Characterization of Π-Manifolds by the Difference Between the Two Levi-Civita Connections of the Structure

1. Introduction

In 1976, Ichiro Sato introduced in [13] the notion of an (almost) paracontact structure compatible with a Riemannian metric on differentiable manifolds of arbitrary dimension, as an analogue of almost contact Riemannian manifolds. Over the following years, this topic was studied by several authors, e. g. in [14–1,12,16].

One of the directions for the further development of this research is the combination of this structure with an almost paracomplex structure ([2]) on the paracontact distribution, together with a compatibility condition on the given metric that generates another metric. In the case when the given metric is Riemannian, a classification of these manifolds was proposed in 2001 by the first author and M. Staikova [8]. Later, these manifolds, briefly called Riemannian $\mathrm{\Pi }$-manifolds, were studied extensively by the first author of the present article together with S. Ivanov, H. Manev, V. Tavkova, e. g. [4,5,6,9,10].

A characteristic property of the manifolds under study with the so-called structure $\mathrm{\Pi }$ is that every metric g compatible with $\mathrm{\Pi }$ gives rise to another metric $\tilde{g}$, which is compatible with $\mathrm{\Pi }$ in the same way and similarly generates g. Each of these metrics uniquely determines the corresponding Levi-Civita connection, denoted by ∇ and $\tilde{\mathrm{\nabla }}$, respectively.

As is well known, the difference between the pair of Levi-Civita connections $(\mathrm{\nabla },\tilde{\mathrm{\nabla }})$, generated by the pair of metrics $(g,\tilde{g})$ on a manifold, is not an affine connection but a tensor field of type $(1,2)$. The tensor of the difference of $(\mathrm{\nabla },\tilde{\mathrm{\nabla }})$, defined by $\mathrm{\Phi }=\tilde{\mathrm{\nabla }}-\mathrm{\nabla }$, characterizes how geometric parallel transport changes between the two metrics, or measures how the geodesics of one metric differ from those of the other. It is formally known as the deformation tensor or connection difference tensor. This tensor should not be confused with similar tensors arising from the difference of two affine connections, one or both of which are not symmetric, i. e. possess torsion.

Let us compare $\mathrm{\Pi }$-manifolds with the more widely studied almost paracontact metric manifolds (cf. [17]). On the former, the metric g acts as an isometry on the paracontact distribution $\mathcal{H}$, whereas on the latter, g acts as an anti-isometry on $\mathcal{H}$. For this reason, the tensor associated with g through the structure is a 2-form rather than a metric, as in the case considered here. Therefore, there is no associated metric with its own Levi-Civita connection, and consequently a difference tensor such as $\mathrm{\Phi }$ cannot be defined on almost paracontact metric manifolds.

In [3], a difference tensor of type $\mathrm{\Phi }$ is considered on almost contact manifolds with B-metric, also known as almost contact complex Riemannian manifolds, which are classified in terms of $\mathrm{\Phi }$ in [11]. For these manifolds, there also exists a pair of mutually related metrics compatible with the structure, but both metrics are necessarily pseudo-Riemannian. This property is not shared by the $\mathrm{\Pi }$-manifolds discussed in the present paper.

The aim of the present paper is to study the tensor $\mathrm{\Phi }$ on the considered manifolds and, more specifically, to determine the characteristics of the basic classes of $\mathrm{\Pi }$-manifolds in terms of this tensor. The paper is organized as follows. Section 2 recalls the basic definitions and properties of the manifolds under study and comments on the possible definiteness of the two basic metrics. Particular attention is paid to the classification of $\mathrm{\Pi }$-manifolds used in the paper and to the corresponding properties of the fundamental tensor F in each basic class. The main results are presented in Section 3. They concern the properties of $\mathrm{\Phi }$, as well as their relation with the properties of F, which provide a characterization of the basic classes of $\mathrm{\Pi }$-manifolds in terms of $\mathrm{\Phi }$. Finally, in Section 4, we consider explicit examples of 3-dimensional Lie groups regarded as $\mathrm{\Pi }$-manifolds belonging to all basic classes. The corresponding characteristics are computed in terms of the components of $\mathrm{\Phi }$, thereby illustrating the obtained theoretical results.

2. $\mathrm{\Pi }$-manifolds

A real differentiable manifold $\mathcal{M}$ is said to have an almost paracontact structure $(\varphi ,\xi ,\eta )$ or for short a $(\varphi ,\xi ,\eta )$-structure, if $\mathcal{M}$ admits a tensor field $\varphi$ of type $(1,1)$, a vector field $\xi$ and 1-form $\eta$ satisfying the conditions [13]:

$$\eta \left(\xi \right)=1,{\varphi }^2=i-\eta \otimes \xi ,$$

(1)

where i denotes the identity on the tangent bundle of $\mathcal{M}$.

Further, $x,y,z$ will stand for arbitrary differentiable vector fields on $\mathcal{M}$ or arbitrary differentiable vectors in the tangent space $T_p\mathcal{M}$ at $p\in \mathcal{M}$.

If $\mathcal{M}$ has dimension $2n+1$ and for its $(\varphi ,\xi ,\eta )$-structure the condition $\mathrm{tr}\left(\varphi \right)=0$ is satisfied, then it is called an almost paracontact almost paracomplex manifold and is denoted by $(\mathcal{M},\varphi ,\xi ,\eta )$ [8,9].

If $(\mathcal{M},\varphi ,\xi ,\eta )$ admits a metric g with the following property

$$g(\varphi x,\varphi y)=g(x,y)-\eta \left(x\right)\eta \left(y\right),$$

(2)

then g is called a compatible metric with the $(\varphi ,\xi ,\eta )$-structure of $\mathcal{M}$.

In view of (2), compatible metrics can be either Riemannian or pseudo-Riemannian. We study the case when g is Riemannian [8]. For this reason, $\mathcal{M}$ is sometimes called an almost paracontact almost paracomplex Riemannian manifold [9].

The equalities (1) and (2) imply immediately

$$g(x,\xi )=\eta \left(x\right),\eta \circ \varphi =0,\varphi \xi =0,g(x,\varphi y)=g(\varphi x,y).$$

(3)

Then there also exists a metric $\tilde{g}$ associated with g through the $(\varphi ,\xi ,\eta )$-structure, which is defined by:

$$\tilde{g}(x,y)=g(x,\varphi y)+\eta \left(x\right)\eta \left(y\right).$$

(4)

As a consequence of (4), we have $\tilde{g}(x,\xi )=\eta \left(x\right)$, which means that the paracontact form $\tilde{\eta }(·)=\tilde{g}(·,\xi )$ of $\xi$ through $\tilde{g}$ coincides with $\eta$, the paracontact form with respect to g, i. e. $\tilde{\eta }=\eta$.

Similarly, g is associated with $\tilde{g}$ via the $(\varphi ,\xi ,\eta )$-structure, i. e.

$$g(x,y)=\tilde{g}(x,\varphi y)+\eta \left(x\right)\eta \left(y\right).$$

In [8] it is shown that $\tilde{g}$ is a compatible metric with $(\mathcal{M},\varphi ,\xi ,\eta )$ as is g, i. e. the analogue of (2) for $\tilde{g}$ holds, but $\tilde{g}$ is a pseudo-Riemannian metric of signature $(n+1,n)$.

For brevity, we denote by $\mathrm{\Pi }$ the tensor structure introduced above on $\mathcal{M}$, setting $\mathrm{\Pi }=(\varphi ,\xi ,\eta ,g,\tilde{g})$. Then we will refer to this manifold as a Π-manifold and denote it by $(\mathcal{M},\mathrm{\Pi })$.

On each $(\mathcal{M},\mathrm{\Pi })$, two complementary tangent distributions are defined that are mutually orthogonal with respect to g and $\tilde{g}$. One is the horizontal (or paracontact) distribution $\mathcal{H}=ker\left(\eta \right)$, having dimension $2n$, and the other is the vertical distribution $\mathcal{V}=span\left(\xi \right)$ and it is 1-dimensional.

 Proposition 1.

Let us consider $(\mathcal{M},\varphi ,\xi ,\eta )$ with compatible metrics g and $\tilde{g}$ introduced by (2) and (4). If one of the metrics g and $\tilde{g}$ is pseudo-Riemannian with index $2k$, $k\in \{0,1,\dots ,n\}$ (in particular, Riemannian at $k=0$), then the other is pseudo-Riemannian of signature $(n+1,n)$ and therefore its restriction to $\mathcal{H}$ is neutral.

