Some Flatness Conditions on Normal Metric Contact Pairs

TL;DR

This paper investigates the geometric properties of normal metric contact pair manifolds under various flatness conditions of curvature tensors, revealing their Einstein nature and specific curvature characteristics.

Contribution

It establishes new results linking flatness of conformal, concircular, and quasi-conformal curvature tensors to Einstein manifolds with specific scalar curvatures in the context of normal metric contact pairs.

Findings

Conformal flatness implies Einstein manifold with negative scalar curvature and positive sectional curvature.

Concircular flatness implies Einstein manifold.

Quasi-conformal flatness implies Einstein manifold with positive scalar curvature and constant curvature.

Abstract

In this paper the geometry of normal metric contact pair manifolds is studied under the flatness of conformal, concircular and quasi-conformal curvature tensors. It is proved that a conformal flat normal metric contact pair manifold is an Einstein manifold with a negative scalar curvature and has positive sectional curvature. It is also shown that a concircular flat normal metric contact pair manifold an Einstein manifold. Finally it is obtained that a quasi-conformal flat normal metric contact pair manifold is an Einstein manifold with a positive scalar curvature and, is a space of constant curvature.

Equations

$$\mathcal{D}=\{X:\eta (X)=0,X\in \Gamma (TM)\}.$$

$${\phi }^2=-I+\eta \otimes \xi ,\eta (\xi )=1,,g(\phi \bullet ,\bullet )=-g(\bullet ,\phi \bullet )$$

$${\phi }^2=(\phi J{)}^2=-I+\eta \otimes \xi -(\eta \circ J)\otimes (J\circ \xi )$$

$$\eta (\xi )=1,\eta (-J\circ \xi )=0,(\eta \circ J)(-J\circ \xi )=1,(\eta \circ J)(\xi )=0$$

$$g(\phi \bullet ,\bullet )=-g(\bullet ,\phi \bullet ),g((\phi \circ J)\bullet ,\bullet )=-g(\bullet ,(\phi \circ J)\bullet )$$

$$\mathcal{H}=ker{\alpha }_1\cap ker{\alpha }_2.$$

$${\alpha }_1(Z_1)={\alpha }_2(Z_2)=1,{\alpha }_1(Z_2)={\alpha }_2(Z_1)=0$$

$$i_{Z_1}d{\alpha }_1=i_{Z_1}d{\alpha }_2=i_{Z_2}d{\alpha }_2=0$$

$$T{\mathcal{G}}_i=kerd{\alpha }_i\cap ker{\alpha }_1\cap ker{\alpha }_2,i=1,2$$

$$T{\mathcal{F}}_i=T{\mathcal{G}}_i\oplus \mathbb{R}{Z}_1.$$

$$X=X^{1^h}+X^{2^h}+{\alpha }_1(X^2)Z_1+{\alpha }_2(X^1)Z_2$$

$${\alpha }_1(X^{1^h})={\alpha }_1(X^{2^h})=0,{\alpha }_2(X^{1^h})={\alpha }_2(X^{2^h})=0.$$

$${\varphi }^2=-I+{\alpha }_1\otimes Z_1+{\alpha }_2\otimes Z_2,\varphi Z_1=\varphi Z_2=0.$$

$$g(Z_i,X)={\alpha }_i(X),g(Z_i,Z_j)={\delta }_{ij}$$

$${\nabla }_{Z_i}{Z}_j=0,{\nabla }_{Z_i}\varphi =0$$

$${\nabla }_X{Z}_1=-{\varphi }_{1}X,{\nabla }_X{Z}_2=-{\varphi }_{2}X$$

$$J=\varphi -{\alpha }_2\otimes Z_1+{\alpha }_1\otimes Z_2,T=\varphi +{\alpha }_2\otimes Z_1-{\alpha }_1\otimes Z_2$$

$$g(({\nabla }_{X_1}\varphi )X_2,X_3)=\sum _{i=1}^2(d{\alpha }_i(\varphi X_2,X_1){\alpha }_i(X_3)-d{\alpha }_i(\varphi X_3,X_1){\alpha }_i(X_2))$$

