From a single Sasakian manifold to a family of Sasakian manifolds

Abstract

The aim of this paper is to construct a 1-parameter family of Sasakian manifold starting from a single Sasakian manifold. Concrete examples are given.

1 Introduction

The product of manifolds plays a very important role in the construction of the new manifolds and also for the structures on manifolds. Bishop and O’Neill introduced the notion of warped product as a generalization of Riemannian product (Bishop and ONeill 1969).

The product of an almost contact manifold M and the real line ${\mathbf {R}}$ carries a natural almost complex structure. When this structure is integrable the almost contact structure is said to be normal.

By means of warped product there is a one-to-one correspondence between Sasakian and Kählerian structures [see Oubiña (1985)].

In Blair (2012), building on the work of Tanno (1969) (the homothetic deformation on contact metric manifold), Blair introduced the notion of ${\mathcal {D}}$-homothetic warping and he used it to prove the above correspondence by another way.

Recently, Beldjilali and Belkhelfa (2016) introduced the ${\mathcal {D}}$-homothetic bi-warping and they proved that every sasakian manifold M generates a 1-parameter family of Kählerian manifolds. Thereby generalizing the results of Oubiña (1985) and Blair (2012).

One of the goals of this paper is to explore other ways for constructing Sasakian manifolds. Results in our paper can be divided in three parts. In the first part, we state and proof our first construction result [see Beldjilali and Belkhelfa (2016)] by another way and we give an example. In the second part, we construct Sasakian structures from Kählerian structures built above using a new technique and we construct a concrete example. Finally, we combine the two previous steps together to provide a way to create a 1-parameter family of Sasakian manifolds from just one Sasakian manifold directly. This text is organized in the following way.

Section 2 is devoted to the background of the structures which will be used in the sequel.

In Sect. 3, we construct a 1-parameter family of Kählerian manifolds using the ${\mathcal {D}}$-homothetic bi-warping but in a different way than (Beldjilali and Belkhelfa 2016) and we give an example. In Sect. 4 We introduce certain Riemannian metric on the product of real line and the Kählerian manifold built in the Sect. 3 and we use it to construct a 1-parameter family of Sasakian manifolds. Finaly, in Sect. 5, we merge the two previous constructions to give a direct way to get a 1-parameter family of Sasakian manifolds from a single Sasakian manifold and we give a concrete example.

2 Preliminaries

2.1 Almost contact metric structures and Sasakian structures

An odd-dimensional Riemannian manifold $(M^{2n+1},g)$ is said to be an almost contact metric manifold if there exist on M a (1, 1) tensor field $\varphi$, a vector field $\xi$ (called the structure vector field) and a 1-form $\eta$ such that

$$\begin{aligned} \eta (\xi )=1, \quad \varphi ^{2}(X) = -X+\eta (X)\xi , \quad and \quad g(\varphi X,\varphi Y) = g(X,Y)-\eta (X)\eta (Y), \end{aligned}$$

for any vector fields X,Y on M. In particular, in an almost contact metric manifold we also have $\varphi \xi =0$ and $\eta \circ \varphi =0$.

Such a manifold is said to be a contact metric manifold if $d\eta =\phi$, where $\phi (X,Y)=g(X,\varphi Y)$ is called the fundamental 2-form of M. If, in addition, $\xi$ is a Killing vector field, then M is said to be a K-contact manifold. It is well-known that a contact metric manifold is a K-contact manifold if and only if $\nabla _{X}\xi = -\varphi X$, for any vector field X on M.

On the other hand, the almost contact metric structure of M is said to be normal if $[\varphi ,\varphi ](X,Y)=-2d\eta ~(X,Y)\xi$, for any X, Y, where $[\varphi ,\varphi ]$ denotes the Nijenhuis torsion of $\varphi$, given by

$$\begin{aligned}{}[\varphi ,\varphi ](X,Y)=\varphi ^{2}[X,Y]+[\varphi X,\varphi Y]-\varphi [\varphi X,Y]-\varphi [X,\varphi Y] \end{aligned}$$

A normal contact metric manifold is called a Sasakian manifold. It can be proved that a Sasakian manifold is K-contact, and that an almost contact metric manifold is Sasakian if and only if

$$\begin{aligned} (\nabla _{X}\varphi )Y = g(X,Y)\xi -\eta (Y)X, \end{aligned}$$

(1)

for any X,Y. Moreover, for a Sasakian manifold the following equation holds:

$$\begin{aligned} R(X,Y)\xi = \eta (Y)X - \eta (X)Y . \end{aligned}$$

From the formula (1) easily obtains

$$\begin{aligned} \nabla _{X}\xi = - \varphi X, \qquad (\nabla _{X}\eta )Y = -g(\varphi X,Y). \end{aligned}$$

(2)

For more background on almost contact metric manifolds and recent study of $\eta$-Einstein manifolds, we recommend the reference Blair (2002) and Boyer et al. (2006).

2.2 Almost complex structures and Kählerian structures

An almost complex manifold with a Hermitian metric is called an almost Hermitian manifold. For an almost Hermitian manifold $({\overline{M}}^{2n},J,{\overline{g}})$ we thus have

$$\begin{aligned} J^{2} = -1,\qquad \qquad {\overline{g}}(JX,JY)={\overline{g}}(X,Y). \end{aligned}$$

An almost complex stucture J is integrable, and hence the manifold is a complex manifold, if and only if its Nijenhuis tensor $N_j$ vanishes, with

$$\begin{aligned} N_J(X,Y)=[JX,JY]-[X,Y]-J[X,JY]-J[JX,Y]. \end{aligned}$$

For an almost Hermitian manifold $({\overline{M}},J,{\overline{g}})$, we define the fundamental Kähler form ${\overline{\varOmega }}$ as:

$$\begin{aligned} {\overline{\varOmega }}(X,Y)={\overline{g}}(X,JY). \end{aligned}$$

$({\overline{M}},J,{\overline{g}})$ is then called almost Kähler if ${\overline{\varOmega }}$ is closed i.e. ${ \mathrm d} {\overline{\varOmega }} = 0$. It can be shown that this condition for $({\overline{M}},J,{\overline{g}})$ to be almost Kähler is equivalent to

$$\begin{aligned} {\overline{g}}\big (({\overline{\nabla }}_X J)Y,Z \big )+{\overline{g}}\big ( ({\overline{\nabla }}_Y J)Z,X\big ) +{\overline{g}}\big (({\overline{\nabla }}_Z J)X,Y \big )=0. \end{aligned}$$

An almost Kähler manifold with integrable J is called a Kähler manifold, and thus is characterized by the conditions: $d{\overline{\varOmega }}=0$ and $N_J=0$. One can prove that these both conditions combined are equivalent with the single condition

$$\begin{aligned} {\overline{\nabla }} J=0. \end{aligned}$$

For more background on almost complex structure manifolds, we recommend the reference Yano and Kon (1984).