 Proof. 

We present the arguments assuming an arbitrary signature for g and derive the signature of $\tilde{g}$. The reverse direction is analogous.

Let $e=\{e_i;\varphi e_i;e_j;\varphi e_j;\xi \}$, $i\in \{1,\dots ,k\}$, $j\in \{k+1,\dots ,n\}$, $k\in \{0,1,\dots ,n\}$ be an arbitrary $(\varphi ,\xi ,\eta )$-adapted basis of $T_p\mathcal{M}$ at $p\in \mathcal{M}$, which is orthogonal with respect to g and is pseudonormalized as follows:

$$g(e_i,e_i)=g(\varphi e_i,\varphi e_i)=-1,g(e_j,e_j)=g(\varphi e_j,\varphi e_j)=1.$$

This means that g has index $2k$, i. e. signature $(2n+1-2k,2k)$.

We construct an eigenbasis of $\varphi$, denoted by $b=\{b_i;b_{k+i};b_j;b_{k+j};\xi \}$, as follows:

$$\begin{array}{cc}{b}_i={\displaystyle \frac{1}{\sqrt{2}}}\left(e_i+\varphi e_i\right),\hfill & b_{k+i}={\displaystyle \frac{1}{\sqrt{2}}}\left(e_i-\varphi e_i\right),\hfill \\ b_j={\displaystyle \frac{1}{\sqrt{2}}}\left(e_j+\varphi e_j\right),\hfill & b_{k+j}={\displaystyle \frac{1}{\sqrt{2}}}\left(e_j-\varphi e_j\right).\hfill \end{array}$$

It is easy to compute the following

$$\begin{array}{c}\varphi b_i=b_i,\varphi b_{k+i}=-b_{k+i},\varphi b_j=b_j,\varphi b_{k+j}=-b_{k+j},\\ g(b_i,b_i)=g(b_{k+i},b_{k+i})=-1,g(b_j,b_j)=g(b_{k+j},b_{k+j})=1,\\ \tilde{g}(b_i,b_i)=\tilde{g}(b_{k+j},b_{k+j})=-1,\tilde{g}(b_j,b_j)=\tilde{g}(b_{k+i},b_{k+i})=1,\\ g(b_s,b_t)=g(b_s,\xi )=\tilde{g}(b_s,b_t)=\tilde{g}(b_s,\xi )=0,s,t\in \{1,2,\dots ,2n\},s\ne t.\end{array}$$

Then, $\mathrm{diag}\left(\varphi \right)=(I_k,-I_k,I_{n-k},-I_{n-k},1)$ is the form of the matrix of $\varphi$ with respect to b, where $I_k$ denotes the identity matrix of size k.

It is obvious that the signatures of g and $\tilde{g}$ are $\left(2\right(n-k)+1,2k)$ and $(n+1,n)$, respectively, i. e. the signature of $\tilde{g}$ does not depend on the choice of k. □

 Corollary 1.

Under the conditions of Proposition 1, the two metrics g and $\tilde{g}$ can have the same signature when $dim\mathcal{M}=4k+1$, i. e. $n=2k$ for $k\in \mathbb{N}$ is satisfied. Then this signature is $(2k+1,2k)$, i. e. both metrics are neutral on $\mathcal{H}$.

 Proof. 

This follows directly from Proposition 1. □

Let us note the following regarding Proposition 1 and Corollary 1 for the two lowest dimensions of the considered manifolds. In the case $dim\mathcal{M}=3$, i. e. $n=1$, then $k\in \{0,1\}$ is possible and we have $\mathrm{sign}\left(g\right)=(3-2k,2k)$ (in particular $(3,0)$ for $k=0$ and $(1,2)$ for $k=1$) and $\mathrm{sign}\left(\tilde{g}\right)=(2,1)$. When $dim\mathcal{M}=5$ holds, i. e. $n=2$, the possible values of the index $2k$ of g correspond to $k\in \{0,1,2\}$. Therefore, the signatures are the following $\mathrm{sign}\left(g\right)=(5-2k,2k)$ (in particular $(3,0)$ for $k=0$, $(3,2)$ for $k=1$ and $(1,4)$ for $k=2$) and $\mathrm{sign}\left(\tilde{g}\right)=(3,2)$.

2.1. The Classification of $\mathrm{\Pi }$-Manifolds Regarding the Riemannian Metric of the $\mathrm{\Pi }$-Structure

The tensor F of type $(0,3)$ plays a fundamental role in the geometry of the considered manifolds. For the Levi-Civita connection ∇ of g, it is defined as follows:

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

(5)

The basic properties of F with respect to the structure are the following:

$$\begin{array}{c}F(x,y,z)=F(x,z,y)=-F(x,\varphi y,\varphi z)+\eta \left(y\right)F(x,\xi ,z)+\eta \left(z\right)F(x,y,\xi ).\end{array}$$

(6)

The following relations are immediate consequences of (1), (3) and (6):

$$\begin{array}{c}F(x,y,\varphi z)=-F(x,\varphi y,z)+\eta \left(z\right)F(x,\varphi y,\xi )+\eta \left(y\right)F(x,\varphi z,\xi ),\\ F(x,\varphi y,\varphi z)=-F(x,{\varphi }^{2}y,{\varphi }^{2}z),F(x,\varphi y,{\varphi }^{2}z)=-F(x,{\varphi }^{2}y,\varphi z),\\ F(x,\xi ,\xi )=0.\end{array}$$

(7)

The relations between $\mathrm{\nabla }\xi$, $\mathrm{\nabla }\eta$, and F are:

$$\left({\mathrm{\nabla }}_x\eta \right)\left(y\right)=g({\mathrm{\nabla }}_x\xi ,y)=-F(x,\varphi y,\xi ).$$

(8)

If $\{e_i,\xi \}$$(i=1,2,\dots ,2n)$ is a basis of the tangent space $T_p\mathcal{M}$ at an arbitrary point $p\in \mathcal{M}$ and $\left(g^{ij}\right)$ is the inverse of the matrix $\left(g_{ij}\right)$ of g, then the following 1-forms are associated with F:

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

(9)

These 1-forms are also known as the Lee forms of the considered manifolds and the following identities are always valid:

$${\theta }^{*}\circ \varphi =-\theta \circ {\varphi }^2,{\theta }^{*}\circ {\varphi }^2=-\theta \circ \varphi ,\omega \left(\xi \right)=0.$$

(10)

 Lemma 1.

On every Π-manifold there are other traces of F using the basis $\{e_i,\xi \}$ and ϕ, besides those used for the Lee forms, but they are always zero, namely:

$$\begin{array}{c}{g}^{ij}F(x,e_i,e_j)=g^{ij}F(x,\varphi e_i,\varphi e_j)\hfill \\ \phantom{g^{ij}F(x,e_i,e_j)}=g^{ij}F(x,e_i,\varphi e_j)=g^{ij}F(x,\varphi e_i,e_j)=0.\hfill \end{array}$$

(11)

 Proof. 

From the second property in (6), by substituting $y=e_i$ and $z=e_j$ and then contracting with the inverse metric tensor $g^{ij}$, taking into account (7), we obtain the equality $g^{ij}F(x,e_i,e_j)=0$.

Similarly, using the third property in (6), and substituting again $y=e_i$ and $z=e_j$, followed by contraction with $g^{ij}$, we derive $g^{ij}F(x,\varphi e_i,e_j)=0$.

Combining these relations with the properties of F in (6), it follows that all traces in (11) vanish. □

A classification of $\mathrm{\Pi }$-manifolds in terms of F is given in [8]. This classification is derived by decomposing the vector space $\mathbb{F}$ of all tensors F with the properties (6). The decomposition of $\mathbb{F}$ is orthogonal and invariant under the action of the structure group of these manifolds, which is $\mathcal{O}\left(n\right)\times \mathcal{O}\left(n\right)\times 1$, where $\mathcal{O}\left(n\right)$ is the orthogonal group. This decomposition consists of 11 subspaces ${\mathbb{F}}_{\ell }$, $\ell \in \{1,\dots ,11\}$.

A $\mathrm{\Pi }$-manifold is said to belong to a basic class${\mathcal{F}}_{\ell }$, $\ell \in \{1,2,\dots ,11\}$, or is briefly called an ${\mathcal{F}}_{\ell }$-manifold if the tensor F belongs to the corresponding subspace ${\mathbb{F}}_{\ell }$. Thus, the classification consists of 11 basic classes ${\mathcal{F}}_{\ell }$. The intersection of all basic classes is the special class ${\mathcal{F}}_0$ determined by the condition $F(x,y,z)=0$.