$$R(X_1,X_2)X_3={\nabla }_{X_1}{\nabla }_{X_2}{X}_3-{\nabla }_{X_2}{\nabla }_{X_1}{X}_3-{\nabla }_{[X_1,X_2]}{X}_3,$$

$$\mathcal{R}(X_1,X_2,X_3,X_4)=g(R(X_1,X_2)X_3,X_4).$$

$$QX=\sum _{i=1}^{dim(M)}R(X,E_i)E_i,$$

$$Ric(X_1,X_2)=g(Q{X}_1,X_2),$$

$$scal=\sum _{i=1}^{dim(M)}Ric(E_i,E_i).$$

$$R(X_1,Z)X_2$$

$$\mathcal{R}(X_1,X_2,Z,X_3)$$

$$R(X_1,Z)Z=-{\varphi }^2{X}_1.$$

$$\{E_1,E_2,...,E_p,\varphi E_1,\varphi E_2,...,\varphi E_p,E_{p+1},E_{p+2},...,E_{p+q},\varphi E_{p+1},\varphi E_{p+2},...,\varphi E_{p+q},Z_1,Z_2\}$$

$$Ric(X_1,Z)=\sum _{i=1}^{2p+2q}d{\alpha }_1(\varphi E_i,E_i){\alpha }_1(X)+d{\alpha }_2(\varphi E_i,E_i){\alpha }_2(X).$$

$$Ric(X,Z)=0,\text{for }X\in \Gamma (\mathcal{H}),$$

$$Ric(Z,Z)=2p+2q.$$

$$Ric(Z_1,Z_1)=2p,Ric(Z_2,Z_2)=2q,Ric(Z_1,Z_2)=0.$$

$$\mathcal{C}(X_1,X_2)X_3$$

$$\mathcal{W}(X_1,X_2)X_3$$

$$\tilde{\mathcal{C}}(X_1,X_2)X_3$$

$$g({\phi }_i{X}_1,X_2)=-g(X_1,{\phi }_i{X}_2),\text{for }i=1,2$$

$$J{\xi }_1={\xi }_2,J{\xi }_2=-{\xi }_1$$

Full text

1 Introduction

Contact transformations were defined as a geometric tool to study system of differential equations in 1872 by S.Lie [1]. The subject contact manifold has many applications to other fields of pure mathematics and some applied areas such as mechanics, optics, thermodynamics, or control theory. Also contact manifolds have several application in theoretical physics [2]. The Riemannian geometry of contact manifolds give us geometric interpretation about Einstein manifolds which arise from general relativity. Also the curvature concept has center role in Riemannian geometry and curvature properties of manifolds present geometric aspects from algebraic relations.

A conformal transformation is a map which converts a metric to an other with preserving angle between two vector fields. Conformal curvature tensor is a $(1,3)-$tensor on a Riemanian manifold which is invariant under conformal transformations. This tensor gives important information about the Riemann geometry of the manifold. If it vanishes the manifold is said to be conformally flat, that’s mean the manifold is flat under conformal transformations. A concircular transformation is a special conformal transformation,and preserve the geodesic circle. This type transformations and their applications to differential geometry were studied by Yano [3]. Concircular curvature tensor was defined by Yano [3] and it is invariant under concircular transformations. Also a manifold is called concircularly flat if this tensor vanishes. Yano and Sawaski [4] introduced quasi-conformal curvature which includes both concircular and conformal curvature as special cases. The Riemannian geometry of contact manifolds is examined with these tensors via flatness and symmetries.

Blair, Ludden and Yano [5] studied on complex manifolds consider the results on Calabi-Eckman manifolds $S^{2p+1}\times S^{2q+1}$. By consider two Sasakian structure on $S^{2p+1}$ and $S^{2q+1}$ they gave the second fundamental form on Calabi-Eckman manifold, defined Hermitian bicontact manifold and obtained an $f-$structure on bicontact manifolds. Also normality of bicontact manifolds was given. Bande and Hadjar [6] studied on bicontact manifolds under the name contact pairs. Further they considered a special type of f-structure with complementary frames related to a contact pair and called the contact pair structure. Also the normality of contact pair structures were given by same authors [7, 8, 9].

