, Em is the identity matrix,


0 1
0 
−1 0




e1
0


0 1


 e2




−1
0
 and e∗ = 

bab = 

 , ea = ±1.
.

.


.
..


.




0
en−2m

0 1
0
−1 0
Thus, we proved the following theorem.
Theorem 6.1 In the holomorphically projective flat parabolically-Kähleo(m,c)
there exists a coordinate system y in which the metric
rian space Kn
tensor has the form (15).
Not neglecting generality of reasons, on basic of theorem 4.1 we can consider
o(m,−1)
o(m,+1)
o(m,0)
.
and Kn
, Kn
c = 0, ±1 that is, spaces Kn
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