In the present work, we use the following expressions equivalent to the defining conditions of ${\mathcal{F}}_{\ell }$ given in [8]:

$$\begin{array}{c}{\mathcal{F}}_1:F(x,y,z)={\displaystyle \frac{1}{2n}}\{g(\varphi x,\varphi y)\theta \left({\varphi }^{2}z\right)+g(\varphi x,\varphi z)\theta \left({\varphi }^{2}y\right)\hfill \\ \phantom{{\mathcal{F}}_1:F(x,y,z)={\displaystyle \frac{1}{2n}}}-g(x,\varphi y)\theta \left(\varphi z\right)-g(x,\varphi z)\theta \left(\varphi y\right)\};\hfill \\ {\mathcal{F}}_2:F(\xi ,y,z)=0,F(x,\xi ,z)=0,\hfill \\ \phantom{{\mathcal{F}}_2:}F(x,y,\varphi z)+F(y,z,\varphi x)+F(z,x,\varphi y)=0,\theta =0;\hfill \\ {\mathcal{F}}_3:F(\xi ,y,z)=0,F(x,\xi ,z)=0,\hfill \\ \phantom{{\mathcal{F}}_3:}F(x,y,z)+F(y,z,x)+F(z,x,y)=0;\hfill \\ {\mathcal{F}}_4:F(x,y,z)={\displaystyle \frac{1}{2n}}\theta \left(\xi \right)\left\{g(\varphi x,\varphi y)\eta \left(z\right)+g(\varphi x,\varphi z)\eta \left(y\right)\right\};\hfill \\ {\mathcal{F}}_5:F(x,y,z)={\displaystyle \frac{1}{2n}}{\theta }^{*}\left(\xi \right)\left\{g(x,\varphi y)\eta \left(z\right)+g(x,\varphi z)\eta \left(y\right)\right\};\hfill \\ {\mathcal{F}}_6:F(x,y,z)=F(x,y,\xi )\eta \left(z\right)+F(x,z,\xi )\eta \left(y\right),\hfill \\ \phantom{{\mathcal{F}}_6:}F(x,y,\xi )=F(y,x,\xi )=F(\varphi x,\varphi y,\xi ),\theta \left(\xi \right)={\theta }^{*}\left(\xi \right)=0;\hfill \\ {\mathcal{F}}_7:F(x,y,z)=F(x,y,\xi )\eta \left(z\right)+F(x,z,\xi )\eta \left(y\right),\left[4pt\right]\hfill \\ \phantom{{\mathcal{F}}_7:}F(x,y,\xi )=-F(y,x,\xi )=F(\varphi x,\varphi y,\xi );\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 \\ \phantom{{\mathcal{F}}_8:}F(x,y,\xi )=F(y,x,\xi )=-F(\varphi x,\varphi y,\xi );\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 \\ \phantom{{\mathcal{F}}_9:}F(x,y,\xi )=-F(y,x,\xi )=-F(\varphi x,\varphi y,\xi );\hfill \\ {\mathcal{F}}_{10}:F(x,y,z)=-\eta \left(x\right)F(\xi ,\varphi y,\varphi z)\hfill \\ {\mathcal{F}}_{11}:F(x,y,z)=\eta \left(x\right)\left\{\eta \right(y\left)w\right(z)+\eta (z\left)w\right(y\left)\right\}.\hfill \end{array}$$

(12)

The Lee forms of $\mathrm{\Pi }$-manifolds belonging to each basic class ${\mathcal{F}}_{\ell }$, $\ell \in \{1,\dots ,11\}$ are described in [7] as follows:

$$\begin{array}{cccc}\theta =\theta \circ {\varphi }^2,\hfill & {\theta }^{*}=-{\theta }^{*}\circ {\varphi }^2,\hfill & \omega =0\hfill & \mathrm{for}\ell =1;\hfill \\ \theta =\theta \left(\xi \right)\eta ,\hfill & {\theta }^{*}=0,\hfill & \omega =0\hfill & \mathrm{for}\ell =4;\hfill \\ \theta =0,\hfill & {\theta }^{*}={\theta }^{*}\left(\xi \right)\eta ,\hfill & \omega =0\hfill & \mathrm{for}\ell =5;\hfill \\ \theta =0,\hfill & {\theta }^{*}=0,\hfill & \omega =\omega \circ {\varphi }^2\hfill & \mathrm{for}\ell =11;\hfill \\ \theta =0,\hfill & {\theta }^{*}=0,\hfill & \omega =0\hfill & \mathrm{for}\mathrm{the}\mathrm{remaining}\hfill \\ \hfill & \hfill & & \mathrm{values}\mathrm{of}\ell .\hfill \end{array}$$

(13)

Note that $(\mathcal{M},\mathrm{\Pi })$ belongs to the class ${\mathcal{F}}_{\ell }$, $\ell \in \{1,\dots ,11\}$, if and only if the equality $F=F_{\ell }$ holds. Moreover, $(\mathcal{M},\mathrm{\Pi })$ belongs to a direct sum of two or more basic classes, i. e. we denote e. g. $(\mathcal{M},\mathrm{\Pi })\in {\mathcal{F}}_{{\ell }_1}\oplus {\mathcal{F}}_{{\ell }_2}$ for ${\ell }_1,{\ell }_2\in \{1,\dots ,11\}$, ${\ell }_1\ne {\ell }_2$, if and only if the expression $F=F_{{\ell }_1}+F_{{\ell }_2}$ holds.

3. Characterization of $\mathrm{\Pi }$-Manifolds with Respect to the Difference Tensor of the Two Levi-Civita Connections

The presence of two Levi-Civita connections ∇ and $\tilde{\mathrm{\nabla }}$, corresponding to the metrics g and $\tilde{g}$, motivates the study of the deviation of the associated connection $\tilde{\mathrm{\nabla }}$ from the initial connection ∇.

We introduce a tensor field $\mathrm{\Phi }$ of type $(1,2)$ by the equality

$$\mathrm{\Phi }(x,y)={\tilde{\mathrm{\nabla }}}_{x}y-{\mathrm{\nabla }}_{x}y,$$

and call it the difference tensor between ∇ and $\tilde{\mathrm{\nabla }}$.

This tensor is also known as the deformation tensor of $\tilde{\mathrm{\nabla }}$ with respect to ∇, due to the relation ${\tilde{\mathrm{\nabla }}}_{x}y={\mathrm{\nabla }}_{x}y+\mathrm{\Phi }(x,y)$, which is equivalent to the above definition.

Since both Levi-Civita connections are torsion-free, it follows that $\mathrm{\Phi }$ is symmetric with respect to its two arguments, i. e.

$$\mathrm{\Phi }(x,y)=\mathrm{\Phi }(y,x).$$

(14)

The corresponding tensor of $\mathrm{\Phi }$ with respect to g, which is of type $(0,3)$, is denoted by the same letter and is defined by:

$$\mathrm{\Phi }(x,y,z)=g\left(\mathrm{\Phi }\right(x,y),z).$$

(15)

Due to (14), the following property holds:

$$\mathrm{\Phi }(x,y,z)=\mathrm{\Phi }(y,x,z).$$

(16)

 Proposition 2.

