Classification of Π-Manifolds Regarding the Pseudo-Riemannian Metric Associated Through the Structure

Abstract

The study of the so-called $\Pi$-manifolds has been continued. This is a short name for almost paracontact almost paracomplex manifolds with a pair of metrics—Riemannian and pseudo-Riemannian—that are mutually related and compatible with the structure of the manifold. The well-known classification of these manifolds with respect to the Riemannian metric has been used to derive an alternative classification of the same manifolds with respect to the pseudo-Riemannian metric. The interrelations between the two classifications and the corresponding classification tensors are investigated. Finally, a three-dimensional explicit example on a Lie group is given, which illustrates the most interesting case of the transition from one classification to the other.

1. Introduction

The notion of an almost paracontact structure on a differentiable manifold of arbitrary dimension was introduced by I. Satō in 1976 as an analog of the almost contact structure [1]. The differential geometry of manifolds admitting such a structure was developed immediately by T. Adati, T. Miyazawa and I. Satō in [2,3,4]. A few years later, Ref. [5] was published, where S. Sasaki introduced the notion of an almost paracontact Riemannian manifold of type $(p,q)$. Here p and q denote, respectively, the multiplicities of the eigenvalues $+1$ and $-1$ of the structure endomorphism, and in addition there is also a simple eigenvalue 0.

On the paracontact distribution, the almost paracontact structure induces an almost product structure. A classification of Riemannian almost product manifolds with respect to the covariant derivative of the almost product structure was obtained in 1983 by A. M. Naveira in [6]. In the case of $p=q$, almost product manifolds are known as almost paracomplex manifolds. The origins of almost paracomplex geometry can be traced back to the pioneering work of P. K. Rashevskii [7] (1948), followed by the contributions of P. Libermann [8] (1952) and E. M. Patterson [9] (1954).

Later, in the period 1987–1991, these manifolds were studied by V. Mihova, M. Staikova, K. Gribachev and D. Mekerov, e.g., in [10,11,12,13]. Using Naveira’s classification, M. Staikova and K. Gribachev presented a classification of Riemannian almost paracomplex manifolds in 1992 [13]. A detailed review of paracomplex geometry was done by V. Cruceanu, P. Fortuny and P. M. Gadea in 1996 [14].

Recent developments in related areas include studies of geometric solitons, curvature properties, and classification problems on various almost contact and paracontact manifolds (see, e.g., [15,16,17,18,19,20,21]).

A differentiable manifold endowed with an almost paracontact structure admits two different types of compatible metrics according to the action of the structure endomorphism on the paracontact distribution. In the case when this action is an isometry, the manifold with the main metric is said to have an almost paracontact Riemannian structure. If the induced transformation acts as an anti-isometry, then the compatible metric is pseudo-Riemannian of signature $(n+1,n)$. Both cases have been studied by many authors.

In 2001, M. Manev and M. Staikova gave in [22] a classification of almost paracontact Riemannian manifolds of the Sasaki type $(n,n)$. This classification generalizes the one given in [13] by applying it to the paracomplex distribution. The decomposition is orthogonal with respect to the Riemannian metric and invariant with respect to the structure group of the studied manifolds, i.e., $\mathcal{O}\left(n\right)\times \mathcal{O}\left(n\right)\times 1$, where $\mathcal{O}\left(n\right)$ is the orthogonal group. The classification conditions are given in terms of the tensor F, dual to the covariant derivative of the almost paracontact endomorphism with respect to the Riemannian metric g.

Note that the dimension of these manifolds is odd-dimensional and the induced almost product structure P on the paracontact distribution is traceless, which means that P is an almost paracomplex structure.

The considered manifolds have been studied in recent years by the first author of this paper, S. Ivanov, H. Manev and V. Tavkova under the name almost paracontact almost paracomplex Riemannian manifolds or, for short, Riemannian $\Pi$-manifolds, see, e.g., [23,24].

Let us emphasize that in addition to the compatible Riemannian metric g, these manifolds also have an associated pseudo-Riemannian metric $\tilde{g}$, generated by g through the structure tensors. The metric $\tilde{g}$ is also compatible, as g is, and is similarly related to g, so these metrics form a pair and both are equally important for the geometry of the manifolds under study.

Therefore, the natural question arises of whether the geometric classification determined by one metric is preserved under the passage to the other compatible metric. The answer to this question reveals which geometric properties are intrinsic to the $\Pi$-structure itself and which depend on the particular choice of compatible metric.

