Two Lie Groups with Π-Structures That Exchange Their Membership in Two Basic Classes

1. Preliminaries for $\Pi$-Manifolds

Let $(\mathcal{M},\varphi ,\xi ,\eta ,g)$ be an almost paracontact almost paracomplex Riemannian manifold (in short, $\Pi$-manifold). This means that $\mathcal{M}$ is a $(2n+1)$-dimensional real differentiable manifold, $\varphi$ is a paracontact endomorphism of the tangent bundle $T\mathcal{M}$ of $\mathcal{M}$, $\xi$ is a characteristic vector field and $\eta$ is its dual 1-form, g and $\tilde{g}$ are compatible Riemannian metrics, such that:

$$\begin{array}{cc}& {\varphi }^2=\iota -\eta \otimes \xi ,\eta \left(\xi \right)=1,\eta \circ \varphi =0,\varphi \xi =0,\mathrm{tr}\varphi =0,\hfill \\ & g(\varphi x,\varphi y)=g(x,y)-\eta \left(x\right)\eta \left(y\right),\tilde{g}(x,y)=g(x,\varphi y)+\eta \left(x\right)\eta \left(y\right),\hfill \end{array}$$

where $\iota$ denotes the identity on $T\mathcal{M}$ [1,2].

It has been established that $\tilde{g}$ is, similarly to g, a compatible metric with $(\varphi ,\xi ,\eta )$, while $\tilde{g}$ is a pseudo-Riemannian metric of signature $(n+1,n)$. We use the notation $\Pi =(\varphi ,\xi ,\eta ,g,\tilde{g})$ for the tensor structure on $\mathcal{M}$ defined above. Therefore, the corresponding manifold is called a $\Pi$-manifold and is denoted by $(\mathcal{M},\Pi )$.

The tensor F of type $(0,3)$, defined by $F(x,y,z)=g\left(\left({\nabla }_x\varphi \right)y,z\right)$ for the Levi-Civita connection ∇ corresponding to g, plays an fundamental role in the differential geometry of the studied manifolds. The fundamental properties of F with respect to the underlying structure are expressed as

$$F(x,y,z)=F(x,z,y)=-F(x,\varphi y,\varphi z)+\eta \left(y\right)F(x,\xi ,z)+\eta \left(z\right)F(x,y,\xi ).$$

The Lee forms associated with the structure $(\varphi ,\xi ,\eta )$ are the following 1-forms determined by F:

$$\theta \left(z\right)=g^{ij}F(e_i,e_j,z),{\theta }^{*}\left(z\right)=g^{ij}F(e_i,\varphi e_j,z),\omega \left(z\right)=F(\xi ,\xi ,z),$$

where $g^{ij}$ denote the components of the inverse matrix of g with respect to a basis $\{\xi ;e_i\}$, $i=1,2,\dots ,2n$, of $T_p\mathcal{M}$ at an arbitrary point $p\in \mathcal{M}$.

The classification of $\Pi$-manifolds was introduced in [2]. It comprises 11 basic classes ${\mathcal{F}}_1,\dots ,{\mathcal{F}}_{11}$ and is based on the properties of F. In this paper we focus on three of them: ${\mathcal{F}}_8$, ${\mathcal{F}}_9$ and ${\mathcal{F}}_{10}$, which have the following definitions:

$$\begin{array}{cc}\hfill {\mathcal{F}}_8:& F(x,y,z)=F(x,y,\xi )\eta \left(z\right)+F(x,z,\xi )\eta \left(y\right),\hfill \\ & F(x,y,\xi )=F(y,x,\xi )=-F(x̑,y̑,\xi );\hfill \\ \hfill {\mathcal{F}}_9:& F(x,y,z)=F(x,y,\xi )\eta \left(z\right)+F(x,z,\xi )\eta \left(y\right),\hfill \\ & F(x,y,\xi )=-F(y,x,\xi )=-F(x̑,y̑,\xi );\hfill \\ \hfill {\mathcal{F}}_{10}:& F(x,y,z)=-\eta \left(x\right)F(\xi ,y̑,z̑);\hfill \end{array}$$

The corresponding components $F_i$ of F for the classes ${\mathcal{F}}_i$ are described in [3]:

Let $\tilde{\nabla }$ be the Levi-Civita connection induced by the associated metric $\tilde{g}$ on $\Pi$-manifold. The tensor field $\tilde{F}$ of type $(1,2)$ is defined as follows:

$$\tilde{F}(x,y,z)=\tilde{g}(\left({\tilde{\nabla }}_x)y,z\right).$$

In [4], the following expression of $\tilde{F}$ in terms of F is given:

$$\begin{array}{cc}\hfill 2\tilde{F}(x,y,z)=& F(y̑,z,x)+F(z̑,y,x)-F(y,z̑,x)-F(z,y̑,x)\hfill \\ & +\eta \left(x\right)\left\{F\right(y,z,\xi )-F(z̑,y̑,\xi )\hfill \\ & \phantom{+\eta \left(x\right)\{}+F(z,y,\xi )-F(y̑,z̑,\xi )\}\hfill \\ & +\eta \left(y\right)\left\{F\right(x,z,\xi )-F(z̑,x̑,\xi )+F(x,z̑,\xi \left)\right\}\hfill \\ & +\eta \left(z\right)\left\{F\right(x,y,\xi )-F(y̑,x̑,\xi )+F(x,y̑,\xi \left)\right\}.\hfill \end{array}$$

(1)

Then, in [5] it is established that, on every $\Pi$-manifold, the corresponding Lee forms associated with F and $\tilde{F}$ coincide. In particular, the following equalities hold:

$$\tilde{\theta }=\theta ,{\tilde{\theta }}^{*}={\theta }^{*},\tilde{\omega }=\omega .$$

In [5], it is obtained the relations between the fundamental tensors F and $\tilde{F}$ associated with the Levi-Civita connections of g and $\tilde{g}$, as follows:

$$\begin{array}{cc}\hfill 2F(x,y,z)& =\tilde{F}(\varphi y,z,x)-\tilde{F}(y,\varphi z,x)+\tilde{F}(\varphi z,y,x)-\tilde{F}(z,\varphi y,x)\hfill \\ & +\eta \left(x\right)\{\tilde{F}(y,z,\xi )-\tilde{F}(\varphi z,\varphi y,\xi )\hfill \\ & \phantom{+\eta \left(x\right)\{}+\tilde{F}(z,y,\xi )-\tilde{F}(\varphi y,\varphi z,\xi )\}\hfill \\ & +\eta \left(y\right)\{\tilde{F}(x,z,\xi )-\tilde{F}(\varphi z,\varphi x,\xi )+\tilde{F}(x,\varphi z,\xi )\}\hfill \\ & +\eta \left(z\right)\{\tilde{F}(x,y,\xi )-\tilde{F}(\varphi y,\varphi x,\xi )+\tilde{F}(x,\varphi y,\xi )\}.\hfill \end{array}$$

(2)

Subsequently, these relations are applied to investigate the form of $\tilde{F}$ within the basic classes of $\Pi$-manifolds arising from the alternative classification. The basic classes discussed here have the following definitional conditions:

$$\begin{array}{cc}\hfill {\mathcal{F}}_8:& \tilde{F}(x,y,z)=\tilde{\eta }\left(z\right)\tilde{F}(x,y,\xi )+\tilde{\eta }\left(y\right)\tilde{F}(x,z,\xi ),\hfill \\ & \tilde{F}(x,y,\xi )=\tilde{F}(y,x,\xi )=-\tilde{F}(x̑,y̑,\xi );\hfill \\ \hfill {\mathcal{F}}_9:& \tilde{F}(x,y,z)=\tilde{\eta }\left(z\right)\tilde{F}(x,y,\xi )+\tilde{\eta }\left(y\right)\tilde{F}(x,z,\xi ),\hfill \\ & \tilde{F}(x,y,\xi )=-\tilde{F}(y,x,\xi )=-\tilde{F}(x̑,y̑,\xi );\hfill \\ \hfill {\mathcal{F}}_{10}:& \tilde{F}(x,y,z)=-\tilde{\eta }\left(x\right)\tilde{F}(\xi ,y̑,z̑).\hfill \end{array}$$

The relationship between the two classifications of a $\Pi$-manifold with respect to the metrics g and $\tilde{g}$ is characterized by the following correspondence. The classes ${\mathcal{F}}_s$ and ${\tilde{\mathcal{F}}}_s$ coincide for $s=0,1,\dots ,7,11$, while the classes ${\mathcal{F}}_9$ and ${\mathcal{F}}_{10}$ are interchanged under the change of metric. The class ${\mathcal{F}}_8$ has a more complex correspondence: with respect to the other metric, it may be associated with a direct sum of the classes ${\tilde{\mathcal{F}}}_8$, ${\tilde{\mathcal{F}}}_9$, and ${\tilde{\mathcal{F}}}_{10}$, and conversely. These direct sums are generally irreducible into their individual components, with the exception of the trivial class ${\mathcal{F}}_0\equiv {\tilde{\mathcal{F}}}_0$. The direct sums ${\mathcal{F}}_9\oplus {\mathcal{F}}_{10}$ and ${\mathcal{F}}_8\oplus {\mathcal{F}}_9\oplus {\mathcal{F}}_{10}$ are preserved when the metric is changed from g to $\tilde{g}$, and vice versa. Hence, they are invariant under the transition between the two classifications, according to [5].