3 From Sasakian structure to Kählerian structures

In Beldjilali and Belkhelfa (2016), the authors introduced the notion of ${\mathcal {D}}$-homothetically bi-warped metric on ${\overline{M}}= M' \times M$ where $M'$ is a Riemannian manifolds and M is an almost contact metric manifold by

$$\begin{aligned} {\overline{g}}=g' + f^2 g + f^2 (h^2 - 1) \eta \otimes \eta . \end{aligned}$$

(3)

where $fh\ne 0$ everywhere.

Using the Koszul formula for the Levi-Civita connection of a Riemannian metric, one can obtain the following:

Proposition 1

Beldjilali and Belkhelfa (2016) Let $\nabla ',\nabla \; and\; {\overline{\nabla }}$ denote the Riemannian connections of $g', g, \; and\; {\overline{g}}$ respectively. For all $X',Y'$ vector fields tangent to $M'$ and independent of M and similarly for X, Y, we have

$$\begin{aligned} {\overline{\nabla }}_{X'} Y'= & {} \nabla '_{X'} Y',\\ {\overline{g}}({\overline{\nabla }}_{X'} Y,Z)= & {} {\overline{g}}({\overline{\nabla }}_{Y} X',Z)=- {\overline{g}}({\overline{\nabla }}_{Y} Z,X')\\= & {} f X'(f)g(Y,Z) + f\Big ((h^2 -1)X'(f) +fh X'(h) \Big ) \eta (Y)\eta (Z),\\ {\overline{g}}({\overline{\nabla }}_{X} Y,Z)= & {} {\overline{g}}(\nabla _{X} Y,Z) + f^2 (h^2 -1) \Big ( \frac{1}{2}\big ( g( \nabla _X \xi ,Y) +g( \nabla _Y \xi , X) \big ) \eta (Z)\\&+ d\eta (X,Z)\eta (Y) +d\eta (Y,Z)\eta (X) \Big ). \end{aligned}$$

Our motivation is to consider the case where $M'={\mathbf {R}}$ and M is a Sasakian manifold. For brevity we denote the unit tangent field to ${\mathbf {R}}$ by $\partial _t$ In this case the proposition (1) gives

Proposition 2

Let $(M,\varphi , \xi , \eta , g)$ be a Sasakian manifold. Let $\nabla \; and\; {\overline{\nabla }}$ denote the Riemannian connections of $g \; and\; {\overline{g}}$ respectively. For all X, Y vector fields tangent to M and independent of ${\mathbf {R}}$, we have

$$\begin{aligned} {\overline{\nabla }}_{\partial _t} X= & {} {\overline{\nabla }}_{X} \partial _t=\frac{f'}{f}X +\frac{h'}{h}\eta (X)\xi ,\\ {\overline{\nabla }}_{X} Y= & {} \nabla _{X} Y + (1- h^2)\big ( \eta (X)\varphi Y + \eta (Y) \varphi X \big ) \\&- \Big ( ff'g(\varphi X , \varphi Y) +fh(fh)' \eta (X) \eta (Y) \Big )\partial _t. \end{aligned}$$

Next, we introduce a class of almost complex structure J on manifold ${\overline{M}}$ :

$$\begin{aligned} J(a \partial _t , X)= & {} \Big ( fh \eta (X)\partial _t \; , \; \varphi X - \frac{a}{fh} \xi \Big ), \end{aligned}$$

(4)

for any vector filds X of M where f, h,  are functions on ${\mathbf {R}}$ and $fh\ne 0$ everywhere.

That $J^2=-I$ is easily checked and for all ${\overline{X}}=(a\partial _t , X),{\overline{Y}} =(b\partial _t,Y)$ on ${\overline{M}}$ we can see that ${\overline{g}}$ is an almost Hermitian with respect to J i.e.

$$\begin{aligned} {\overline{g}}\big (J{\overline{X}},J{\overline{Y}}\big ) ={\overline{g}}({\overline{X}},{\overline{Y}}). \end{aligned}$$

Knowing that $({\overline{\nabla }}_{{\overline{X}}} J){\overline{Y}}= {\overline{\nabla }}_{{\overline{X}}} (J{\overline{Y}}) - J {\overline{\nabla }}_{{\overline{X}}} {\overline{Y}}$ with using the proposition (2) and formulas (1) and (2), we get the following proposition:

Proposition 3

Let $(M,\varphi , \xi , \eta , g)$ be a Sasakian manifold. For all X, Y vectors fields on M, the non-zero components ${\overline{\nabla }}J$ are

$$\begin{aligned} ({\overline{\nabla }}_{X} J)\partial _t= & {} \frac{1}{f}(h-f')\varphi X \\ ({\overline{\nabla }}_{X} J)Y= & {} (h-f')\Big ( \frac{1}{h}g(X,Y)\xi -h \eta (Y)X+\frac{h^2 -1}{h}\eta (X)\eta (Y)\xi +f g(X,\varphi Y) \partial _t \Big ). \end{aligned}$$

From the above proposition we have immediately that

Proposition 4

Let $(M,\varphi , \xi , \eta , g)$ be a Sasakian manifold. The Hermitian structure constructed in (3) and (4) is Kählerian if and only if $h=f'$

Example 1

For this example, we rely on the example of Blair Blair (2002). We know that ${\mathbf {R}}^{2n+1}$ with coordinates $(x^{i}, y^i, z)$, $i=1..n$, admits the Sasakian structure

$$\begin{aligned} g= & {} \frac{1}{4}\left( \begin{array}{lll} \delta _{ij}+y^{i}y^{j} &{}\quad 0 &{}\quad -y^{i} \\ 0 &{}\quad \delta _{ij} &{}\quad 0 \\ -y^{j} &{}\quad 0 &{}\quad 1 \end{array} \right) ,\qquad \varphi =\left( \begin{array}{lll} 0 &{}\quad \delta _{ij} &{}\quad 0 \\ -\delta _{ij} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad y^{j} &{}\quad 0 \end{array} \right) ,\\ \xi= & {} 2\left( \frac{\partial }{\partial z}\right) ,\qquad \eta = \frac{1}{2}( dz-y^{i} dx^{i}). \end{aligned}$$

