Normal complex contact metric manifolds admitting a semi symmetric metric con- nection

In this paper, we study on normalθcomplexθcontactθmetricθmanifold admitting a semi symmetric metric connection. We obtain curvature properties of a normalθcomplexθ contact metric manifold admitting a semi symmetric metric connection. We also prove that this type of manifold is not conformal flat, concircular flat, and conharmonic flat. Finally, we examine complex Heisenberg group with the semi symmetric metric connection.


Introduction
The Riemannian geometry of complex contact manifolds has been studied since 1970s. In the early 1980s some important developments were presented by Ishihara-Konishi. They obtained the normality conditions and curvature properties [9,10]. Due to some important features that are different from real contact geometry, in 2000s some researchers have taken their attention to this notion. Blair, Korkmaz and Foreman gave results for the Riemannian geometry of complex contact manifolds [2,4,5,8,12] . Also two of presented authors examined curvature and symmetry notions [15,16].
In Riemannian geometry the notion of connection gives information about transporting data along a curve or family of curves in a parallel and consistent manner. Affine connections and Levi-Civita connections are commonly used for to understand the geometry of manifolds. Levi-Civita connection is symmetric, i.e, has zero torsion, and also it is metric, i.e, the covariant derivation of metric vanish. In recent years some different connections were defined and worked on manifolds. One of them is semi symmetric metric connection. This type of connection were defined by Hayden and this was developed by Yano [13].
In this paper, we study on normalθ complexθ contactθ metricθ manifold with a semi symmetric metric connection. Firstly, we give some basic properties. Our starting point was the non-vanishing of special curvature tensors (conformal, concircular, quasi-conformal etc.) on normalθ complexθ contact manifolds with canonic connection. We research the flatness conditions of these special tensors on normalθ complexθ contactθ metric manifold with a semi symmetric metric connection. We proved that a normalθ complexθ contactθ metricθ manifold admitting the semi symmetric metric connection is not conformal flat, concircular flat and conharmonic flat. Finally we apply our results to complex-Heisenberg group as a wellknown example of normalθ complexθ contactθ metricθ manifolds.

Preliminaries
In 1959 Kobyasahi [11] gave the definition of a complex contact manifold. A complex contact manifold is a (2m + 1)−complex dimensional complex manifold with a holomorphic 1−form ω such that ω ∧ (dω) m = 0. ω is not globally defined. For an open covering by coordinate neighborhoods Complexθ almostθ contact structure on a complexθ contactθ manifold were given by Ishihara-Konishi [10]. For a Hermitian metric g and complex structure J, we have 1-forms u and v = u • J, with dual vector fields U and V = −JU, and (1, 1) tensor fields G and H = GJ such that Also there are functions a and b on O ∩ O = / 0 such that u = au − bv, v = bu + av, With these properties M is said to be a complex almost contact metric manifold. On the other hand the verical subbundle of T M is spanned by U,V i.e V = sp{U,V }. Thus we have T M ∼ = H ⊕ V . Also 2−forms du, dv are defined as follow; where σ (Z) = g(∇ Z U,V ) [10]. σ is called IK-connection [7]. Also if complex contact 1-form is globally defined then σ vanishes.
There are two normality notions for a complex almost contact metric manifold in literature. The fundamental difference between of these normality notions is to be a Kähler manifold. IK-tensors are given as below; As similar to φ −sectional curvature in real contact geometry, in complex contact geometry the definition of G H −sectional curvature were given.

Definition 2.
[12] Let M be a normalθ complexθ contactθ metricθ manifold. Z be an unit horizontal vector field on M and a 2 + b 2 = 1. A G H −section is a plane which is spanned by Z and T = aGZ + bHZ and the sectional curvature of this plane is called G H −sectional curvature .

Normal Complex Contact Metric Manifolds Admitting a Semi Symmetric Metric Connection
In this section the definition of a semi symmetric metric connection are given for normalθ complexθ contact metricθ manifolds. Some basic equalities are computed via this connection. Let M be normalθ complexθ contactθ metricθ manifold and define ∇ : where ∇ is Levi-Cevita connection on M, U,V are the structure vector fields and u, v are dual 1−forms. It can be easily showed that ∇ is an linear connection. Also we could write the torsion tensor field of ∇ as follow; As we see ∇ is not torsion free and it is also a semi symmetric metric connection. In addition we have Lie bracket operator [Z, T ] = [Z, T ]. For brevity we use a abbreviation "NCCMM" for normalθ complexθ contactθ metricθ manifold , and (M, ∇) for a normalθ complexθ contactθ metricθ manifold M admitting a semi symmetric metric connection ∇.
for an arbitrary vector field T .
Proof. Let T be an arbitrary vector field on (M, ∇). From (1) we have Similarly we get Corollary 2. Onθ (M, ∇)θ weθ have for arbitrary vector field Z on M.

