Download symmetry properties of sasakian space forms

SOOCHOW JOURNAL OF MATHEMATICS
Volume 31, No. 4, pp. 611-616, October 2005
SYMMETRY PROPERTIES OF SASAKIAN SPACE FORMS
BY
MOHAMED BELKHELFA, RYSZARD DESZCZ AND LEOPOLD VERSTRAELEN
Dedicated to Professor Dr. Zbigniew Olszak on his 60th birthday
Abstract. The only pseudo-symmetric Kählerian manifolds of dimension n ≥ 6 are
the semi-symmetric ones ([5], [6]). A similar result was obtained for Ricci pseudosymmetry ([10]). Therefore Olszak introduced a variant of pseudo-symmetric
Kählerian manifolds [10]. For dimension 4 Olszak gave an example of non semisymmetric pseudo-symmetric Kählerian manifold [11]. In contrast however there
do exist pseudo-symmetric Sasakian manifolds which are not semi-symmetric. In
the present paper, we investigate the pseudo-symmetry of Sasakian space forms.
1. Sasakian Manifold
Let M = (M 2n+1 , g) be a (2n + 1)-dimensional Riemannian manifold and
let (φ, ξ, η) be tensor fields of type (1, 1), (1, 0) and (0, 1) respectively on M ,
such that: φ2 (X) = −X + η(X)ξ, η ◦ φ = 0, η(ξ) = 1, g(φX, φY ) = g(X, Y ) −
η(X)η(Y ), for all vector field X, Y of M . If in addition, dη(X, Y ) = g(X, φY ),
then M is called contact Riemannian manifold. If, moreover M is normal, i.e. if
φ2 [X, Y ]+[φX, φY ]−φ[φX, Y ]−φ[X, φY ]+2dη⊗ξ = 0, then M is called Sasakian
manifold, for more details we refer to [3], [4], [17]. The sectional curvature of the
plane section spanned by the unit tangent vector field X orthogonal to ξ and φX
is called a φ-sectional curvature. If M has a constant φ-sectional curvature c,
then M is called a Sasakian space forms and denoted by M 2n+1 (c) ([3], [4], [17]).
The Riemannian curvature tensor of Sasakian space forms is given by the
following formula
R(X, Y )Z =
c+3
(g(Y, Z)X − g(X, Z)Y )
4
Received May 3, 2004; revised January 4, 2005.
AMS Subject Classification. 53B20, 53C25.
Key words. semi-symmetry, Ricci-symmetry, pseudo-symmetry, Sasakian space forms.
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MOHAMED BELKHELFA, RYSZARD DESZCZ AND LEOPOLD VERSTRAELEN
c−1
(η(X)η(Z)Y − η(Y )η(Z)X)
4
c−1
{g(X, Z)η(Y )ξ − g(Y, Z)η(X)ξ
+
4
+g(Z, φY )φX − g(Z, φX)φY
+
+2g(X, φY )φZ}.
(1.1)
The Ricci curvature S and the scalar curvature κ are given by
S(X, Y ) =
n(c + 3) + c − 1
(n + 1)(c − 1)
g(X, Y ) −
η(X)η(Y )
2
2
and
1
(n(2n + 1)(c + 3) + n(c − 1)).
2
Example 1.1. We consider R2n+1 with coordinates (xi , y i , z), i = 1, . . . , n
P
and its usual contact form 12 (dz − ni=1 y i dxi ). The characteristic vector field ξ is


0 δij 0


∂

given by ξ = 2 ∂z
, the tensor field φ is given by the matrix 
 −δij 0 0  and the
κ=
0
2
yj 0
2
Riemannian metric g = η ⊗ η + 41 ni=1 ((dxi ) + (dy i ) ) is an associated metric for
η, with this metric R2n+1 is a Sasakian form with φ-sectional curvature c = −3
denoted by R2n+1 (−3).
P
For more details, see [3], [4].
2. Pseudo-Symmetry
In the following, by R we will also denote the curvature operator, such that
R(X, Y ) = ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] .