So, using this structure, we can define a family of Kählerian structures $(J, {\overline{g}})$ on ${\mathbf {R}}^{2n+2}$ as follows

$$\begin{aligned} {\overline{g}}= & {} \left( \begin{array}{llll} 1 &{}\quad 0 &{} \quad 0 &{}\quad 0 \\ 0 &{}\quad \frac{1}{4}f^2 (\delta _{ij}+f'^2 y^i y^j) &{}\quad 0 &{}\quad -\frac{1}{4}f^2 f'^2 y^i \\ 0 &{}\quad 0 &{}\quad \frac{\delta _{ij}}{4}f^2 &{}\quad 0 \\ 0 &{} \quad -\frac{1}{4}f^2 f'^2 y^j &{}\quad 0 &{}\quad \frac{1}{4}f^2 f'^2 \\ \end{array} \right) \\ J= & {} \left( \begin{array}{llll} 0 &{} \quad -\frac{1}{2}y^jff' &{}\quad 0 &{}\quad \frac{1}{2}ff' \\ 0 &{}\quad 0 &{}\quad \delta _{ij} &{}\quad 0 \\ 0 &{}\quad -\delta _{ij} &{}\quad 0 &{}\quad 0 \\ -\frac{2 }{ff'} &{}\quad 0 &{}\quad y^j &{}\quad 0 \\ \end{array} \right) \end{aligned}$$

4 From Kählerian strucrures to Sasakian structures

Now, consider $(M,\varphi , \xi , \eta , g)$ is a Sasakian manifold and $({\overline{M}}={\mathbf {R}} \times M,J, {\overline{g}})$ the 1-parameter family of Kählerian structures defined above by

$$\begin{aligned} \left\{ \begin{array}{ll} {\overline{g}}=dt^2 + f^2 g + f^2(f'^2 -1) \eta \otimes \eta ,\\ J(a \frac{\partial }{\partial t} , X) = \Big ( ff' \eta (X)\frac{\partial }{\partial t} \; , \; \varphi X - \frac{a}{ff'} \xi \Big ), \end{array} \right. \end{aligned}$$

(5)

where $f=f(t) \in C^{\infty }({\mathbf {R}})$ and $ff' \ne 0$ anywhere.

The fundamental 2-form ${\overline{\varOmega }}$ of $( J, {\overline{g}})$ is

$$\begin{aligned} {\overline{\varOmega }}\Big ( \big (a \frac{\partial }{\partial t}, X\big ) , \big (b \frac{\partial }{\partial t}, Y\big )\Big )={\overline{g}}\Big ( \big (a \frac{\partial }{\partial t}, X\big ) , J\big (b \frac{\partial }{\partial t}, Y\big )\Big ), \end{aligned}$$

we can check that is very simply as follows:

$$\begin{aligned} {\overline{\varOmega }}= & {} f(2f' \;{ \mathrm d} t \wedge \eta + f \phi )\\= & {} { \mathrm d}\big ( f^2 \eta \big ), \end{aligned}$$

where ${ \mathrm d}$ denotes the exterior derivative and $\phi$ the fundamental 2-form of $( \varphi , \xi , \eta , g)$.

Putting

$$\begin{aligned} \theta = f^2 \eta , \end{aligned}$$

and define on ${\tilde{M}}={\mathbf {R}}\times {\overline{M}}$ a Riemannian metric ${\tilde{g}}$ by

$$\begin{aligned} {\tilde{g}}= {\overline{g}} + {\tilde{\eta }}\otimes {\tilde{\eta }} \qquad with \qquad {\tilde{\eta }}=dr + \theta , \end{aligned}$$

(6)

where r is the standard coordinate with respect to the frame $\partial _r$ on ${\mathbf {R}}$. For all X, Y vector fields on M, we have:

$$\begin{aligned} {\tilde{g}}(X,Y)={\overline{g}} (X,Y)+\theta (X)\theta (Y),\quad {\tilde{g}}(X,\partial _r)=\theta (X),\quad {\tilde{g}}(\partial _r,\partial _r)=1. \end{aligned}$$

We denote by ${\overline{\nabla }}$ (resp. ${\tilde{\nabla }}$) the covariant derivative with respect to the metric ${\overline{g}}$ on ${\overline{M}}$ (resp. ${\tilde{g}}$ on ${\tilde{M}}$). From the Koszul formula, we have the following:

Proposition 5

For all X, Y, Z vector fields on ${\overline{M}}$, we have:

• 1) ${\tilde{g}}({\tilde{\nabla }}_{\partial _r}\partial _r,\partial _r)={\tilde{g}}({\tilde{\nabla }} _{\partial _r}\partial _r,X)=0$;

• 2) ${\tilde{g}}({\tilde{\nabla }}_{\partial _r}X,\partial _r)={\tilde{g}}({\tilde{\nabla }} _{X}\partial _r,\partial _r)=0$;

• 3) ${\tilde{g}}({\tilde{\nabla }}_{\partial _r}X,Y)={\tilde{g}}({\tilde{\nabla }} _{X}\partial _r,Y)={\overline{g}}(X,JY)$;

• 4) ${\tilde{g}}({\tilde{\nabla }}_{X}Y,\partial _r)=\frac{1}{2}\big [X\theta (Y)+Y\theta (X)+\theta ([X,Y])\big ]$;

• 5) ${\tilde{g}}({\tilde{\nabla }}_{X}Y,Z)={\overline{g}}({\overline{\nabla }} _X Y,Z)+\frac{1}{2}\big [X(\theta (Y))+Y(\theta (X))+\theta ([X,Y])\big ]\theta (Z)$                   $+ {\overline{g}}(X,JZ)\theta (Y)+{\overline{g}}(Y,JZ)\theta (X)$.

Proof

For 1), we have:

$$\begin{aligned} {\tilde{g}}({\tilde{\nabla }}_{\partial _r} \partial _r, \partial _r) =\frac{1}{2}\partial _r {\tilde{g}}(\partial _r , \partial _r)=0, \end{aligned}$$

because ${\tilde{g}}(\partial _r , \partial _r)=1$. We compute

$$\begin{aligned} 2{\tilde{g}}({\tilde{\nabla }}_{\partial _r} \partial _r , X)= & {} \partial _r {\tilde{g}}(\partial _r ,X)+\partial _r {\tilde{g}}(X,\partial _r )-X{\tilde{g}}(\partial _r , \partial _r ) \\&+{\tilde{g}}(X,[\partial _r , \partial _r ])+{\tilde{g}}( \partial _r ,[X,\partial _r ])-{\tilde{g}}(\partial _r ,[\partial _r ,X]), \end{aligned}$$

as ${\tilde{g}}(\partial _r,X)={\tilde{\eta }}(X)$, ${\tilde{g}}(\partial _r , \partial _r)=1$, $[\partial _r , \partial _r]=0$ and $[X,\partial _r]=[\partial _r,X]=0$, we obtain

$$\begin{aligned} {\tilde{g}}({\tilde{\nabla }}_{\partial _r} \partial _r , X)=2\partial _r{\tilde{\eta }}(X)=0, \end{aligned}$$

because ${\tilde{\eta }}(X)= \theta (X)\in C^{\infty }({\overline{M}})$ does not depend on r.