Since the pair $(g,\tilde{g})$ is canonically determined by the same $\Pi$-structure, it is natural to investigate whether the geometric information encoded by one of these metrics is preserved under the replacement by the other. In particular, the corresponding classifications describe how the intrinsic geometry of $\Pi$-manifolds behaves under the transition between the Riemannian and pseudo-Riemannian settings. Moreover, the two metrics induce corresponding curvature characteristics and geometric invariants, whose interrelations constitute a natural direction for further research.

The purpose of this paper is to classify $\Pi$-manifolds in terms of a tensor $\tilde{F}$, dual to the corresponding covariant derivative of the almost paracontact endomorphism with respect to $\tilde{g}$. We also aim to explore the interrelations between these two classifications.

The paper is organized as follows. In Section 2, we recall basic notions and preliminary results concerning $\Pi$-manifolds. The main results are presented in Section 3. First, we derive the relations between the fundamental tensors F and $\tilde{F}$ generated by the Levi-Civita connections of the two metrics g and $\tilde{g}$, respectively. Then we use the obtained relations to study the form of $\tilde{F}$ in the basic classes of $\Pi$-manifolds in the alternative classification. In Section 4, we present an example of a Lie group considered as a $\Pi$-manifold of the lowest dimension. This explicit example illustrates what we consider to be the most interesting case and supports the obtained results.

2. Preliminaries for $\Pi$-Manifolds

2.1. $\Pi$-Structure

A $(2n+1)$-dimensional real differentiable manifold $\mathcal{M}$ is said to have an almost paracontact structure, also denoted as a $(\varphi ,\xi ,\eta )$-structure, if $\mathcal{M}$ admits a tensor field $\varphi$ of type $(1,1)$, a vector field $\xi$ and its dual one-form $\eta$, satisfying the conditions [1]

$$\begin{array}{c}\hfill \eta \left(\xi \right)=1,{\varphi }^2=ı-\eta \otimes \xi ,\end{array}$$

(1)

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

If, in addition, the structure endomorphism $\varphi$ is traceless, i.e., the requirement $tr\varphi =0$ is satisfied, then the resulting manifold is called an almost paracontact almost paracomplex manifold, denoted from now on by $(\mathcal{M},\varphi ,\xi ,\eta )$ [22,23].

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

$$\begin{array}{c}\hfill g(\varphi x,\varphi y)=g(x,y)-\eta \left(x\right)\eta \left(y\right),\end{array}$$

(2)

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

Bearing in mind (2), compatible metrics can be Riemannian or pseudo-Riemannian. We study the case when g is Riemannian [22]. For this reason, we sometimes call $\mathcal{M}$ an almost paracontact almost paracomplex Riemannian manifold [23].

The equalities (1) and (2) immediately imply the following properties:

$$\begin{array}{c}\hfill g(x,\xi )=\eta \left(x\right),\eta \circ \varphi =0,\varphi \xi =0,{\varphi }^3=\varphi ,g(x,\varphi y)=g(\varphi x,y).\end{array}$$

(3)

Then, on $\mathcal{M}$ there also exists a metric $\tilde{g}$ associated with g via the $(\varphi ,\xi ,\eta )$-structure, and $\tilde{g}$ is defined by

$$\begin{array}{c}\hfill \tilde{g}(x,y)=g(x,\varphi y)+\eta \left(x\right)\eta \left(y\right).\end{array}$$

(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.,

$$\begin{array}{c}\hfill g(x,y)=\tilde{g}(x,\varphi y)+\eta \left(x\right)\eta \left(y\right).\end{array}$$

(5)

In [22] it is shown that $\tilde{g}$ is a compatible metric with $(\mathcal{M},\varphi ,\xi ,\eta )$, as is g; i.e., the analog of (2) for $\tilde{g}$ is valid, but $\tilde{g}$ is a pseudo-Riemannian metric of signature $(n+1,n)$.

Denoting briefly the tensor structure introduced on $\mathcal{M}$ as above by $\Pi$, i.e., we have $\Pi =(\varphi ,\xi ,\eta ,g,\tilde{g})$, we will call the studied manifold a $\Pi$-manifold for short and denote it by $(\mathcal{M},\Pi )$.