As is well known, every (1, 1)-tensor field A on a differential manifold determines
a derivation A· of the tensor algebra on this manifold, which commutes with
contractions. In particular, the anti-symmetric (1, 1)-tensor field R(X, Y ) induce
derivations R(X, Y )·, thus associating with any (0, k)-tensor fields T, the (0, 2+k)tensor field R · T defined by
(R · T )(X1 , X2 , . . . , Xk ; X, Y ) = R(X, Y ) · (T (X1 , X2 , . . . , Xk ))
= −T (R(X, Y )X1 , X2 , . . . , Xk ) − · · · − T (X1 , X2 , . . . , R(X, Y )Xk )
(2.1)
SYMMETRY PROPERTIES OF SASAKIAN SPACE FORMS
613
one has
R(X, Y ) · T = ∇X (∇Y T ) − ∇Y (∇X T ) − ∇[X,Y ] T.
By R · R we denote in this way the (0, 6)-tensor field obtained by the derivation
of the second R, the (0, 4)-curvature tensor, using the curvature operator, R·.
When
R · R = 0,
(2.2)
then M is called a semi-symmetric space. Clearly all locally symmetric spaces
and all 2-dimensional Riemannian space are semi-symmetric. Through studies of
semi-symmetric Riemmannian manifolds were done by Z.I. Szabó [12], [13], [14].
To state the definition of the pseudo-symmetry, we first determine a specific
field (0, k + 2)-tensor field Q(g, T ) associated with any (0, k)-tensor field T on
Riemannian manifold M by
Q(g, T )(X1 , X2 , . . . , Xk ; X, Y ) = (X ∧g Y ) · (T (X1 , X2 , . . . , Xk ))
(2.3)
= −T ((X ∧g Y )X1 , X2 , . . . , Xk ) − · · · − T (X1 , X2 , . . . , Xk−1 , (X ∧g Y )Xk ),
where X ∧g Y is the endomorphism given by
(X ∧g Y )Z = g(Y, Z)X − g(X, Z)Y.
A Riemannian manifolds M is said to be pseudo-symmetric (in the sense of R.
Deszcz) if the (0, 6)-tensor fields R · R and Q(g, R) on M are linearly dependent,
i.e. if there exists a function
LR : M −→ R
such that
R · R = LR Q(g, R)
holds on UR = {x ∈ M | R −
κ
n(n−1) G
(2.4)
6= 0 at x} where G is the (0, 4) tensor field
of M defined by G(X1 , X2 , X3 , X4 ) = g((X1 ∧g X2 )X3 , X4 ) ([1], [2], [7], [16]).
Clearly, all semi-symmetric manifolds are pseudo-symmetric. The converse
is not true: there are many proper pseudo-symmetric manifolds, i.e. pseudosymmetric manifolds which are not semi-symmetric. In this paper we give other
examples.
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MOHAMED BELKHELFA, RYSZARD DESZCZ AND LEOPOLD VERSTRAELEN
Theorem 2.1.([9]) If a normal contact space is a symmetric one, it is a
space of constant curvature one.
Theorem 2.2.([15]) A Sasakian manifold is semi-symmetric if and only if
it is a space of constant curvature one.
Theorem 2.3. Every Sasakian space forms M 2n+1 (c) is pseudo-symmetric,
more precisely for every Sasakian space forms:
R · R = Q(g, R).
Proof. As is well known that the Riemannian curvature tensor of every
Sasakian manifold satisfies the following property R(X, ξ)Z = (X ∧ ξ)Z. Therefore
R(X, ξ) · R = (X ∧ ξ) · R.
(2.5)
We remark that in a Sasakian space forms M which is not a real space form, both
the left and right hand side of (2.5) are different from zero, so that if M is pseudosymmetric the coefficient function necessarily has to be 1. Then to proof that M is
effectively pseudo-symmetric it remains to prove that R(X, Y ) · R = (X ∧ Y ) · R,
for all X, Y orthogonal to ξ. First we remark that the Riemannian curvature
tensor of M can be written as follows:
R(X, Y ) = X ∧ Y +
c−1 2
{φ X ∧ φ2 Y + φX ∧ φY + 2g(X, φY )φ}.