2) we have:

$$\begin{aligned} {\tilde{g}}({\tilde{\nabla }}_{\partial _r} X,\partial _r)=\partial _r {\tilde{g}}(X,\partial _r)-{\tilde{g}}(X,{\tilde{\nabla }}_{\partial _r} \partial _r)=0, \end{aligned}$$

from 1). Since $[X , \partial _r]=0$,

$$\begin{aligned} {\tilde{g}}({\tilde{\nabla }} _X \partial _r,\partial _r) ={\tilde{g}}({\tilde{\nabla }} _{\partial _r} X,\partial _r) =0. \end{aligned}$$

3) By the Koszul formula and the definition of ${\tilde{g}}$, we have:

$$\begin{aligned} 2{\tilde{g}}({\tilde{\nabla }}_{\partial _r} X,Y)= & {} \partial _r {\tilde{g}}(X,Y)+X{\tilde{g}}(Y, \partial _r)-Y{\tilde{g}}( \partial _r,X) \\&+{\tilde{g}}(Y,[ \partial _r,X])+{\tilde{g}}(X,[Y, \partial _r])-{\tilde{g}}( \partial _r,[X,Y])\\= & {} \partial _r {\overline{g}}(X,Y)+ \partial _r(\theta (X)\theta (Y))+X(\theta (Y))-Y(\theta (X))-\theta ([X,Y])\\= & {} X(\theta (Y))-Y(\theta (X))-\theta ([X,Y]), \end{aligned}$$

here $[ \partial _r,X]=[Y, \partial _r]=0$ and $\partial _r {\overline{g}}(X,Y)=\partial _r(\theta (X)\theta (Y))=0$. We conclude that:

$$\begin{aligned} {\tilde{g}}({\tilde{\nabla }}_{\partial _r} X,Y)={ \mathrm d}\theta (X,Y)={\overline{\varOmega }}(X,Y)={\overline{g}}(X,JY). \end{aligned}$$

We compute:

$$\begin{aligned} 2{\tilde{g}}({\tilde{\nabla }}_X \partial _r,Y)= & {} X {\tilde{g}}(\partial _r,Y)+\partial _r{\tilde{g}}(Y,X)-Y{\tilde{g}}(X,\partial _r)\\&+{\tilde{g}}(Y,[X,\partial _r])+{\tilde{g}}(\partial _r,[Y,X])-{\tilde{g}}(X,[\partial _r,Y])\\= & {} X(\theta (Y))-Y(\theta (X))-\theta ([X,Y])\\= & {} 2{ \mathrm d}\theta (X,Y), \end{aligned}$$

that is, ${\tilde{g}}({\tilde{\nabla }}_X \partial _r ,Y)={\overline{g}}(X,JY)$.

4) We have:

$$\begin{aligned} {\tilde{g}}({\tilde{\nabla }}_X Y,\partial _r)= & {} X{\tilde{g}}(Y,\partial _r)-{\tilde{g}}(Y,{\tilde{\nabla }}_X \partial _r)\\= & {} X(\theta (Y))-{\overline{g}}(X,JY)\\= & {} X(\theta (Y))-{ \mathrm d}\theta (X,Y)\\= & {} X(\theta (Y))-\frac{1}{2}\big [X(\theta (Y))-Y(\theta (X))-\theta ([X,Y])\big ]\\= & {} \frac{1}{2}\big [X(\theta (Y))+Y(\theta (X))+\theta ([X,Y])\big ]. \end{aligned}$$

5) First we shall use the Koszul formula for a Riemann metric ${\tilde{g}}$ and the Levi-Civita connection ${\tilde{\nabla }}$:

$$\begin{aligned} 2{\tilde{g}}({\tilde{\nabla }}_X Y,Z)= & {} X{\tilde{g}}(Y,Z)+Y{\tilde{g}}(Z,X)-Z{\tilde{g}}(X,Y) \\&+{\tilde{g}}(Z,[X,Y])+{\tilde{g}}(Y,[Z,X])-{\tilde{g}}(X,[Y,Z]), \end{aligned}$$

by the definition of the metric ${\tilde{g}}$, we have:

$$\begin{aligned} 2{\tilde{g}}({\tilde{\nabla }}_X Y,Z)= & {} X{\overline{g}}(Y,Z)+X(\theta (Y)\theta (Z)) +Y{\overline{g}}(Z,X)+Y(\theta (Z)\theta (X))\\&-Z{\overline{g}}(X,Y)-Z(\theta (X)\theta (Y)) +{\overline{g}}(Z,[X,Y])+\theta (Z)\theta ([X,Y])\\&+{\overline{g}}(Y,[Z,X])+\theta (Y)\theta ([Z,X]) -{\overline{g}}(X,[Y,Z])-\theta (X)\theta ([Y,Z]), \end{aligned}$$

by the Koszul formula for ${\overline{g}}$ and ${\overline{\nabla }}$, we get:

$$\begin{aligned} 2{\tilde{g}}({\tilde{\nabla }}_X Y,Z)= & {} 2{\overline{g}}({\overline{\nabla }}_X Y,Z)+X(\theta (Y))\theta (Z)+\theta (Y)X(\theta (Z))\\&+Y(\theta (Z))\theta (X)+\theta (Z)Y(\theta (X))-Z(\theta (X))\theta (Y)\\&-\theta (X)Z(\theta (Y))+\theta (Z)\theta ([X,Y])+\theta (Y)\theta ([Z,X])\\&-\theta (X)\theta ([Y,Z]), \end{aligned}$$

from the formula:

$$\begin{aligned} 2 { \mathrm d}\theta (X,Y) = X(\theta (Y))-Y(\theta (X))-\theta ([X,Y]), \end{aligned}$$

we obtain:

$$\begin{aligned} 2{\tilde{g}}({\tilde{\nabla }}_X Y,Z)= & {} 2{\overline{g}}({\overline{\nabla }}_X Y,Z)+[X(\theta (Y))+Y(\theta (X))+\theta ([X,Y])]\theta (Z)\\&+2\theta (Y)d\theta (X,Z)+2\theta (X)d\theta (Y,Z), \end{aligned}$$

finally, by the condition ${\tilde{\varOmega }}=d\theta$ and the definition of ${\tilde{\varOmega }}$, we obtain:

$$\begin{aligned} {\tilde{g}}({\tilde{\nabla }}_X Y,Z)= & {} {\overline{g}}({\overline{\nabla }}_X Y,Z)+\theta (X){\overline{g}}(Y,JZ)+\theta (Y){\overline{g}}(X,JZ)\\&+\frac{1}{2}[X(\theta (Y))+Y(\theta (X))+\theta ([X,Y])]\theta (Z). \end{aligned}$$