The $(1,3)$-curvature tensor R of the connection ∇ is defined, in the usual manner, by $R=\left[\nabla ,\nabla \right]-{\nabla }_{\left[\right]}$. Using the metric g, the corresponding $(0,4)$-tensor, denoted by the same symbol R, is given by the equality $R(x,y,z,w)=g\left(R\right(x,y)z,w)$.

The Ricci tensor $\rho$ and the scalar curvature $\tau$ for R as well as their associated quantities are defined as follows:

$$\begin{array}{cc}\rho (y,z)=g^{ij}R(e_i,y,z,e_j),\hfill & \tau =g^{ij}\rho (e_i,e_j),\hfill \\ {\rho }^{*}(y,z)=g^{ij}R(e_i,y,z,{ȇ}_j),\hfill & {\tau }^{*}=g^{ij}{\rho }^{*}(e_i,e_j).\hfill \end{array}$$

Let $\alpha$ be a non-degenerate 2-plane in $T_p\mathcal{M}$, $p\in \mathcal{M}$, having a basis $\{x,y\}$. The sectional curvature $k(\alpha ;p)$ is determined by:

$$k(\alpha ;p)={\displaystyle \frac{R(x,y,y,x)}{{\pi }_1(x,y,y,x)}}.$$

2. Example: Two Lie Groups with $\Pi$-Structures That Exchange the Corresponding Basic Classes Under the Replacement of the Compatible Metric

Let $\mathcal{L}$ be a three-dimensional real connected Lie group and $\mathfrak{l}$ its associated Lie algebra, with $\{E_0,E_1,E_2\}$ a basis of left-invariant vector fields on $\mathfrak{l}$. Then a $\Pi$-structure $(\varphi ,\xi ,\eta ,g,\tilde{g})$ can be determined in the following way:

$$\begin{array}{cc}& \varphi E_0=0,\varphi E_1=E_2,\varphi E_2=E_1,\hfill \\ & \xi =E_0,\eta \left(E_0\right)=1,\eta \left(E_1\right)=\eta \left(E_2\right)=0,\hfill \\ & g(E_i,E_j)={\delta }_{ij},\tilde{g}(E_i,E_j)=g(E_i,\varphi E_j)+\eta \left(E_i\right)\eta \left(E_j\right),i,j\in \{0,1,2\}.\hfill \end{array}$$

(3)

In [6] it is proved that the manifold $(\mathcal{L},\xi ,\eta ,g,\tilde{g})$, introduced as above, is a $\Pi$-manifold and belongs to each of the basic classes if and only if the commutators of the basis vector fields are defined in a certain way. For the basic classes ${\mathcal{F}}_9$ and ${\mathcal{F}}_{10}$ we have the following:

$$\begin{array}{cccc}{\mathcal{F}}_9:\hfill & [E_0,E_1]=\mu E_1,\hfill & [E_0,E_2]=-\mu E_2,\hfill & [E_1,E_2]=0,\hfill \\ {\mathcal{F}}_{10}:\hfill & [E_0,E_1]=-{\displaystyle \frac{1}{2}}\nu E_2,\hfill & [E_0,E_2]={\displaystyle \frac{1}{2}}\nu E_1,\hfill & [E_1,E_2]=0,\hfill \end{array}$$

(4)

where $\mu$, $\nu$ are arbitrary real parameters.

In [6] it is also shown that

$$\mu =F_{120}=F_{102}=-F_{210}=-F_{201},\nu =F_{011}=-F_{022}$$

are the only non-zero components of F with respect to the basis $\{E_0,E_1,E_2\}$.