The conformal flatness of a normal metric contact pair manifold were studied by Bande, Blair and Hadjar [10]. They proved that a conformal flat normal metric contact pair manifold is locally isometric to Hopf manifold $S^{2p+1}(1)\times S^{1}$. On the other hand the flatness condition of conformal, concircular and quasi-conformal curvature tensors on contact manifolds has many geometric and physical applications. A conformal flat Sasakian manifold is of constant curvature [11]. At the same time a normal complex contact metric manifold is not conformal, concircular and quasi-conformal flat [12].

In this paper we studied on conformal, concircular and quasi-conformal curvature flatness of normal contact pair manifolds. We prove that a conformal flat normal metric contact pair manifold is an Einstein manifold with a negative scalar curvature and, has positive sectional curvature. Also we show that a concircular flat normal metric contact pair manifold  is an Einstein manifold. Finally we prove that a quasi-conformal flat normal metric contact pair manifold  is an Einstein manifold with a positive scalar curvature and, is a space of constant curvature.

2 Preliminaries

In this section a short survey is given for contact manifolds and contact pair structures. For detail about contact manifolds we refer to reader [13], and [6, 7, 8] for contact pairs.

2.1 Real and Complex Contact Manifolds

A real contact manifold is defined by a contact form $\eta$ which is a volume form on a real $(2p+1)-$ dimensional differentiable manifold $M$. The kernel of $\eta$ defines $2p-$dimensional a non-integrable distribution of $TM$:

[TABLE]

We also recall $\mathcal{D}$ contact or horizontal distribution. Let take a vector field $\xi$ on $M$ which is dual vector of $\eta$. Then for $(1,1)-$tensor field $\varphi$, $M$ is called an almost contact metric manifold if it satisfied following conditions:

[TABLE]

where $I$ is identity map on $M$ and $g$ is a Riemannian metric. Also we recall $g$ by compatible metric. As the similar to Kähler manifold we have a second fundamental form on an almost contact metric manifold $\Omega(\bullet,\bullet)=d\eta(\bullet,\bullet)$. Also $d\eta(\bullet,\bullet)=g(\bullet,\varphi\bullet)$ and in this case we recall $g$ is an associated metric.

The geometry of contact manifold is studied in different classes. One of them is Sasakian manifold which has a Kähler form on Riemannian cone $M\times\mathbb{R^{+}}.$ A Sasakian manifold has also an almost contact metric structure. The almost contact structure on a Sasakian manifold is normal i.e $N(\varphi\bullet,\varphi\bullet)+2d\eta(\bullet,\bullet)\xi=0$ where $N(\varphi\bullet,\varphi\bullet)$ is the Nijenhuis tensor field of $\varphi$.

In 1959 Kobayashi [14] defined complex analogue of a real contact manifold. Therefore the concept of complex contact manifold entered to the literature. 1980s Ishihara and Konishi [15] construct almost contact structure on a complex contact manifold and they defined compatible metric. A complex almost contact metric manifold is a complex odd $(2p+1)-$dimensional complex manifold with $(J,\varphi,\varphi\circ J,\xi,-J\circ\xi,\eta,\eta\circ J,g)$ structure such that

[TABLE]

where $g$ is Hermitian metric on $M$, $J$ is an almost complex structure. The normality of complex almost contact metric manifolds were given by Ishihara and Konishi and Korkmaz [15, 16]. Normal complex contact metric manifolds were studied by several authors [12, 16, 17].

2.2 Metric Contact Pair Manifold

Definition 2.1**.**

Let $M$ be a $2p+2q+2$-dimensional differentiable manifold. A pair of $(\alpha_{1},\alpha_{2})$ on $M$ is said to be a contact pair of type $(p,q)$ if

• •

$\alpha_{1}\wedge(d\alpha_{1})^{p}\wedge\alpha_{2}\wedge(d\alpha_{2})^{q}\neq 0$**

• •

$(d\alpha_{1})^{p+1}=0$* and $(d\alpha_{2})^{q+1}=0$*

where $p,q$ are positive integers [6].