The general expression of Φ in terms of F is the following

$$\begin{array}{c}2\mathrm{\Phi }(x,y,z)=F(x,y,\varphi z)+F(y,x,\varphi z)-F(\varphi z,x,y)\hfill \\ \phantom{2\mathrm{\Phi }(x,y,z)=}-\eta \left(x\right)\left\{F\right(y,z,\xi )-F(\varphi z,\varphi y,\xi \left)\right\}\hfill \\ \phantom{2\mathrm{\Phi }(x,y,z)=}-\eta \left(y\right)\left\{F\right(x,z,\xi )-F(\varphi z,\varphi x,\xi \left)\right\}\hfill \\ \phantom{2\mathrm{\Phi }(x,y,z)=}-\eta \left(z\right)\left\{F\right(\xi ,x,y)-F(x,y,\xi )+F(x,\varphi y,\xi )\hfill \\ \phantom{2\mathrm{\Phi }(x,y,z)=-\eta \left(z\right)\left\{F\right(\xi ,x,y)}-F(y,x,\xi )+F(y,\varphi x,\xi )\}\hfill \\ \phantom{2\mathrm{\Phi }(x,y,z)=}+\eta \left(z\right)\left\{\omega \right(\varphi x\left)\eta \right(y)+\omega (\varphi y\left)\eta \right(x\left)\right\}.\hfill \end{array}$$

(17)

It has the following immediate consequences:

$$\begin{array}{c}2\mathrm{\Phi }(x,\xi ,z)=F(\xi ,x,\varphi z)+F(x,\varphi z,\xi )-F(\varphi z,x,\xi )-\eta \left(x\right)\omega \left(z\right)\hfill \\ \phantom{2\mathrm{\Phi }(x,\xi ,z)=F(\xi ,x,\varphi z)}-F(x,z,\xi )+F(\varphi z,\varphi x,\xi ),\hfill \\ 2\mathrm{\Phi }(x,y,\xi )=-F(\xi ,x,y)+F(x,y,\xi )-F(x,\varphi y,\xi )+\eta \left(x\right)\omega \left(\varphi y\right)\hfill \\ \phantom{2\mathrm{\Phi }(x,y,\xi )=-F(\xi ,x,y)}+F(y,x,\xi )-F(y,\varphi x,\xi )+\eta \left(y\right)\omega \left(\varphi x\right),\hfill \\ 2\mathrm{\Phi }(\xi ,y,z)=F(\xi ,y,\varphi z)+F(y,\varphi z,\xi )-F(\varphi z,y,\xi )-\eta \left(y\right)\omega \left(z\right),\hfill \\ \phantom{2\mathrm{\Phi }(\xi ,y,z)=F(\xi ,y,\varphi z)}-F(y,z,\xi )+F(\varphi z,\varphi y,\xi ),\hfill \\ \mathrm{\Phi }(x,\xi ,\xi )=0,\mathrm{\Phi }(\xi ,\xi ,z)=\omega \left(\varphi z\right)-\omega \left(z\right).\hfill \end{array}$$

(18)

 Proof. 

We start with the well-known Koszul formula for a metric and its Levi-Civita connection, in this case for $\tilde{g}$ and $\tilde{\mathrm{\nabla }}$:

$$\begin{array}{c}2\tilde{g}\left({\tilde{\mathrm{\nabla }}}_{x}y,z\right)=x\left(\tilde{g}(y,z)\right)+y\left(\tilde{g}(x,z)\right)-z\left(\tilde{g}(x,y)\right)\hfill \\ \phantom{2\tilde{g}\left({\tilde{\mathrm{\nabla }}}_{x}y,z\right)=}+\tilde{g}\left([x,y],z\right)+\tilde{g}\left([z,x],y\right)+\tilde{g}\left([z,y],x\right).\hfill \end{array}$$

(19)

Then we use the relation between g and $\tilde{g}$ in (4), the dependency between $\mathrm{\nabla }\eta$ and F in (8) and the properties of F in (6). Thus, after straightforward but lengthy computations, we derive the formula for $\mathrm{\Phi }$ in terms of F, given in (17).

The consequences in (18) of the last expression follow immediately by replacing some of the arguments of $\mathrm{\Phi }$ with $\xi$, taking into account (1) and (3). □

 Proposition 3.

For any Π-manifold the following properties of Φ are valid:

$$\begin{array}{c}\mathrm{\Phi }(x,\varphi y,\varphi z)+\mathrm{\Phi }(x,{\varphi }^{2}y,{\varphi }^{2}z)+\mathrm{\Phi }(x,\varphi z,\varphi y)+\mathrm{\Phi }(x,{\varphi }^{2}z,{\varphi }^{2}y)=0,\end{array}$$

(20)

$$\begin{array}{c}\mathrm{\Phi }(\xi ,x,y)+\mathrm{\Phi }(\xi ,x,\varphi y)+\mathrm{\Phi }(x,y,\xi )+\mathrm{\Phi }(x,\varphi y,\xi )=0,\end{array}$$

(21)

$$\begin{array}{c}\mathrm{\Phi }(x,y,\xi )+\mathrm{\Phi }(\xi ,x,\varphi y)=F(x,y,\xi )-F(x,\varphi y,\xi ),\end{array}$$

(22)

$$\begin{array}{c}\mathrm{\Phi }(x,{\varphi }^{2}y,\varphi z)+\mathrm{\Phi }(x,{\varphi }^{2}z,\varphi y)+\mathrm{\Phi }(x,\varphi y,{\varphi }^{2}z)+\mathrm{\Phi }(x,\varphi z,{\varphi }^{2}y)=0,\end{array}$$

(23)

$$\begin{array}{c}\begin{array}{c}\mathrm{\Phi }(x,y,\varphi z)+\mathrm{\Phi }(x,\varphi y,z)+\mathrm{\Phi }(x,z,\varphi y)+\mathrm{\Phi }(x,\varphi z,y)\hfill \\ =-\left\{\mathrm{\Phi }(\xi ,x,y)+\mathrm{\Phi }(x,y,\xi )\right\}\eta \left(z\right)-\left\{\mathrm{\Phi }(\xi ,x,z)+\mathrm{\Phi }(x,z,\xi )\right\}\eta \left(y\right),\hfill \end{array}\end{array}$$

(24)

$$\begin{array}{c}\mathrm{\Phi }(\xi ,\xi ,\varphi z)=-\mathrm{\Phi }(\xi ,\xi ,z).\end{array}$$

(25)

 Proof. 

Using (17), we derive

$$\begin{array}{c}\mathrm{\Phi }(x,\varphi y,\varphi z)+\mathrm{\Phi }(x,{\varphi }^{2}y,{\varphi }^{2}z)+\mathrm{\Phi }(x,\varphi z,\varphi y)+\mathrm{\Phi }(x,{\varphi }^{2}z,{\varphi }^{2}y)\hfill \\ =F(x,\varphi y,{\varphi }^{2}z)+F(x,{\varphi }^{2}y,\varphi z).\hfill \end{array}$$

(26)

Then, by applying the third equality in (7) to (26), we deduce (20).

We compute the expression $2\mathrm{\Phi }(x,y,\varphi z)+2\mathrm{\Phi }(x,z,\varphi y)$ by using (17) and obtain

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

(27)

Replacing z with $\xi$ in the last formula, we consistently obtain the following, due to the penultimate property in (18)

$$\begin{array}{c}\mathrm{\Phi }(\xi ,x,\varphi y)=-\mathrm{\Phi }(x,y,\xi )+F(x,y,\xi )-F(x,\varphi y,\xi ),\hfill \\ \mathrm{\Phi }(\xi ,x,y)=-\mathrm{\Phi }(x,\varphi y,\xi )+F(x,\varphi y,\xi )-F(x,y,\xi ).\hfill \end{array}$$

(28)

The sum of the equalities in (28) yields the identity in (21).

On the other hand, the equalities in (28) imply the following

$$F(x,y,\xi )-F(x,\varphi y,\xi )=\mathrm{\Phi }(x,y,\xi )+\mathrm{\Phi }(\xi ,x,\varphi y)=-\mathrm{\Phi }(x,\varphi y,\xi )-\mathrm{\Phi }(\xi ,x,y),$$

which proves (22).

Then, combining (27) and (28), we get

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

(29)

Substituting $z=\xi$ into (29), we obtain the identity $0=0$. Therefore, $F(x,y,\xi )$, and consequently $F(x,y,z)$, cannot be expressed solely in terms of $\mathrm{\Phi }$.

Taking into account (6), the equality in (29) can be reformulated as

$$F(x,\varphi y,\varphi z)=-\mathrm{\Phi }(x,y,\varphi z)-\mathrm{\Phi }(x,z,\varphi y)+\eta \left(y\right)\mathrm{\Phi }(\xi ,x,\varphi z)+\eta \left(z\right)\mathrm{\Phi }(\xi ,x,\varphi y)$$

or in an equivalent form as

$$F(x,\varphi y,\varphi z)=-\mathrm{\Phi }(x,{\varphi }^{2}y,\varphi z)-\mathrm{\Phi }(x,{\varphi }^{2}z,\varphi y).$$

(30)

Because of (6), the formula in (30) implies the mutually equivalent identities (23) and (24) via (1).

The identity in (25) follows immediately from the last equality in (18). □

Continuing the above reasoning, we observe that there is no formula expressing F with arbitrary arguments solely in terms of $\mathrm{\Phi }$. This means that two different basic classes according to F can be characterized by the same property of $\mathrm{\Phi }$.