Recall from [6] that the components $F_9$ and $F_{10}$ of F in the corresponding basic classes ${\mathcal{F}}_9$ and ${\mathcal{F}}_{10}$ are expressed as follows:

$$\begin{array}{cc}\hfill F_9(x,y,z)& =\mu \left\{x^1\left(y^0{z}^2+y^2{z}^0\right)-x^2\left(y^0{z}^1+y^1{z}^0\right)\right\},\hfill \\ \hfill F_{10}(x,y,z)& =\nu x^0\left(y^1{z}^1-y^2{z}^2\right),\hfill \end{array}$$

(5)

where $x=x^i{E}_i,y=y^i{E}_i,z=z^i{E}_i$ are arbitrary vector fields on $\mathfrak{l}$ and $F_{ijk}$ stands for $F(E_i,E_j,E_k)$. Using the well-known Koszul equality

$$\begin{array}{cc}\hfill 2g({\nabla }_{x}y,z)=& xg(y,z)+yg(z,x)-zg(x,y)\hfill \\ & +g\left(\right[x,y],z)+g\left(\right[z,x],y)+g\left(\right[z,y],x),\hfill \end{array}$$

the Levi-Civita connection ∇ of g satisfies

$$\begin{array}{ccc}{\nabla }_{E_1}{E}_0=-\mu E_1,\hfill & {\nabla }_{E_2}{E}_0=\mu E_2,\hfill & {\nabla }_{E_1}{E}_1=-{\nabla }_{E_2}{E}_2=\mu E_0,\hfill \\ {\nabla }_{E_0}{E}_1=-{\displaystyle \frac{1}{2}}\nu E_2,\hfill & {\nabla }_{E_0}{E}_2={\displaystyle \frac{1}{2}}\nu E_1.\hfill & \hfill \end{array}$$

(6)

Therefore, the only non-zero component of the curvature tensor are

$$\begin{array}{cc}\hfill R_{0101}=R_{0202}& =-R_{1212}=-{\displaystyle \frac{1}{2}}{\rho }_{00}=-{\rho }_{12}^{*}=-{\rho }_{21}^{*}\hfill \\ & =-{\displaystyle \frac{1}{2}}\tau =-k_{01}=-k_{02}=k_{12}={\mu }^2.\hfill \end{array}$$

In [6] it is proved that if $(\mathcal{L},\varphi ,\xi ,\eta ,g)$ belongs to ${\mathcal{F}}_9$, then it has negative scalar curvature, negative sectional curvatures of the basic $\xi$-sections $\left\{E_0,E_1\right\}$, $\left\{E_0,E_2\right\}$ and positive sectional curvature of the basic $\varphi$-holomorphic section $\left\{E_1,E_2\right\}$. If $(\mathcal{L},\varphi ,\xi ,\eta ,g)$ belongs to ${\mathcal{F}}_{10}$, then it is flat.

We use (1) to express the components of $\tilde{F}$ in terms of F:

$$\begin{array}{cc}\hfill 2{\tilde{F}}_{0jk}=& F_{\tilde{j}k0}+F_{\tilde{k}j0}-F_{j\tilde{k}0}-F_{k\tilde{j}0}+{\eta }_j\{F_{00k}+F_{00\tilde{k}}\}\hfill \\ & +F_{jk0}-F_{\tilde{k}\tilde{j}0}+F_{kj0}-F_{\tilde{j}\tilde{k}0}+{\eta }_k\{F_{00j}+F_{00\tilde{j}}\}\hfill \\ \hfill 2{\tilde{F}}_{i0k}=& F_{\tilde{k}i0}+F_{ik0}-F_{\tilde{k}\tilde{i}0}+F_{i\tilde{k}0}.\hfill \end{array}$$

(7)

Similarly, taking into account (2), we express the components of F in terms of $\tilde{F}$:

$$\begin{array}{cc}\hfill 2F_{ijk}=& {\tilde{F}}_{\tilde{j}ki}-{\tilde{F}}_{j\tilde{k}i}+{\tilde{F}}_{\tilde{k}ji}-{\tilde{F}}_{k\tilde{j}i}+{\eta }_i\{{\tilde{F}}_{jk0}-{\tilde{F}}_{\tilde{k}\tilde{j}0}+{\tilde{F}}_{kj0}-{\tilde{F}}_{\tilde{j}\tilde{k}0}\},\hfill \\ & +{\eta }_j\{{\tilde{F}}_{ik0}-{\tilde{F}}_{\tilde{k}\tilde{i}0}+{\tilde{F}}_{i\tilde{k}0}\}+{\eta }_k\{{\tilde{F}}_{ij0}-{\tilde{F}}_{\tilde{j}\tilde{i}0}+{\tilde{F}}_{i\tilde{j}0}\},\hfill \\ \hfill 2F_{0jk}=& {\tilde{F}}_{\tilde{j}k0}-{\tilde{F}}_{j\tilde{k}0}+{\tilde{F}}_{\tilde{k}j0}-{\tilde{F}}_{k\tilde{j}0}+{\tilde{F}}_{jk0}-{\tilde{F}}_{\tilde{k}\tilde{j}0}+{\tilde{F}}_{kj0}-{\tilde{F}}_{\tilde{j}\tilde{k}0}\hfill \\ & +{\eta }_j\{{\tilde{F}}_{00k}-{\tilde{F}}_{00\tilde{k}}\}+{\eta }_k\{{\tilde{F}}_{00j}-{\tilde{F}}_{00\tilde{j}}\},\hfill \\ \hfill 2F_{i0k}=& -{\tilde{F}}_{0i\tilde{k}}+{\tilde{F}}_{\tilde{k}i0}+{\eta }_i{\tilde{F}}_{00k}+{\tilde{F}}_{ik0}-{\tilde{F}}_{\tilde{k}\tilde{i}0}+{\tilde{F}}_{i\tilde{k}0}.\hfill \end{array}$$

(8)

Then, we have the following

Theorem 2.1.

If a manifold $(M,\Pi )$ belongs to a basic class ${\mathcal{F}}_9$ or ${\mathcal{F}}_{10}$, with respect to g or $\tilde{g}$, respectively, then the same manifold belongs to a corresponding class with respect to the other metric, given by the following equivalences:

$$\left(i\right){\mathcal{F}}_9⟺{\tilde{\mathcal{F}}}_{10},\left(ii\right){\mathcal{F}}_{10}⟺{\tilde{\mathcal{F}}}_9.$$

Proof. (i) Suppose $(M,\Pi )$ belongs to ${\mathcal{F}}_9$. We use (7) to obtain that the only non-zero components of $\tilde{F}$ are:

$${\tilde{F}}_{011}=-2\mu ,{\tilde{F}}_{022}=2\mu ,\tilde{\nu }={\tilde{F}}_{011}=-{\tilde{F}}_{022}=-2\mu ,$$

so that $\tilde{\nu }=-2\mu$.

Hence, we obtain ${\mathcal{F}}_9⟹{\tilde{\mathcal{F}}}_{10}$, and therefore $(M,\Pi )$ belongs to ${\tilde{\mathcal{F}}}_{10}$.

Conversely, let $(M,\Pi )$ belong to ${\tilde{F}}_{10}$. Using (8), we obtain the only non-zero components of F as follows:

$$F_{102}=F_{120}=-F_{201}=-F_{210}=-{\displaystyle \frac{1}{2}}\tilde{\nu },$$

where $\mu =-{\displaystyle \frac{1}{2}}\tilde{\nu }$. So the implication ${\tilde{\mathcal{F}}}_{10}⟹{\mathcal{F}}_9$ is obtained.

We then summarize the last two implications as the equivalence ${\mathcal{F}}_9⟺{\tilde{\mathcal{F}}}_{10}$.

(ii) The equivalence ${\mathcal{F}}_{10}⟺{\tilde{\mathcal{F}}}_9$ is obtained in an analogous way. As obtained in [6], if $(M,\Pi )$ belongs to ${\mathcal{F}}_{10}$, then the only nonzero components of F are $\nu =F_{011}=-F_{022}$.

Using (7), we obtain that the only non-zero components of $\tilde{F}$ are:

$${\tilde{F}}_{102}={\tilde{F}}_{120}=-{\tilde{F}}_{201}=-{\tilde{F}}_{210}=-{\displaystyle \frac{1}{2}}\nu ,$$

where $\tilde{\mu }=-{\displaystyle \frac{1}{2}}\nu$. Hence, we have ${\mathcal{F}}_{10}⟹{\tilde{\mathcal{F}}}_9$.

Conversely, if $(M,\Pi )$ belongs to ${\tilde{F}}_9$, by using (8), we calculate the only non-zero components of F as follows:

$$F_{011}=-F_{022}=-2\tilde{\mu },$$

where $\nu =-2\tilde{\mu }$. So, we have established that $(M,\Pi )$ belongs to ${\mathcal{F}}_{10}$.

Finally, this proves the equivalence ${\mathcal{F}}_{10}⟺{\tilde{F}}_9$ and completes the proof. □