For $1-$forms $\alpha_{1}$ and $\alpha_{2}$ we have two integrable subbundle of $TM$; $\mathcal{D}_{1}=\{X:\alpha_{1}(X)=0,X\in\Gamma(TM)\}$ and $\mathcal{D}_{2}=\{X:\alpha_{2}(X)=0,X\in\Gamma(TM)\}$. Then we have two characteristic foliations of $M$, denoted $\mathcal{F}_{1}=\mathcal{D}_{1}\cap kerd\alpha_{1}$ and $\mathcal{F}_{2}=\mathcal{D}_{2}\cap kerd\alpha_{2}$ respectively. $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are $(2p+1)$ and $(2q+1)-$dimensional contact manifolds with contact form induced by $\alpha_{1}$ and $\alpha_{2}$. Thus we can define $(2p+2q)-$dimensional horizontal subbundle $\mathcal{H}$

[TABLE]

To a contact pair $(\alpha_{1},\alpha_{2})$ of type $(p,q)$ there are associated two commuting vector fields $Z_{1}$ and $Z_{2}$, called Reeb vector fields of the pair, which are uniquely determined by the following equations:

[TABLE]

where $i_{X}$ is the contraction with the vector field X. In particular, since the Reeb vector fields commute, they determine a locally free $\mathbb{R}^{2}$-action, called the Reeb action.

The tangent bundle of $(M,(\alpha_{1},\alpha_{2}))$ can be split into in a different way. For the two subbundle of $TM$

[TABLE]

and we can write

[TABLE]

Therefore we get $TM=T\mathcal{G}_{1}\oplus T\mathcal{G}_{2}\oplus\mathbb{R}Z_{1}\oplus\mathbb{R}Z_{2}$. The horizontal subbundle can be written as $\mathcal{H}=T\mathcal{G}_{1}\oplus T\mathcal{G}_{2}$ and $\mathcal{V}=\mathbb{R}Z_{1}\oplus\mathbb{R}Z_{2}$, we call $\mathcal{V}$ is vertical subbundle of $TM.$

Let $X$ be an arbitrary vector field on $M$. We can write $X=X^{\mathcal{H}}+X^{\mathcal{V}}$, where $X^{\mathcal{H}},X^{\mathcal{V}}$ horizontal and vertical component of $X$ respectively. For $X^{1}\in T\mathcal{F}_{1}$ and $X^{2}\in T\mathcal{F}_{2}$ we have $X=X^{1}+X^{2}$. Also we can write $X^{1}=X^{1^{h}}+\alpha_{2}(X^{1})Z_{2}$ and $X^{2}=X^{2^{h}}+\alpha_{1}(X^{2})Z_{1}$, where $X^{1^{h}}$ and $X^{2^{h}}$ are horizontal parts of $X^{1},X^{2}$ respectively. From all these decomposition of $X$ finally we get

[TABLE]

Since we have two different $1-$form by above decomposition we understand the components of $X\in TM$ in which distributions.

Definition 2.2**.**

An almost contact pair structure on a $(2p+2q+2)-$dimensional manifold $M$ is a triple $\alpha_{1},\alpha_{2},\phi$, where $(\alpha_{1},\alpha_{2})$ is a contact pair and $\phi$ a $(1,1)$ tensor field such that:

[TABLE]

The rank of $\phi$ is $(2p+2q)$ and $\alpha_{i}(\phi)=0$ for $i=1,2$.

The endomorphism $\phi$ is said to be decomposable if $T\mathcal{F}_{i}$ is invariant under $\phi$. If $\phi$ is decomposable then $(\alpha_{i},Z_{i},\phi)$ induce an almost contact structure on $\mathcal{F}_{i}$ for $i=1,2$ [6]. Unless otherwise stated we assume that $\phi$ is decomposable.