We introduce the following 1-forms, which are defined as the fundamental traces of $\mathrm{\Phi }$ with respect to g and the $\mathrm{\Pi }$-structure for the basis $\{e_i,\xi \}$$(i=1,2,\dots ,2n)$:

$$\begin{array}{c}\alpha \left(z\right)=g^{ij}\mathrm{\Phi }(e_i,e_j,z),{\alpha }^{*}\left(z\right)=g^{ij}\mathrm{\Phi }(e_i,\varphi e_j,z),\widehat{\alpha }\left(z\right)=\mathrm{\Phi }(\xi ,\xi ,z).\hfill \end{array}$$

(31)

Note that the remaining possible traces of $\mathrm{\Phi }(x,y,z)$ are always zero, i. e. we have

$$g^{ij}\mathrm{\Phi }(x,e_i,e_j)=g^{ij}\mathrm{\Phi }(x,e_i,\varphi e_j)=g^{ij}\mathrm{\Phi }(x,\varphi e_i,e_j)=g^{ij}\mathrm{\Phi }(x,\varphi e_i,\varphi e_j)=0.$$

Indeed, calculating $g^{ij}\mathrm{\Phi }(x,e_i,e_j)$ from (17), we get that it vanishes, given (11). The other traces are obtained in a similar way.

 Proposition 4.

The 1-forms α, ${\alpha }^{*}$ and $\widehat{\alpha }$ associated with Φ are related to the Lee forms as follows:

$$\begin{array}{c}\alpha =-{\theta }^{*}+\theta \left(\xi \right)\eta ,{\alpha }^{*}=-\theta +{\theta }^{*}\left(\xi \right)\eta ,\end{array}$$

(32)

$$\begin{array}{c}\widehat{\alpha }=\omega \circ \varphi -\omega .\end{array}$$

(33)

 Proof. 

We contract (18), taking into account (31) and (9), as well as the properties of F from (6) and those of the $\mathrm{\Pi }$-structure in (1)–(3). We also use the relations in (10) to obtain

$$\begin{array}{c}\begin{array}{c}\alpha =\theta \circ \varphi +\{\theta \left(\xi \right)-{\theta }^{*}\left(\xi \right)\}\eta ,\\ \alpha \circ {\varphi }^2=\theta \circ \varphi =-{\theta }^{*}\circ {\varphi }^2,\alpha \left(\xi \right)=\theta \left(\xi \right)-{\theta }^{*}\left(\xi \right),\end{array}\end{array}$$

(34)

$$\begin{array}{c}\begin{array}{c}{\alpha }^{*}={\theta }^{*}\circ \varphi -\{\theta \left(\xi \right)-{\theta }^{*}\left(\xi \right)\}\eta ,\\ {\alpha }^{*}\circ {\varphi }^2={\theta }^{*}\circ \varphi =-\theta \circ {\varphi }^2,{\alpha }^{*}\left(\xi \right)={\theta }^{*}\left(\xi \right)-\theta \left(\xi \right),\end{array}\end{array}$$

(35)

as well as (33).

From (34) and (35) it follows that $\alpha$ and ${\alpha }^{*}$ coincide respectively with ${\theta }^{*}$ and $-\theta$ on $\mathcal{H}$, while $\alpha$ and ${\alpha }^{*}$ are opposite and equal to the difference between $\theta$ and ${\theta }^{*}$ on $\mathcal{V}$, namely we have:

$$\begin{array}{c}\alpha {|}_{\mathcal{H}}=-{\theta }^{*}{|}_{\mathcal{H}}=-{\theta }^{*}\circ {\varphi }^2,{\alpha }^{*}{|}_{\mathcal{H}}=-\theta {|}_{\mathcal{H}}={\alpha }^{*}\circ {\varphi }^2,\\ \alpha {|}_{\mathcal{V}}=-{\alpha }^{*}{|}_{\mathcal{V}}=\left(\theta -{\theta }^{*}\right){|}_{\mathcal{V}}=\left\{\theta \left(\xi \right)-{\theta }^{*}\left(\xi \right)\right\}\eta .\end{array}$$

(36)

Considering (36), we obtain the expressions of $\alpha$ and ${\alpha }^{*}$ in terms of $\theta$ and ${\theta }^{*}$, given in (32), which are simpler than those in (34) and (35). □

On the other hand, considering the relations in Proposition 4, we can eliminate the use of Lee forms in expressing the dependencies between $\alpha$ and ${\alpha }^{*}$.

 Proposition 5.

The 1-forms α, ${\alpha }^{*}$ and $\widehat{\alpha }$ are interconnected as follows:

$$\begin{array}{c}\alpha =-{\alpha }^{*}\circ \varphi -{\alpha }^{*}\left(\xi \right)\eta ,{\alpha }^{*}=-\alpha \circ \varphi -\alpha \left(\xi \right)\eta ,\end{array}$$

(37)

$$\begin{array}{c}\widehat{\alpha }\circ \varphi =-\widehat{\alpha }.\end{array}$$

(38)

 Proof. 

Taking into account (34) and (35), we derive the following identities:

$$\begin{array}{c}\alpha \circ {\varphi }^2=-{\alpha }^{*}\circ \varphi ,\alpha \circ \varphi =-{\alpha }^{*}\circ {\varphi }^2,\alpha \left(\xi \right)=-{\alpha }^{*}\left(\xi \right).\end{array}$$

Hence, each of the 1-forms $\alpha$ and ${\alpha }^{*}$ is expressed in terms of the other as in (37).

In addition, (33) implies the property $\widehat{\alpha }\left(\xi \right)=0$.

Therefore, their horizontal and vertical components are as follows:

$$\begin{array}{ccc}\alpha {|}_{\mathcal{H}}=-{\alpha }^{*}\circ \varphi ,\hfill & {\alpha }^{*}{|}_{\mathcal{H}}=-\alpha \circ \varphi ,\hfill & \widehat{\alpha }{|}_{\mathcal{H}}=\widehat{\alpha },\hfill \\ \alpha {|}_{\mathcal{V}}=-{\alpha }^{*}\left(\xi \right)\eta ,\hfill & {\alpha }^{*}{|}_{\mathcal{V}}=-\alpha \left(\xi \right)\eta ,\hfill & \widehat{\alpha }{|}_{\mathcal{V}}=0.\hfill \end{array}$$

Moreover, due to (33) and the third identity in (10), we have (38). □

 Theorem 1.

The difference tensor Φ has the following properties on a Π-manifold belonging to a basic class ${\mathcal{F}}_{\ell }$, $\ell \in \{1,\dots ,11\}$:

$$\begin{array}{c}\ell =1:\mathrm{\Phi }(x,y,z)={\displaystyle \frac{1}{2n}}\left\{g(\varphi x,\varphi y)\alpha \left({\varphi }^{2}z\right)-g(x,\varphi y)\alpha \left(\varphi z\right)\right\};\hfill \\ \ell =2:\mathrm{\Phi }(\xi ,y,z)=\mathrm{\Phi }(x,y,\xi )=0,\mathrm{\Phi }(x,\varphi y,\varphi z)=-\mathrm{\Phi }(x,y,z),\hfill \\ \phantom{\ell =2:}\alpha =0;\hfill \\ \ell =3:\mathrm{\Phi }(\xi ,y,z)=\mathrm{\Phi }(x,y,\xi )=0,\mathrm{\Phi }(\varphi x,\varphi y,z)=-\mathrm{\Phi }(x,y,z);\hfill \\ \ell =4:\mathrm{\Phi }(x,y,z)={\displaystyle \frac{1}{2n}}\alpha \left(\xi \right)\left\{g(\varphi x,\varphi y)-g(x,\varphi y)\right\}\eta \left(z\right);\hfill \\ \ell =5:\mathrm{\Phi }(x,y,z)={\displaystyle \frac{1}{2n}}\alpha \left(\xi \right)\left\{g(\varphi x,\varphi y)-g(x,\varphi y)\right\}\eta \left(z\right);\hfill \\ \ell =6:\mathrm{\Phi }(x,y,z)=\mathrm{\Phi }(x,y,\xi )\eta \left(z\right),\hfill \\ \phantom{\ell =2:}\mathrm{\Phi }(x,y,\xi )=\mathrm{\Phi }(\varphi x,\varphi y,\xi )=-\mathrm{\Phi }(x,\varphi y,\xi ),\alpha \left(\xi \right)=0;\hfill \\ \ell =7:\mathrm{\Phi }(x,y,z)=\eta \left(x\right)\mathrm{\Phi }(\xi ,y,z)+\eta \left(y\right)\mathrm{\Phi }(x,\xi ,z),\hfill \\ \phantom{\ell =7:}\mathrm{\Phi }(\xi ,x,y)=-\mathrm{\Phi }(\xi ,y,x)=\mathrm{\Phi }(\xi ,\varphi x,\varphi y);\hfill \\ \ell =8:\mathrm{\Phi }(x,y,z)=\eta \left(x\right)\mathrm{\Phi }(\xi ,y,z)+\eta \left(y\right)\mathrm{\Phi }(x,\xi ,z)+\eta \left(z\right)\mathrm{\Phi }(x,y,\xi ),\hfill \\ \phantom{\ell =8:}\mathrm{\Phi }(\xi ,y,z)=\mathrm{\Phi }(\xi ,z,y)=-\mathrm{\Phi }(\xi ,\varphi y,\varphi z)\hfill \\ \phantom{\ell =8:\mathrm{\Phi }(\xi ,y,z)}=-\mathrm{\Phi }(y,z,\xi )=\mathrm{\Phi }(\varphi y,\varphi z,\xi );\hfill \\ \ell =9:\mathrm{\Phi }(x,y,z)=\eta \left(x\right)\mathrm{\Phi }(\xi ,y,z)+\eta \left(y\right)\mathrm{\Phi }(x,\xi ,z)+\eta \left(z\right)\mathrm{\Phi }(x,y,\xi ),\hfill \\ \phantom{\ell =9:}\mathrm{\Phi }(\xi ,y,z)=\mathrm{\Phi }(\xi ,z,y)=-\mathrm{\Phi }(\xi ,\varphi y,\varphi z)\hfill \\ \phantom{\ell =9:\mathrm{\Phi }(\xi ,y,z)}=-\mathrm{\Phi }(y,z,\xi )=\mathrm{\Phi }(\varphi y,\varphi z,\xi );\hfill \\ \ell =10:\mathrm{\Phi }(x,y,z)=\eta \left(x\right)\mathrm{\Phi }(\xi ,y,z)+\eta \left(y\right)\mathrm{\Phi }(x,\xi ,z)+\eta \left(z\right)\mathrm{\Phi }(x,y,\xi ),\hfill \\ \phantom{\ell =10:}\mathrm{\Phi }(\xi ,y,z)=-\mathrm{\Phi }(\xi ,z,y)=-\mathrm{\Phi }(\xi ,\varphi y,\varphi z)\hfill \\ \phantom{\ell =9:\mathrm{\Phi }(\xi ,y,z)}=-\mathrm{\Phi }(y,\varphi z,\xi )=\mathrm{\Phi }(\varphi y,z,\xi );\hfill \\ \ell =11:\mathrm{\Phi }(x,y,z)=-\eta \left(x\right)\eta \left(y\right)\widehat{\alpha }\left(\varphi z\right).\hfill \end{array}$$

(39)

 Proof. 

First, we will consider the main classes ${\mathcal{F}}_{\ell }$, $\ell \in \{1,4,5,11\}$, i. e. there F is expressed explicitly through the metrics and the Lee forms. Using the definition conditions (12) of ${\mathcal{F}}_{\ell }$ in the formula in (17), we derive an initial expression for $\mathrm{\Phi }$. Then, in this expression, we apply its consequences for the corresponding 1-forms. Thus we obtain the final expressions in (39) in these cases.

Second, we consider the classes ${\mathcal{F}}_{\ell }$, $\ell \in \{2,3\}$. Since $F(\xi ,y,z)=F(x,y,\xi )=0$ hold, the second and third formulas in (18) imply the first equalities in (39) for $\ell \in \{2,3\}$. On the other hand, applying (17) with $x\to \varphi x$ and $y\to \varphi y$, we obtain

$$2\mathrm{\Phi }(\varphi x,\varphi y,z)=F(y,x,\varphi z)-F(\varphi y,x,z)$$

(40)

after applying (6) and the last equality in (12) for ${\mathcal{F}}_2$. Then we put $y\to \varphi y$ and $z\to \varphi z$ in (40) and get

$$2\mathrm{\Phi }(\varphi x,y,\varphi z)=F(\varphi y,x,z)-F(y,x,\varphi z).$$

(41)

Therefore, (40) and (41) imply

$$\mathrm{\Phi }(\varphi x,\varphi y,z)=-\mathrm{\Phi }(\varphi x,y,\varphi z).$$

(42)

The property $\mathrm{\Phi }(x,\varphi y,\varphi z)=-\mathrm{\Phi }(x,y,z)$, given in (39) for $\ell =2$, is obtained after substituting $x\to \varphi x$ and $z\to \varphi z$ in (42). This property does not give us restrictions on $\alpha$. But, the first equality in (32) together with (13) for ${\mathcal{F}}_2$ implies that $\alpha$ is zero for this class. Thus, the characteristics for ${\mathcal{F}}_2$ in (39) are obtained. The computations and reasoning for ${\mathcal{F}}_3$ are performed in a similar manner.

Thirdly, we consider the case ${\mathcal{F}}_{\ell }$, $\ell \in \{6,7,8,9\}$, and then (17) takes the form:

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

(43)

Due to the properties of $F(x,y,\xi )$ in the definitions in (12) of ${\mathcal{F}}_{\ell }$ for $\ell \in \{6,7,8,9\}$, the expression in (43) simplifies. For $\ell =6$ it has the following form

$$\mathrm{\Phi }(x,y,z)=\mathrm{\Phi }(x,y,\xi )\eta \left(z\right),\mathrm{\Phi }(x,y,\xi )=F(x,y,\xi )-F(x,\varphi y,\xi ).$$

Then, from the last equality and the properties of $F(x,y,\xi )$ in ${\mathcal{F}}_6$, we obtain the properties of $\mathrm{\Phi }(x,y,\xi )$ given in (39) for ${\mathcal{F}}_6$. The additional condition there, namely $\alpha \left(\xi \right)=0$, follows from the vanishing of $\theta \left(\xi \right)$ and ${\theta }^{*}\left(\xi \right)$, according to (12) for $\ell =6$. The argumentation of the conditions for $\mathrm{\Phi }$ in (39) for $\ell \in \{7,8,9\}$ is similar, and there, no additional condition is required for the vanishing of the 1-form $\alpha$, because this fact follows from the properties of $\mathrm{\Phi }$ in the corresponding class.

Finally, the case ${\mathcal{F}}_{10}$ is similar to those for $\ell =6,7,8,9$, and therefore we do not discuss it separately. □

Let us note that according to Theorem 1, in the two pairs of basic classes: ${\mathcal{F}}_4$ and ${\mathcal{F}}_5$ on the one hand, and ${\mathcal{F}}_8$ and ${\mathcal{F}}_9$ on the other hand, the same properties of $\mathrm{\Phi }$ are satisfied, respectively. Therefore, when characterizing $\mathrm{\Pi }$-manifolds with respect to $\mathrm{\Phi }$, these pairs of basic classes cannot be distinguished.

 Corollary 2.

The 1-forms of Φ on an ${\mathcal{F}}_{\ell }$-manifold, $\ell \in \{1,\dots ,11\}$, satisfy the following equalities:

$$\begin{array}{cccc}\alpha =\alpha \circ {\varphi }^2=-{\alpha }^{*}\circ \varphi ,\hfill & {\alpha }^{*}={\alpha }^{*}\circ {\varphi }^2=-\alpha \circ \varphi ,\hfill & \widehat{\alpha }=0\hfill & \mathrm{for}\ell =1;\hfill \\ \alpha =\alpha \left(\xi \right)\eta =-{\alpha }^{*}\left(\xi \right)\eta ,\hfill & {\alpha }^{*}={\alpha }^{*}\left(\xi \right)\eta =-\alpha \left(\xi \right)\eta ,\hfill & \widehat{\alpha }=0\hfill & \mathrm{for}\ell \in \{4,5\};\hfill \\ \alpha =0,\hfill & {\alpha }^{*}=0,\hfill & \widehat{\alpha }=\widehat{\alpha }\circ {\varphi }^2\hfill & \mathrm{for}\ell =11;\hfill \\ \alpha =0,\hfill & {\alpha }^{*}=0,\hfill & \widehat{\alpha }=0\hfill & \mathrm{for}\mathrm{the}\mathrm{remaining}\mathrm{values}\mathrm{of}\ell .\hfill \end{array}$$

 Proof. 