4
(2.6)
We observe that in any Sasakian manifold: φ · R = 0, as can be verified using
the property that R(X, Y ) − X ∧ Y = R(φX, φY ) − φX ∧ φY ([17]) or can be
obtained from [3, p.93]. From (2.6) we have thus
R(X, Y ) · R = (X ∧ Y ) · R +
c−1
{(φ2 X ∧ φ2 Y ) · R + (φX ∧ φY ) · R}. (2.7)
4
Therefore it is needed to prove that for any X, Y perpendicular to ξ, the following
holds (φX ∧ φY ) · R = −(X ∧ Y ) · R, which can be verified by a long but straight
forward computations.
Corollary 2.4. The Sasakian space forms R 2n+1 (−3) is pseudo-symmetric
but not semi-symmetric.
SYMMETRY PROPERTIES OF SASAKIAN SPACE FORMS
615
Denoting X ∧±φ Y = X ∧ Y ± φX ∧ φY, respectively, we obtain the following
characterization of Sasakian space forms.
Proposition 2.5. Let M 2n+1 , n ≥ 2, be a Sasakian manifold. Then, (X ∧ +φ
Y ) · R = 0 for all X, Y orthogonal to ξ if and only if M 2n+1 is a Sasakian space
form.
Proof. We already proved that in a Sasakian space forms we have ((X ∧ +φ
Y )·R) = 0. Next we assume that (X ∧+φ Y )·R = 0, for all X, Y orthogonal to ξ in
M 2n+1 . Assuming that X, Y are orthonormal, by straight-forward computations,
we find that
((X ∧+φ Y ) · R)(Y, φX, X, φX)
(2.8)
= R(X, φX, φX, X) + R(Y, φX, Y, φX) + R(Y, φY, X, φX) + R(Y, φX, X, φY )
((X ∧+φ Y ) · R)(X, φY, Y, φY )
(2.9)
= −R(Y, φY, φY, Y ) − R(X, φY, X, φY ) − R(X, φX, Y, φY ) − R(X, φY, Y, φX).
By (2.8), (2.9) and the following property of the Riemannian curvature of any
Sasakian manifold: R(φX, Y ) + R(X, φY ) = φX ∧ Y − φY ∧ X. We obtain that:
R(X, φX, φX, X) = R(Y, φY, φY, Y ). This implies that M 2n+1 has constant φsectional curvature.
Remark 2.6. As it is well-known that, if a Sasakian manifold M 2n+1 ,
n > 3 is conformally flat or semi-symmetric it is necessary a space of constant
curvature, ([9], [15]). It is known (see e.g. [5]) that on every Riemannian manifold
of dimension ≥ 5 the conditions R · R = 0 and R · C = 0 are equivalent on the set
UC = {x ∈ M | C 6= 0 at x} ,
1
κ
where C(X, Y ) = R(X, Y ) − n−2
(X ∧g SY + SX ∧g Y − n−1
X ∧g Y ) is the Weyl
tensor.
Proposition 2.7. Every (2n + 1)-dimensional Sasakian manifold M 2n+1 ,
n > 1 is Weyl semi-symmetric if and only if it is a space of constant curvature
one.
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MOHAMED BELKHELFA, RYSZARD DESZCZ AND LEOPOLD VERSTRAELEN
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Centre Universitaire Mustapha Stombouli, Laboratoire de Physique Quantique de la Matiere et
Modelisations Mathmatiques - L.P.Q. 3M - Route de Mamounia, Mascara 29000 (Algerie).
E-mail: [email protected]
Department of Mathematics, Agricultural University of Wroclaw, Grunwaldzka 53, PL - 50-357
Wroclaw, Poland.
E-mail: [email protected]
Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B Heverlee Belgium.
E-mail: [email protected]