Lemma 1

We choose an orthonormal basis $\{e_1,...,e_{2n}\}$ of the tangent space $T_x {\overline{M}}$ at each point $x\in {\overline{M}}$, then

$$\begin{aligned} \{ \partial _r, e_{1}-\theta (e_{1}) \partial _r,..., e_{2n}-\theta (e_{2n}) \partial _r\} \end{aligned}$$

is an orthonormal basis of $T_{(r,x)}{\tilde{M}}$, where $r\in {\mathbf {R}}$.

Proof

For any $i,j\in \{1,...,2n\}$, we have:

$$\begin{aligned} {\tilde{g}}\big (e_{i}-\theta (e_{i})\partial _r,e_{j}-\theta (e_{j})\partial _r \big )= & {} {\tilde{g}}(e_i,e_j)-\theta (e_{j}) {\tilde{g}}(e_i,\partial _r)-\theta (e_{i}) {\tilde{g}}(\partial _r ,e_j)\\&+\theta (e_{i})\theta (e_{j})g(\partial _r , \partial _r) \\= & {} {\overline{g}}(e_i,e_j)+\theta (e_i)\theta (e_j)-\theta (e_i)\theta (e_j)-\theta (e_i)\theta (e_j)\\&+\theta (e_i)\theta (e_j) \\= & {} {\overline{g}}(e_i,e_j)=\delta _{ij}. \end{aligned}$$

We compute:

$$\begin{aligned} {\tilde{g}}\big (e_{i}-\theta (e_{i})\partial _r , \partial _r \big )= & {} {\tilde{g}}(e_i,\partial _r)-\theta (e_{i}) {\tilde{g}}(\partial _r , \partial _r)\\= & {} \theta (e_i)-\theta (e_{i})=0. \end{aligned}$$

So that, for all vector $v\in T_{(r,x)}{\tilde{M}}$, there exist constants $a,b_1,...,b_{2n}$ such that:

$$\begin{aligned} v=a\partial _r + \sum _{i=1}^{2n}b_i\big (e_{i}-\theta (e_{i})\partial _r \big ). \end{aligned}$$

(7)

Note that, $a={\tilde{g}}(v,\partial _r)$ and $b_i={\tilde{g}}\big (v,e_{i}-\theta (e_{i})\partial _r \big )$ for all $i=1,..,2n.$ From the Proposition 5, and the Lemma 1, we get the following:

Proposition 6

For all X, Y vector fields on M, we have:

• 1) ${\tilde{\nabla }}_{\partial _r}\partial _r=0$;

• 2) ${\tilde{\nabla }}_{\partial _r}X={\tilde{\nabla }}_{X}\partial _r=-J X + \theta (J X) \partial _r$;

• 3) ${\tilde{\nabla }}_{X}Y = {\overline{\nabla }}_X Y-\theta (Y) \big ( JX-\theta (JX)\partial _r \big ) -\theta (X) \big ( JY-\theta (JY)\partial _r\big )$

  $\qquad \qquad + \frac{1}{2}\big [({\overline{\nabla }}_{X}\theta )Y +({\overline{\nabla }}_{Y}\theta )X\big ]\partial _r.$

Proof

Let $\{e_1,...,e_{2n}\}$ be an orthonormal frame on ${\overline{M}}$. From the proposition 5, we have: 1)

$$\begin{aligned}&{\tilde{g}}\big ({\tilde{\nabla }}_{\partial _r}\partial _r , \partial _r \big )=0,\\&{\tilde{g}}\big ({\tilde{\nabla }}_{\partial _r}\partial _r, e_{i}-\theta (e_{i})\partial _r \big )={\tilde{g}}\big ({\tilde{\nabla }}_{\partial _r}\partial _r,e_{i})- \theta (e_{i}){\tilde{g}}({\tilde{\nabla }}_{\partial _r}\partial _r , \partial _r \big )=0. \end{aligned}$$

2)

$$\begin{aligned}&{\tilde{g}}\big ({\tilde{\nabla }}_{\partial _r}X,\partial _r \big )=0,\\&{\tilde{g}}\big ({\tilde{\nabla }}_{\partial _r}X,e_{i}-\theta (e_{i})\partial _r \big ) = {\tilde{g}}({\tilde{\nabla }}_{\partial _r}X,e_{i})- \theta (e_{i}){\tilde{g}}({\tilde{\nabla }}_{\partial _r}X,\partial _r)\\&\qquad \qquad \qquad = {\overline{g}}(X,Je_i)=-{\overline{g}}(JX,e_i), \end{aligned}$$

so that:

$$\begin{aligned} {\tilde{\nabla }}_{\partial _r }X= & {} -{\overline{g}}(JX,e_i)e_{i}+{\overline{g}}(JX,e_i)\theta (e_{i})\partial _r \\= & {} -JX +\theta (JX)\partial _r, \end{aligned}$$

with the same method we find that ${\tilde{\nabla }}_{X}\partial _r =-JX +\theta (JX)\partial _r.$