Definition 2.3**.**

Let $(\alpha_{1},\alpha_{2},Z_{1},Z_{2},\phi)$ be an almost contact pair structure on a manifold $M$. A Riemannian metric $g$ is called

compatible if $g(\phi X_{1},\phi X_{2})=g(X_{1},X_{2})-\alpha_{1}(X_{1})\alpha_{1}(X_{2})-\alpha_{2}(X_{1})\alpha_{2}(X_{2})$ for all $X_{1},X_{2}\in TM$, 2. 2.

associated if $g(X_{1},\phi X_{2})=(d\alpha_{1}+d\alpha_{2})(X_{1},X_{2})$ and $g(X_{1},Z_{i})=\alpha_{i}(X_{1}),$ for $i=1,2$ and for all $X_{1},X_{2}\in\Gamma(TM)$

$4-$tuple $(\alpha_{1},\alpha_{2},\phi,g)$ is called a metric almost contact pair on a manifold $M$ and g is an associated metric with respect to contact pair structure $(\alpha_{1},\alpha_{2},\phi)$. We recall $(M,\phi,Z_{1},Z_{2},\alpha_{1},\alpha_{2},g)$ is a metric contact pair manifold.

We have following properties for a metric almost contact pair manifold $M$ [6]:

[TABLE]

and for every $X$ tangent to $\mathcal{F}_{i}\,\ i=1,2$ we have

[TABLE]

where $\phi=\phi_{1}+\phi_{2}$. Normality of metric contact pair manifold is given by Bande and Hadjar [7]. Let $(M,\phi,Z_{1},Z_{2},\alpha_{1},\alpha_{2},g)$ be metric contact pair manifold then we have two almost complex structure:

[TABLE]

Definition 2.4**.**

A metric contact pair manifold is said to be normal if $J$ and $T$ are integrable [7].

Theorem 2.1**.**

Let $(M,\phi,Z_{1},Z_{2},\alpha_{1},\alpha_{2},g)$ be a normal metric contact pair manifold then we have

[TABLE]

where $X_{1},X_{2},X_{3}$ are arbitrary vector fields on $M$ [8].

We can consider a natural question: could any metric contact pair structure be considered locally the product of two contact metric manifold? An example of metric contact pair were given in [18], which is not locally product of two contact metric manifold. So metric contact pair structure has some different properties from contact metric manifolds and their results will have useful interpretation for the geometry of contact and complex manifolds.

On a normal metric contact pair manifold we have $\nabla_{X}Z=-\phi X$ for $X\in\Gamma(TM)$ and $Z=Z_{1}+Z_{2}$.

2.3 Curvature Properties of Normal Metric Contact Pair Manifolds

We use the following statements for the Riemann curvature;

[TABLE]

for all $X_{1},X_{2},X_{3},X_{4}\in\Gamma(TM)$. Also the Ricci operator is defined by

[TABLE]

the Ricci curvature and scalar curvature is given by

[TABLE]

In [9] the curvature of contact pairs were examined. Let $(M,\phi,Z_{1},Z_{2},\alpha_{1},\alpha_{2},g)$ be a normal metric contact pair manifold. Then for $X_{1},X_{2},X_{3}\in\Gamma(\mathcal{H})$ and $Z=Z_{1}+Z_{2}$ Reeb vector field we have :

[TABLE]

[TABLE]

[TABLE]

Let take an orthonormal basis of $M$

[TABLE]

then for all $X_{1}\in\Gamma(TM)$ we get the Ricci curvature of $M$ as

[TABLE]

So, we obtain the following result:

[TABLE]

Conformal $\mathcal{C}$, concircular $\mathcal{W}$ and quasi-conformal curvature tensor $\widetilde{\mathcal{C}}$ of a $(2p+2q+2)$-dimensional normal contact metric pair manifold are given by followings, respectively:

[TABLE]

where $X_{1},X_{2},X_{3}\in\Gamma(TM)$, $a$ and $b$ are constants.

3 Hermitian Contact Pair Manifold

As known, the product of two contact metric manifolds is a contact pair metric manifold. In this section we give an almost contact pair structure on a Hermitian manifold.

Let $(M,g,J)$ be $(2p+2q+2)-$dimensional Hermitian manifold and $(\varphi_{1},\eta_{1},\xi_{1})$ and $(\varphi_{2},\eta_{2},\xi_{2})$ be two almost contact structures on $M$ with following properties.

[TABLE]

where $X_{1},X_{2}$ are two arbitrary vector fields on $M$ [19].