Curvature Properties of Normal Complex Contact Metric Manifolds Admitting a Semi Symmetric Metric Connection
The Riemannian and Ricci curvature properties of (M, ∇) is given in this section.
where T,W, Z are arbitrary vector fields on M and R, R are the Riemannian curvature tensor of ∇ and ∇ , respectively.
Proof. It is known that for arbitrary vector fields X,Y, Z on M, the Riemannian curvature R is given by From (1) we obtain ∇ T ∇ W Z, ∇ W ∇ T Z and ∇ [W,W ] Z as below: By consider all these equalities we get 6.
Also we have As we know that Riemannian curvature tensor R has some symmetric properties. The Riemannian curvature tensor R of (M, ∇) has the following symmetry properties.
Also similar to Bianchi identity for R we have As we know for a normalθ complexθ contactθ metricθ manifold [15], we have These results let us to obtain curvature properties of (M, ∇).
An other geometric important object in the complex contact geometry is dσ . In [15] an equality for dσ . By following Proposition we present a new version of dσ on a normalθ complexθ contactθ metric manifold M was obtained.
for all Z, T ∈ Γ(T M).

For brevity let state
Thus by direct computations the proof is completed.
Also from the above theorem we get following corollaries: Corollary 9. The scalar curvature and Ricci operator of (M, ∇) is given by

Flatness conditions on Normal Complex Contact Metric Manifolds Admitting a Semi Symmetric Metric Connection
A Riemannian manifold is called flat if its Riemannian curvature tensor vanishes. That means manifold is locally Euclidean. Also the flatness of a Riemannian manifold can be provided by some special transformations like conformal, concircular etc. If the manifold is flat under these special transformations, is called conformally flat, concircularly flat etc. On the other hand there are several tensors which are invariant of these special transformations, and can give flatness of the manifold when vanishes. The three of them are conformal, concircular and quasi-conformal curvature tensor. The flatness of conformal curvature tensor on a NCCMM was studied by Blair and Molina [4]. Two of present authors studied the flatness of concircular and quasi-concircular curvature tensors. A NCCMM is not conformal, concircular and quasi-conformal flat. In this section weθ study on the flatnessθ ofθ theseθ tensors on (M, ∇). Conformal curvature tensor C , concircular curvature tensor Z , quasi-conformal curvature tensorC and conharmonic curvature tensor K of a (2m + 1)-complex dimensional normalθ complexθ contactθ metricθ manifold M is defined by From (20), (21) and (22) we obtain Onθ theθ otherθ handθ for W = Z = U, unitθ andθ mutuallyθ orthogonal T,Y vector fields we have Thus from (20) and (21) we obtain Also for unit T vector field, from (23) G H (T ) is given by From (24) we get Similarly the holomorphic sectional curvature is Thus from (24) we get Therefore we obtain G H (T ) = k(T, JT ) . There is a contradiction from 18 and so our assumption is not true. By following same steps one can shown the non-existence of conformal and concircular flatness.

Iwasawa Manifold Admitting a Semi Symmetric Metric Connection
An Iwasawa manifold is an important example of a compact complex manifold which does not admit any Kähler metric [6]. Fernandez and Gray [6] proved that an Iwasawa manifold has indefinite Kähler structure has symplectic forms each of which is Hermitian with respect to a complex structure .
The Iwasawa manifold is the compact quotient space Γ \ H C formed from the right cosets of the discrete subgroup Γ given by the matrices whose entries z 1 , z 2 , z 3 are Gaussian integers where H C is given by Like realθ Heisenbergθ group is an exampleθ ofθ contactθ manifolds (see [3]), complexθ Heisenberg group has complex almost contact structure. This structure was given by Baikoussis et al. [1] and normality of the structure was obtained by Korkmaz [12]. Also this manifold is the initial point of the work of Korkmaz and it distinguishes Korkmaz's normality from IK-normality.
Blair and Turgut Vanlı [14] worked on corrected energy of Iwasawa manifolds and also Turgut Vanlı and Unal [15] obtained some curvature results. In this section we examine Iwasawa Manifold with a semi symmetric metric connection.
Let {e 1 , e * 1 , e 2 , e * 2 ,U,V } be an orthonormal frame of Iwasawa manifold which is given by Then for and, from Kozsul formula we get ∇ e j e j = ∇ e j e j = ∇ e j e j * = ∇ e * j e * j = 0.
In addition for e i , e j ∈ H we have Ric (e i , e i ) = Ric (e * i , e * i ) = −4 and Ric(U,U) = Ric(V,V ) = 4. By using above equations and from the definition of the semi symmetric metric connection ∇ we get following corollary. T h i s p a g e i s i n t e n t i o n a l l y l e f t b l a n k