3) We compute:

$$\begin{aligned} {\tilde{g}}\big ({\tilde{\nabla }}_{X}Y,\partial _r \big )= & {} \frac{1}{2}\big [X\theta (Y)+Y\theta (X)+\theta ([X,Y])\big ], \\ {\tilde{g}} \big ({\tilde{\nabla }}_{X}Y,e_{i}-\theta (e_{i})\partial _r \big )= & {} {\tilde{g}}({\tilde{\nabla }}_{X}Y,e_{i}) -\theta (e_{i}){\tilde{g}}({\tilde{\nabla }}_{X}Y,\partial _r)\\= & {} {\overline{g}}({\overline{\nabla }}_X Y,e_i)+\frac{1}{2}\big [X(\theta (Y))+Y(\theta (X))+\theta ([X,Y]) \big ]\theta (e_i)\\&+{\overline{g}}(X,Je_i)\theta (Y)+{\overline{g}}(Y,Je_i)\theta (X)\\&-\frac{1}{2}\theta (e_{i})\big [X\theta (Y)+Y\theta (X)+\theta ([X,Y])\big ]\\= & {} {\overline{g}}({\overline{\nabla }}_X Y,e_i)-{\overline{g}}(JX,e_i)\theta (Y)-{\overline{g}}(JY,e_i)\theta (X), \end{aligned}$$

we conclude that:

$$\begin{aligned} {\tilde{\nabla }}_{X}Y= & {} \frac{1}{2}\big [X\theta (Y)+Y\theta (X)+\theta ([X,Y])\big ]\partial _r \\&+{\overline{g}}({\overline{\nabla }}_X Y,e_i)e_i -{\overline{g}}({\overline{\nabla }}_X Y,e_i)\theta (e_{i})\partial _r\\&-{\overline{g}}(JX,e_i)\theta (Y)e_i+{\overline{g}}(JX,e_i)\theta (Y)\theta (e_{i})\partial _r\\&-{\overline{g}}(JY,e_i)\theta (X)e_i+{\overline{g}}(JY,e_i)\theta (X)\theta (e_{i})\partial _r\\= & {} \frac{1}{2}\big [X\theta (Y)+Y\theta (X)+\theta ([X,Y])\big ]\partial _r \\&+{\overline{\nabla }}_X Y -\theta ({\overline{\nabla }}_X Y)\partial _r\\&-\theta (Y)JX+\theta (Y)\theta (JX)\partial _r\\&-\theta (X)JY+\theta (X)\theta (JY)\partial _r. \end{aligned}$$

Now, we define on ${\tilde{M}}$ a structure $({\tilde{\varphi }}, {\tilde{\xi }}, {\tilde{\eta }})$ as follows

$$\begin{aligned} {\tilde{\xi }}= \partial _r,\quad {\tilde{\varphi }} \partial _r =0, \quad {\tilde{\varphi }} X= JX- \theta (JX) \partial _r, \end{aligned}$$

(8)

for all X vector field on ${\overline{M}}$.

Proposition 7

The manifold $({\tilde{M}},{\tilde{\varphi }}, {\tilde{\xi }}, {\tilde{\eta }}, {\tilde{g}})$ constructed as above is an almost contact metric manifold. Specifically, it is a Sasakian manifold as long as $({\overline{M}}, J, {\overline{g}})$ is a Kählerian manifold.

Proof

We have ${\tilde{\eta }}=dr + \theta$ and ${\tilde{\xi }} = \partial _r$, so ${\tilde{\eta }}({\tilde{\xi }})=1$. As ${\tilde{\varphi }}\partial _r=0$, we get ${\tilde{\varphi }}^2 \partial _r=0$, on the other hand, $-\partial _r+\eta (\partial _r)\partial _r=0$. Let X vector field on ${\overline{M}}$, we compute:

$$\begin{aligned} {\tilde{\varphi }}^2X= & {} {\tilde{\varphi }}\big (JX- \theta (JX) \partial _r \big ) \\= & {} {\tilde{\varphi }}(JX)- \theta (JX) {\tilde{\varphi }}\partial _r \\= & {} J^2X- \theta (J^2X)\partial _r\\= & {} -X+\theta (X)\partial _r\\= & {} -X+{\tilde{\eta }}(X){\tilde{\xi }}. \end{aligned}$$

Let X, Y vector fields on ${\overline{M}}$, we have

$$\begin{aligned} {\tilde{g}}({\tilde{\varphi }} X,{\tilde{\varphi }} Y)= & {} {\tilde{g}}\big (JX- \theta (JX) \partial _r, JY- \theta (JY) \partial _r \big ) \\= & {} {\tilde{g}}\big (JX,JY)- \theta (JY){\tilde{g}}(JX,\partial _r )-\theta (JX){\tilde{g}}(\partial _r, JY)\\&+\theta (JX)\theta (JY){\tilde{g}}(\partial _r , \partial _r), \end{aligned}$$

by the definition of the metric ${\tilde{g}}$ with ${\tilde{\eta }}=dr + \theta$, we obtain

$$\begin{aligned} {\tilde{g}}({\tilde{\varphi }} X,{\tilde{\varphi }} Y)= & {} {\overline{g}}(JX,JY) +\theta (JX)\theta (JY)-\theta (JX)\theta (JY)\\&-\theta (JX)\theta (JY)+\theta (JX)\theta (JY)\\= & {} {\overline{g}}(JX,JY), \end{aligned}$$

as ${\overline{g}}(JX,JY)={\overline{g}}(X,Y)$ and ${\tilde{g}}(X,Y)={\overline{g}}(X,Y)+\theta (X)\theta (Y)$, we conclude that

$$\begin{aligned} {\tilde{g}}({\tilde{\varphi }} X,{\tilde{\varphi }} Y)= & {} {\tilde{g}}(X,Y)-\theta (X)\theta (Y)\\= & {} {\tilde{g}}(X,Y)-{\tilde{\eta }}(X){\tilde{\eta }}(Y). \end{aligned}$$

As ${\tilde{\varphi }} \partial _r=0$, ${\tilde{g}}(X,\partial _r)=\theta (X)={\tilde{\eta }}(X)$, ${\tilde{\eta }}(\partial _r)=1$ and ${\tilde{g}}(\partial _r , \partial _r)=1$, we get

$$\begin{aligned}&{\tilde{g}}({\tilde{\varphi }} X,{\tilde{\varphi }}\partial _r)={\tilde{g}}(X,\partial _r)-\tilde{ \eta }(X){\tilde{\eta }}(\partial _r)=0,\\&{\tilde{g}}({\tilde{\varphi }} \partial _r,{\tilde{\varphi }} \partial _r)={\tilde{g}}(\partial _r , \partial _r)-{\tilde{\eta }}(\partial _r){\tilde{\eta }}(\partial _r)=0. \end{aligned}$$

This confirms that $({\tilde{M}},{\tilde{\varphi }} , {\tilde{\xi }} , {\tilde{\eta }}, {\tilde{g}})$ is an almost contact metric manifold.