The above dependencies between the 1-forms associated with $\mathrm{\Phi }$ follow directly from (39) and the properties of the $\mathrm{\Pi }$-structure. □

4. Examples of 3-Dimensional $\mathrm{\Pi }$-Manifolds of the Basic Classes

Let us consider a 3-dimensional real connected Lie group $\mathcal{L}$ and its associated Lie algebra $\mathfrak{l}$. Assume that $\{E_0,E_1,E_2\}$ is a basis of left-invariant vector fields on $\mathfrak{l}$. Then an almost paracontact almost paracomplex structure $(\varphi ,\xi ,\eta )$, together with a Riemannian metric g, can be defined by:

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

(44)

Then the nonzero components of the associated metric (4) with respect to the basis are given by

$$\tilde{g}(E_0,E_0)=\tilde{g}(E_1,E_2)=\tilde{g}(E_2,E_1)=1.$$

It is shown in [9] that the manifold $(\mathcal{L},\varphi ,\xi ,\eta ,g,\tilde{g})$ constructed as above is a 3-dimensional $\mathrm{\Pi }$-manifold. Moreover, it belongs to one of the basic classes ${\mathcal{F}}_{\ell }$, $\ell \in \{1,4,5,8,9,10,11\}$, if and only if the corresponding Lie algebra $\mathfrak{l}$ is determined by the following commutators:

$$\begin{array}{cccc}{\mathcal{F}}_1:\hfill & [E_0,E_1]=0,\hfill & [E_0,E_2]=0,\hfill & [E_1,E_2]={\displaystyle \frac{1}{2}}\{{\theta }_1{E}_1+{\theta }_2{E}_2\};\hfill \\ {\mathcal{F}}_4:\hfill & [E_0,E_1]={\displaystyle \frac{1}{2}}{\theta }_0{E}_2,\hfill & [E_0,E_2]={\displaystyle \frac{1}{2}}{\theta }_0{E}_1,\hfill & [E_1,E_2]=0;\hfill \\ {\mathcal{F}}_5:\hfill & [E_0,E_1]={\displaystyle \frac{1}{2}}{\theta }_0^{*}{E}_1,\hfill & [E_0,E_2]={\displaystyle \frac{1}{2}}{\theta }_0^{*}{E}_2,\hfill & [E_1,E_2]=0;\hfill \\ {\mathcal{F}}_8:\hfill & [E_0,E_1]=\lambda E_2,\hfill & [E_0,E_2]=-\lambda E_1,\hfill & [E_1,E_2]=2\lambda E_0;\hfill \\ {\mathcal{F}}_9:\hfill & [E_0,E_1]=\mu E_1,\hfill & [E_0,E_2]=-\mu E_2,\hfill & [E_1,E_2]=0;\hfill \\ {\mathcal{F}}_{10}:\hfill & [E_0,E_1]=-{\displaystyle \frac{1}{2}}\nu E_2,\hfill & [E_0,E_2]={\displaystyle \frac{1}{2}}\nu E_1,\hfill & [E_1,E_2]=0;\hfill \\ {\mathcal{F}}_{11}:\hfill & [E_0,E_1]={\omega }_2{E}_0,\hfill & [E_0,E_2]={\omega }_1{E}_0,\hfill & [E_1,E_2]=0.\hfill \end{array}$$

(45)

In the above equalities, ${\theta }_i=\theta \left(E_i\right)$, ${\theta }_i^{*}={\theta }^{*}\left(E_i\right)$, ${\omega }_i=\omega \left(E_i\right)$ are the nonzero components of the corresponding Lee forms for $i\in \{0,1,2\}$. Moreover, the coefficients $\lambda$, $\mu$, $\nu$ are used for the basic classes ${\mathcal{F}}_{\ell }$, $\ell \in \{8,9,10\}$, where all Lee forms vanish. The parameters $\lambda$, $\mu$, $\nu$ represent the nonzero components of $F_{ijk}=F(E_i,E_j,E_k)$, $i,j,k\in \{0,1,2\}$, in ${\mathcal{F}}_8$, ${\mathcal{F}}_9$, and ${\mathcal{F}}_{10}$, respectively, i. e. we have

$$\begin{array}{c}\lambda =F_{101}=F_{110}=-F_{202}=-F_{220},\mu =F_{102}=F_{120}=-F_{201}=-F_{210},\\ \nu =F_{011}=-F_{022}.\end{array}$$

The components of ∇ with respect to the basis are computed and the nonzero ones are as follows:

$$\begin{array}{cc}{\mathcal{F}}_1:\hfill & {\mathrm{\nabla }}_{E_1}{E}_1=-{\displaystyle \frac{1}{2}}{\theta }_1{E}_2,{\mathrm{\nabla }}_{E_1}{E}_2={\displaystyle \frac{1}{2}}{\theta }_1{E}_1,\hfill \\ \hfill & {\mathrm{\nabla }}_{E_2}{E}_1=-{\displaystyle \frac{1}{2}}{\theta }_2{E}_2,{\mathrm{\nabla }}_{E_2}{E}_2={\displaystyle \frac{1}{2}}{\theta }_2{E}_1;\hfill \\ {\mathcal{F}}_4:\hfill & {\mathrm{\nabla }}_{E_1}{E}_0=-{\displaystyle \frac{1}{2}}{\theta }_0{E}_2,{\mathrm{\nabla }}_{E_2}{E}_0=-{\displaystyle \frac{1}{2}}{\theta }_0{E}_1,\hfill \\ \hfill & {\mathrm{\nabla }}_{E_1}{E}_2={\mathrm{\nabla }}_{E_2}{E}_1={\displaystyle \frac{1}{2}}{\theta }_0{E}_0;\hfill \\ {\mathcal{F}}_5:\hfill & {\mathrm{\nabla }}_{E_1}{E}_0=-{\displaystyle \frac{1}{2}}{\theta }_0^{*}{E}_1,{\mathrm{\nabla }}_{E_2}{E}_0=-{\displaystyle \frac{1}{2}}{\theta }_0^{*}{E}_2,\hfill \\ \hfill & {\mathrm{\nabla }}_{E_1}{E}_1={\mathrm{\nabla }}_{E_2}{E}_2={\displaystyle \frac{1}{2}}{\theta }_0^{*}{E}_0;\hfill \\ {\mathcal{F}}_8:\hfill & {\mathrm{\nabla }}_{E_1}{E}_0=-\lambda E_2,{\mathrm{\nabla }}_{E_2}{E}_0=\lambda E_1,{\mathrm{\nabla }}_{E_1}{E}_2=-{\mathrm{\nabla }}_{E_2}{E}_1=\lambda E_0;\hfill \\ {\mathcal{F}}_9:\hfill & {\mathrm{\nabla }}_{E_1}{E}_0=-\mu E_1,{\mathrm{\nabla }}_{E_2}{E}_0=\mu E_2,{\mathrm{\nabla }}_{E_1}{E}_1=-{\mathrm{\nabla }}_{E_2}{E}_2=\mu E_0;\hfill \\ {\mathcal{F}}_{10}:\hfill & {\mathrm{\nabla }}_{E_0}{E}_1=-{\displaystyle \frac{1}{2}}\nu E_2,{\mathrm{\nabla }}_{E_0}{E}_2={\displaystyle \frac{1}{2}}\nu E_1;\hfill \\ {\mathcal{F}}_{11}:\hfill & {\mathrm{\nabla }}_{E_0}{E}_1={\omega }_2{E}_0,{\mathrm{\nabla }}_{E_0}{E}_2={\omega }_1{E}_0,{\mathrm{\nabla }}_{E_0}{E}_0=-{\omega }_2{E}_1-{\omega }_1{E}_2.\hfill \end{array}$$

(46)

The components $F_{ijk}=F(E_i,E_j,E_k)$ of F in the basis $\{E_0,E_1,E_2\}$ are calculated, taking into account (46), (5) and (44), and the nozero of them in the corresponding basic class ${\mathcal{F}}_{\ell }$ are:

$$\begin{array}{cc}{\mathcal{F}}_1:\hfill & F_{111}=-F_{122}={\theta }_1,F_{211}=-F_{222}=-{\theta }_2,\hfill \\ {\mathcal{F}}_4:\hfill & F_{101}=F_{110}=F_{202}=F_{220}={\displaystyle \frac{1}{2}}{\theta }_0,\hfill \\ {\mathcal{F}}_5:\hfill & F_{102}=F_{120}=F_{201}=F_{210}={\displaystyle \frac{1}{2}}{\theta }_0^{*},\hfill \\ {\mathcal{F}}_8:\hfill & F_{101}=F_{110}=-F_{202}=-F_{220}=\lambda ,\hfill \\ {\mathcal{F}}_9:\hfill & F_{102}=F_{120}=-F_{201}=-F_{210}=\mu ,\hfill \\ {\mathcal{F}}_{10}:\hfill & F_{011}=-F_{022}=\nu ,\hfill \\ {\mathcal{F}}_{11}:\hfill & F_{001}=F_{010}={\omega }_1,F_{002}=F_{020}={\omega }_2\hfill \end{array}$$