Let take $\phi=\varphi_{1}\circ\varphi_{2}$. Then $\phi$ is a $(1,1)$ tensor field on $M$. By direct computation we get

[TABLE]

Thus we obtain an almost contact pair structure on $M$ with the contact pair $\eta_{1},\eta_{2}$ and we state:

Corollary 3.1**.**

Let $(M^{2p+2q+2},J,g)$ be an almost Hermitian manifold and $(\varphi_{i},\eta_{i},\xi_{i})_{i=0}^{2}$ be two almost contact structure on $M$ with properties are given above. Then $(\eta_{1},\eta_{2},\phi)$ is an almost contact pair structure on $M$ such that

[TABLE]

for all $X_{1}\in\Gamma(TM)$.

Also for $X_{1},X_{2}\in\Gamma(TM)$ we have

[TABLE]

and

[TABLE]

Thus we obtain compatible metric with contact pair structure.

These results show that a contact pair structure on an almost Hermitian manifold could be obtained from two almost contact structure on this manifold. Since contact pair manifolds have some significant properties, some future works could be done for Hermitian and contact structure. Also if the manifold is complex the structure $(\varphi_{1},\varphi_{2},J,\xi_{1},\xi_{2},\eta_{1},\eta_{2},g)$ be a complex almost contact metric manifold. This type of manifolds was studied by several authors [12, 14, 15, 16, 17]

4 Flatness Conditions on Normal Contact Pair Manifold

In this section we give some results on the flatness of conformal, concircular and quasi-conformal curvature tensors.

Theorem 4.1**.**

A conformal flat normal metric contact pair manifold is an Einstein manifold with a negative scalar curvature and has positive sectional curvature.

Proof 4.1**.**

Let $(M,\phi,\alpha_{1},\alpha_{2})$ be normal metric contact pair manifold. Suppose that $M$ is conformal flat. Then we have

[TABLE]

where $A=\frac{scal}{(2p+2q+1)(2p+2q)}$ and $B=\frac{1}{2p+2q}$. Taking $X_{2}=X_{3}=Z$ and $X_{1},X_{4}\in\Gamma(\mathcal{H})$ in (4.1) , since $g(Z,Z)=2$ and from (2.9), (2.11) we obtain

[TABLE]

Also from (2.3) we get

[TABLE]

and thus we obtain

[TABLE]

So, the manifold is Einstein. On the other hand by direct computation from (4.3) the scalar curvature is

[TABLE]

This shows the scalar curvature is negative. Let choose $X_{1}=X_{4}$, $X_{2}=X_{3}$ unit and orthogonal vector fields in (4.1). Then the sectional curvature is obtained by

[TABLE]

Thus, the proof is completed.

An Einstein manifold is also Einstein under concircular transformation. Yano proved that a concircular flat Riemann manifold is Einstein[3]. By similar way we can easily obtain following result.

Theorem 4.2**.**

A concircular flat normal contact pair manifold is Einstein.

Our finally result is about quasi-conformal flatness of normal metric contact pair manifold.

Theorem 4.3**.**

A quasi-conformally flat normal metric contact pair manifold;

is an Einstein manifold with a positive scalar curvature 2. 2.

is a space of constant curvature.

Proof 4.2**.**

Let $M$ be a quasi-conformally flat normal metric contact pair manifold. Then for $X_{1},X_{2},X_{3},X_{4}\in\Gamma(TM)$ we have

[TABLE]

Let write $\frac{scal}{2p+2q+2}[\frac{a}{4m+1}+2b]=K$ for brevity. In ( 4.2), by taking $X_{1}=X_{4}=E_{i}$ and getting sum from $i=1$ to $i=2p+2q+2$ we obtain

[TABLE]

and therefore we get

[TABLE]

Assume that $a+b(2p+2q)\neq 0$. Then

[TABLE]

By taking $X_{2}=X_{3}=Z$ in (4.6) and from (2.12), we get positive scalar curvature is given by

[TABLE]

So, the Ricci curvature has the following form:

[TABLE]

*This shows manifold is Einstein.

On the other hand consider (4.7) in (4.2) we get*

[TABLE]

If $a\neq 0$ we get

[TABLE]

This shows us that the manifold is a space of constant curvature.