On the other hand, the manifold $({\tilde{M}},{\tilde{\varphi }} , {\tilde{\xi }} , {\tilde{\eta }}, {\tilde{g}})$ is Sasakian if it satisfies:

$$\begin{aligned} ({\tilde{\nabla }}_{{\tilde{\xi }}}{\tilde{\varphi }}) {\tilde{\xi }}= & {} {\tilde{g}}({\tilde{\xi }},{\tilde{\xi }}){\tilde{\xi }} - {\tilde{\eta }}({\tilde{\xi }}){\tilde{\xi }} \end{aligned}$$

(9)

$$\begin{aligned} ({\tilde{\nabla }}_{{\tilde{\xi }}}{\tilde{\varphi }}) X= & {} {\tilde{g}}({\tilde{\xi }},X){\tilde{\xi }} - {\tilde{\eta }}(X){\tilde{\xi }} \end{aligned}$$

(10)

$$\begin{aligned} ({\tilde{\nabla }}_{X}{\tilde{\varphi }}) {\tilde{\xi }}= & {} {\tilde{g}}(X,{\tilde{\xi }}){\tilde{\xi }} - {\tilde{\eta }}({\tilde{\xi }})X \end{aligned}$$

(11)

$$\begin{aligned} ({\tilde{\nabla }}_{X}{\tilde{\varphi }})Y= & {} {\tilde{g}}(X,Y){\tilde{\xi }} - {\tilde{\eta }}(Y)X, \end{aligned}$$

(12)

for all X, Y vectors fields on ${\overline{M}}$. It is easy to prove that conditions (9), (10), and (11) are satisfied, and we have:

$$\begin{aligned} ({\tilde{\nabla }}_{{\tilde{\xi }}}{\tilde{\varphi }}) {\tilde{\xi }}={\tilde{\nabla }}_{{\tilde{\xi }}}{\tilde{\varphi }}{\tilde{\xi }}-{\tilde{\varphi }}({\tilde{\nabla }}_{{\tilde{\xi }}}{\tilde{\xi }})= & {} {\tilde{g}}({\tilde{\xi }},{\tilde{\xi }}){\tilde{\xi }} - {\tilde{\eta }}({\tilde{\xi }}){\tilde{\xi }}=0, \\ ({\tilde{\nabla }}_{{\tilde{\xi }}}{\tilde{\varphi }}) X= & {} {\tilde{\nabla }}_{{\tilde{\xi }}}{\tilde{\varphi }} X -{\tilde{\varphi }}({\tilde{\nabla }}_{{\tilde{\xi }}} X)\\= & {} {\tilde{\nabla }}_{{\tilde{\xi }}}JX-{\tilde{\nabla }}_{{\tilde{\xi }}}(\theta (JX){\tilde{\xi }}) +{\tilde{\varphi }}^2(X)\\= & {} -{\tilde{\varphi }}(JX)-\theta (JX){\tilde{\nabla }}_{{\tilde{\xi }}}{\tilde{\xi }}-X+\theta (X){\tilde{\xi }}\\= & {} -(J^2X-\theta (J^2X){\tilde{\xi }})-X+\theta (X){\tilde{\xi }}\\= & {} X+\theta (X){\tilde{\xi }}-X+\theta (X){\tilde{\xi }}=0, \end{aligned}$$

so that, $({\tilde{\nabla }}_{{\tilde{\xi }}}{\tilde{\varphi }}) X = {\tilde{g}}({\tilde{\xi }},X){\tilde{\xi }} - {\tilde{\eta }}(X){\tilde{\xi }}=0$. We compute:

$$\begin{aligned} ({\tilde{\nabla }}_{X}{\tilde{\varphi }}) {\tilde{\xi }}= & {} {\tilde{\nabla }}_{X}{\tilde{\varphi }} {\tilde{\xi }} -{\tilde{\varphi }}({\tilde{\nabla }}_{X} {\tilde{\xi }})\\= & {} {\tilde{\varphi }}^2X=-X+\theta (X){\tilde{\xi }}, \end{aligned}$$

so, $({\tilde{\nabla }}_{X}{\tilde{\varphi }}) {\tilde{\xi }} = {\tilde{g}}(X,{\tilde{\xi }}){\tilde{\xi }} - {\tilde{\eta }}({\tilde{\xi }})X=-X+\theta (X){\tilde{\xi }}$.

For the condition (12), we compute:

$$\begin{aligned} ({\tilde{\nabla }}_{X}{\tilde{\varphi }})Y ={\tilde{\nabla }}_{X}{\tilde{\varphi }} Y -{\tilde{\varphi }}({\tilde{\nabla }}_{X}Y), \end{aligned}$$

(13)

the first term of (13), is given by:

$$\begin{aligned} {\tilde{\nabla }}_{X}{\tilde{\varphi }} Y= & {} {\tilde{\nabla }}_{X}(JY-\theta (JY){\tilde{\xi }})\\= & {} {\tilde{\nabla }}_{X}JY -X(\theta (JY)){\tilde{\xi }} - \theta (JY){\tilde{\nabla }}_{X}{\tilde{\xi }} , \end{aligned}$$

using the proposition 5, we have

$$\begin{aligned} {\tilde{\nabla }}_{X}{\tilde{\varphi }} Y= & {} {\overline{\nabla }}_{X}JY -\theta (JY){\tilde{\varphi }} X -\theta (X){\tilde{\varphi }} JY+\frac{1}{2}\big [({\overline{\nabla }}_{X}\theta )JY +({\overline{\nabla }}_{JY}\theta )X\big ]{\tilde{\xi }} \nonumber \\&-X(\theta (JY)){\tilde{\xi }}+ \theta (JY){\tilde{\varphi }} X\nonumber \\= & {} {\overline{\nabla }}_{X}JY +\theta (X)Y-\theta (X)\theta (Y){\tilde{\xi }} -\frac{1}{2}X(\theta (JY)){\tilde{\xi }}-\frac{1}{2}\theta ( {\overline{\nabla }}_XJY){\tilde{\xi }}\nonumber \\&+\frac{1}{2}(JY)(\theta (X)){\tilde{\xi }}-\frac{1}{2}\theta ( {\overline{\nabla }}_{JY}X){\tilde{\xi }}, \end{aligned}$$

(14)

the second term of (13), is given by:

$$\begin{aligned} -{\tilde{\varphi }}({\tilde{\nabla }}_{X}Y)= & {} -{\tilde{\varphi }}({\overline{\nabla }}_{X}Y) +\theta (Y){\tilde{\varphi }}^{2}X+\theta (X){\tilde{\varphi }}^{2}Y\nonumber \\= & {} -J{\overline{\nabla }}_{X}Y+\theta (J{\overline{\nabla }}_{X}Y){\tilde{\xi }} -\theta (Y)X+2\theta (X)\theta (Y){\tilde{\xi }}-\theta (X)Y, \end{aligned}$$