(47)

and the components of the Lee forms are as follows

$$\begin{array}{ccc}{\theta }_0=F_{110}+F_{220},\hfill & {\theta }_1=F_{111}=-F_{122}=-{\theta }_2^{*},\hfill & {\omega }_1=F_{001},\hfill \\ {\theta }_0^{*}=F_{120}+F_{210},\hfill & {\theta }_2=F_{222}=-F_{211}=-{\theta }_1^{*},\hfill & {\omega }_2=F_{002},\hfill \\ \hfill & \hfill & {\omega }_0=0.\hfill \end{array}$$

(48)

Now, using (19) and (45), we compute the components of $\tilde{\mathrm{\nabla }}$ and obtain the following nonzero components:

$$\begin{array}{cc}{\mathcal{F}}_1:\hfill & {\tilde{\mathrm{\nabla }}}_{E_1}{E}_1=-{\displaystyle \frac{1}{2}}{\theta }_2{E}_1,{\tilde{\mathrm{\nabla }}}_{E_1}{E}_2={\displaystyle \frac{1}{2}}{\theta }_2{E}_2,\hfill \\ \hfill & {\tilde{\mathrm{\nabla }}}_{E_2}{E}_1=-{\displaystyle \frac{1}{2}}{\theta }_1{E}_1,{\tilde{\mathrm{\nabla }}}_{E_2}{E}_2={\displaystyle \frac{1}{2}}{\theta }_1{E}_2;\hfill \\ {\mathcal{F}}_4:\hfill & {\tilde{\mathrm{\nabla }}}_{E_1}{E}_0=-{\displaystyle \frac{1}{2}}{\theta }_0{E}_2,{\tilde{\mathrm{\nabla }}}_{E_2}{E}_0=-{\displaystyle \frac{1}{2}}{\theta }_0{E}_1,\hfill \\ \hfill & {\tilde{\mathrm{\nabla }}}_{E_1}{E}_1={\tilde{\mathrm{\nabla }}}_{E_2}{E}_2={\displaystyle \frac{1}{2}}{\theta }_0{E}_0;\hfill \\ {\mathcal{F}}_5:\hfill & {\tilde{\mathrm{\nabla }}}_{E_1}{E}_0=-{\displaystyle \frac{1}{2}}{\theta }_0^{*}{E}_1,{\tilde{\mathrm{\nabla }}}_{E_2}{E}_0=-{\displaystyle \frac{1}{2}}{\theta }_0^{*}{E}_2,\hfill \\ \hfill & {\tilde{\mathrm{\nabla }}}_{E_1}{E}_2={\tilde{\mathrm{\nabla }}}_{E_2}{E}_1={\displaystyle \frac{1}{2}}{\theta }_0^{*}{E}_0;\hfill \end{array}$$

(49a)

$$\begin{array}{cc}{\mathcal{F}}_8:\hfill & {\tilde{\mathrm{\nabla }}}_{E_0}{E}_1=-\lambda E_1,{\tilde{\mathrm{\nabla }}}_{E_0}{E}_2=\lambda E_2,\hfill \\ \hfill & {\tilde{\mathrm{\nabla }}}_{E_1}{E}_0=-{\tilde{\mathrm{\nabla }}}_{E_2}{E}_0=-\lambda (E_1+E_2),\hfill \\ \hfill & {\tilde{\mathrm{\nabla }}}_{E_1}{E}_1=-{\tilde{\mathrm{\nabla }}}_{E_2}{E}_2={\tilde{\mathrm{\nabla }}}_{E_1}{E}_2=-{\tilde{\mathrm{\nabla }}}_{E_2}{E}_1=\lambda E_0;\hfill \\ {\mathcal{F}}_9:\hfill & {\tilde{\mathrm{\nabla }}}_{E_0}{E}_1=\mu E_1,{\tilde{\mathrm{\nabla }}}_{E_0}{E}_2=-\mu E_2;\hfill \\ {\mathcal{F}}_{10}:\hfill & {\tilde{\mathrm{\nabla }}}_{E_1}{E}_0={\displaystyle \frac{1}{2}}\nu E_2,{\tilde{\mathrm{\nabla }}}_{E_2}{E}_0=-{\displaystyle \frac{1}{2}}\nu E_1,\hfill \\ \hfill & {\tilde{\mathrm{\nabla }}}_{E_1}{E}_1=-{\tilde{\mathrm{\nabla }}}_{E_2}{E}_2=-{\displaystyle \frac{1}{2}}\nu E_0;\hfill \\ {\mathcal{F}}_{11}:\hfill & {\tilde{\mathrm{\nabla }}}_{E_0}{E}_0=-{\omega }_1{E}_1-{\omega }_2{E}_2,{\tilde{\mathrm{\nabla }}}_{E_0}{E}_1={\omega }_2{E}_0,{\tilde{\mathrm{\nabla }}}_{E_0}{E}_2={\omega }_1{E}_0.\hfill \end{array}$$

(49b)

We apply (15) and (16) to (46) and (Section 4) and obtain the following nonzero components ${\mathrm{\Phi }}_{ijk}=\mathrm{\Phi }(E_i,E_j,E_k)$, $i,j,k\in \{0,1,2\}$:

$$\begin{array}{cc}{\mathcal{F}}_1:\hfill & {\mathrm{\Phi }}_{111}={\mathrm{\Phi }}_{221}=-{\mathrm{\Phi }}_{122}=-{\mathrm{\Phi }}_{212}={\displaystyle \frac{1}{2}}{\alpha }_1,{\alpha }_1=-{\theta }_2,\hfill \\ \hfill & {\mathrm{\Phi }}_{112}={\mathrm{\Phi }}_{222}=-{\mathrm{\Phi }}_{121}=-{\mathrm{\Phi }}_{122}={\displaystyle \frac{1}{2}}{\alpha }_2,{\alpha }_2={\theta }_1;\hfill \\ {\mathcal{F}}_4:\hfill & {\mathrm{\Phi }}_{110}={\mathrm{\Phi }}_{220}=-{\mathrm{\Phi }}_{120}=-{\mathrm{\Phi }}_{210}={\displaystyle \frac{1}{2}}{\alpha }_0,{\alpha }_0={\theta }_0;\hfill \\ {\mathcal{F}}_5:\hfill & {\mathrm{\Phi }}_{110}={\mathrm{\Phi }}_{220}=-{\mathrm{\Phi }}_{120}=-{\mathrm{\Phi }}_{210}={\displaystyle \frac{1}{2}}{\alpha }_0,{\alpha }_0={\theta }_0^{*};\hfill \\ {\mathcal{F}}_8:\hfill & {\mathrm{\Phi }}_{011}={\mathrm{\Phi }}_{101}=-{\mathrm{\Phi }}_{110}=-{\mathrm{\Phi }}_{022}=-{\mathrm{\Phi }}_{202}={\mathrm{\Phi }}_{220}=-\lambda ;\hfill \\ {\mathcal{F}}_9:\hfill & {\mathrm{\Phi }}_{011}={\mathrm{\Phi }}_{101}=-{\mathrm{\Phi }}_{110}=-{\mathrm{\Phi }}_{022}=-{\mathrm{\Phi }}_{202}={\mathrm{\Phi }}_{220}=\mu ;\hfill \\ {\mathcal{F}}_{10}:\hfill & {\mathrm{\Phi }}_{012}=-{\mathrm{\Phi }}_{021}={\mathrm{\Phi }}_{102}=-{\mathrm{\Phi }}_{110}=-{\mathrm{\Phi }}_{201}={\mathrm{\Phi }}_{220}={\displaystyle \frac{1}{2}}\nu ;\hfill \\ {\mathcal{F}}_{11}:\hfill & {\mathrm{\Phi }}_{001}={\widehat{\alpha }}_1=-{\mathrm{\Phi }}_{002}=-{\widehat{\alpha }}_2,{\widehat{\alpha }}_1=-{\widehat{\alpha }}_2={\omega }_2-{\omega }_1.\hfill \end{array}$$

(50)

Taking into account (47), (48) and (50), we confirm the validity of the statements in Section 3, namely Theorem 1 and Propositions 2–5.