(15)

Substituting the formulas (14) and (15) in (13), we obtain:

$$\begin{aligned} ({\tilde{\nabla }}_X{\tilde{\varphi }})Y= & {} ({\overline{\nabla }}_X J)Y+\frac{1}{2}\big [(JY)(\theta (X))-X(\theta (JY))-\theta ([JY,X])\big ]{\tilde{\xi }} \\&-\theta ({\overline{\nabla }}_X JY){\tilde{\xi }}-\theta (Y)X+\theta (X)\theta (Y){\tilde{\xi }}+\theta (J{\overline{\nabla }}_XY){\tilde{\xi }}\\= & {} ({\overline{\nabla }}_X J)Y+\mathrm{d}\theta (JY,X){\tilde{\xi }}-\theta (({\overline{\nabla }}_X J)Y){\tilde{\xi }}-\theta (Y)X+\theta (X)\theta (Y){\tilde{\xi }}\\= & {} ({\overline{\nabla }}_X J)Y+{\overline{g}}(X,Y){\tilde{\xi }}-{\tilde{\eta }}(({\overline{\nabla }}_X J)Y){\tilde{\xi }}-{\tilde{\eta }}(Y)X+{\tilde{\eta }}(X){\tilde{\eta }}(Y){\tilde{\xi }}\\= & {} ({\overline{\nabla }}_X J)Y-{\tilde{\eta }}(({\overline{\nabla }}_X J)Y){\tilde{\xi }}+{\tilde{g}}(X,Y){\tilde{\xi }}-{\tilde{\eta }}(Y)X, \end{aligned}$$

that is:

$$\begin{aligned} ({\tilde{\nabla }}_X{\tilde{\varphi }})Y=-{\tilde{\varphi }}^2 ({\overline{\nabla }}_X J)Y+{\tilde{g}}(X,Y){\tilde{\xi }}-{\tilde{\eta }}(Y)X, \end{aligned}$$

and since $({\overline{M}}, J, {\overline{g}})$ is a Kählerian manifold i.e. ${\overline{\nabla }} J=0$ then

$$\begin{aligned} ({\tilde{\nabla }}_X{\tilde{\varphi }})Y={\tilde{g}}(X,Y){\tilde{\xi }}- {\tilde{\eta }}(Y)X. \end{aligned}$$

(16)

This completes the proof.

5 From Sasakian strucrure to 1-parameter family Sasakian structures

In this section, we merge the two previous constructions to give a direct way to get a 1-parameter family of Sasakian manifolds from a single Sasakian manifold i.e. instead of going from the Sasaki case to the Kählerian case and then to the Sasaki case again, one can construct a direct transfer bridge to the family of Sasakienne structures from a single Sasakian structure.

The main theorem in this paper is the following:

Theorem 1

Let $(M,\varphi , \xi , \eta , g)$ be a Sasakian manifold. The product ${\tilde{M}}= {\mathbf {R}}^2 \times M$ provided with the almost contact metric structure $({\tilde{\varphi }} , {\tilde{\xi }} , {\tilde{\eta }}, {\tilde{g}})$ such that

$$\begin{aligned}&{\tilde{g}}=dt^2 + f^2 g + f^2 (f'^2 -1) \eta \otimes \eta + {\tilde{\eta }} \otimes {\tilde{\eta }},\\&{\tilde{\eta }}= dr + f^2 \eta , \qquad {\tilde{\xi }}= \partial _r, \end{aligned}$$

and for all X vector field on M

$$\begin{aligned} \left\{ \begin{array}{ll} {\tilde{\varphi }} \partial _r=0,\\ {\tilde{\varphi }} \partial _t=\frac{-1}{ff'}\big ( \xi - f^2 \partial _r\big ),\\ {\tilde{\varphi }} X=\varphi X + ff' \eta (X)\partial _t,\\ \end{array} \right. \end{aligned}$$

where $f=f(t)$ and $ff'\ne 0$ everywhere, is a 1-parameter family of Sasakian manifold.

Proof

Follows from Propositions (4) and (7).

Example 2

Basing on Example 1 and taking dimension 3, we get the Sasakian structure

$$\begin{aligned} g= & {} \frac{1}{4}\left( \begin{array}{lll} 1+y^2 &{}\quad 0 &{}\quad -y \\ 0 &{}\quad 1 &{}\quad 0 \\ -y &{}\quad 0 &{}\quad 1 \end{array} \right) ,\qquad \varphi =\left( \begin{array}{lll} 0 &{}\quad 1 &{}\quad 0 \\ -1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad y &{}\quad 0 \end{array} \right) ,\\ \xi= & {} 2 \frac{\partial }{\partial z},\qquad \eta = \frac{1}{2}( dz-y dx). \end{aligned}$$

So, using Theorem 1, we can define a 1-parameter family of Sasakian structures $({\tilde{\varphi }} , {\tilde{\xi }} , {\tilde{\eta }}, {\tilde{g}})$ on ${\mathbf {R}}^{4}$ as follows

$$\begin{aligned} {\tilde{g}}= & {} \frac{1}{4}\left( \begin{array}{lllll} 4 &{}\quad 0 &{}\quad -2y f^2 &{}\quad 0 &{}\quad 2f^2 \\ 0 &{}\quad 4 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ -2yf^2 &{}\quad 0 &{}\quad f^2 \big (1+y^2(f^2+f'^2 ) \big ) &{}\quad 0 &{} \quad -y f^2 (f^2+f'^2 )\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad f^2 &{} \quad 0 \\ 2f^2&{}\quad 0 &{}\quad -y f^2 (f^2+f'^2 ) &{} \quad 0 &{}\quad f^2 (f^2+f'^2 ) \end{array} \right) , \\ {\tilde{\varphi }}= & {} \left( \begin{array}{lllll} 0 &{}\quad \frac{f}{f'} &{}\quad 0 &{} \quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{} \quad -\frac{1}{2}ff'y &{}\quad 0 &{}\quad \frac{1}{2}ff'\\ 0 &{} \quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{} \quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 \\ 0&{} \quad -\frac{2}{ff'} &{} \quad 0 &{}\quad y &{}\quad 0 \end{array} \right) , \\ {\tilde{\xi }}= & {} \frac{\partial }{\partial r},\qquad {\tilde{\eta }}= dr +\frac{1}{2}f^2( dz-y dx). \end{aligned}$$