2.2. Classification of $\Pi$-Manifolds Regarding the Riemannian Metric of the $\Pi$-Structure

The tensor F of type $(0,3)$ plays a fundamental role in the differential geometry of the considered manifolds; e.g., the classification given in [22] is made according to it and subsequent studies mainly use it. This tensor is defined by

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

where ∇ is the Levi-Civita connection of g.

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

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

(6)

Further, we also use some immediate consequences of (1), (3) and (6), namely

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

(7)

The relations of $\nabla \xi$ and $\nabla \eta$ with F are [23]

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

(8)

Let $\{e_i,\xi \}$$(i=1,2,\dots ,2n)$ be a basis of the tangent space $T_p\mathcal{M}$ at an arbitrary point $p\in \mathcal{M}$, where $\{e_1,e_2,\dots ,e_{2n}\}$ is an arbitrary basis of the horizontal distribution $\mathcal{H}=ker\eta$. Denoting by $\left(g^{ij}\right)$ the inverse of the matrix $\left(g_{ij}\right)$ of g, we have the following one-forms that 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 one-forms are known also as the Lee forms of the considered manifolds and the following identities for them are always valid: [23]

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

(10)

The other traces of F defined using the basis $\{e_i,\xi \}$ and $\varphi$, except for those used for the Lee forms, vanish.

Lemma 1. 

The following traces of F with respect to the basis $\{e_i,\xi \}$ are equal to zero on every Π-manifold:

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

(11)

Proof. 

The annulment of these traces follows directly from (6) and (7). Note that the values of the last two traces in (11) are known from [25]. □

A classification of $\Pi$-manifolds in terms of F is given in [22]. This classification is derived by decomposing the vector space $\mathcal{F}$ of all tensors F with properties (6). This decomposition is orthogonal and invariant to the structure group of the considered manifolds. It consists of 11 subspaces ${\mathcal{F}}_s$ ($s=1$, 2, $\dots ,11$). Each of these subspaces gives rise to a unique basic class of the manifolds under study, with the corresponding basic class being denoted in the same way as the subspace.

A $\Pi$-manifold is said to belong to a basic class ${\mathcal{F}}_s$ and is called an ${\mathcal{F}}_s$-manifold for short if its tensor F belongs to the corresponding subspace ${\mathcal{F}}_s$. In such a way, it is obtained that this classification consists of 11 basic classes ${\mathcal{F}}_s$. The intersection of the 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 equivalent expressions for the defining conditions of ${\mathcal{F}}_s$ in [22]:

$$\begin{array}{cc}\hfill {\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 \\ \hfill & \phantom{F(x,y,z)={\displaystyle \frac{1}{2n}}}-g(x,\varphi y)\theta (\varphi z)-g(x,\varphi z)\theta (\varphi y)\};\hfill \\ \hfill {\mathcal{F}}_2:& F(\xi ,y,z)=0,F(x,\xi ,z)=0,\hfill \\ \hfill & F(x,y,\varphi z)+F(y,z,\varphi x)+F(z,x,\varphi y)=0,\theta =0;\hfill \\ \hfill {\mathcal{F}}_3:& F(\xi ,y,z)=0,F(x,\xi ,z)=0,\hfill \\ \hfill & F(x,y,z)+F(y,z,x)+F(z,x,y)=0;\hfill \\ \hfill {\mathcal{F}}_4:& F(x,y,z)={\displaystyle \frac{\theta \left(\xi \right)}{2n}}\left\{g(\varphi x,\varphi y)\eta \left(z\right)+g(\varphi x,\varphi z)\eta \left(y\right)\right\};\hfill \\ \hfill {\mathcal{F}}_5:& F(x,y,z)={\displaystyle \frac{{\theta }^{\ast }\left(\xi \right)}{2n}}\left\{g(x,\varphi y)\eta \left(z\right)+g(x,\varphi z)\eta \left(y\right)\right\};\hfill \\ \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 \\ \hfill & F(x,y,\xi )=F(y,x,\xi )=F(\varphi x,\varphi y,\xi ),\theta \left(\xi \right)={\theta }^{\ast }\left(\xi \right)=0;\hfill \\ \hfill {\mathcal{F}}_7:& F(x,y,z)=F(x,y,\xi )\eta \left(z\right)+F(x,z,\xi )\eta \left(y\right),\hfill \\ \hfill & F(x,y,\xi )=-F(y,x,\xi )=F(\varphi x,\varphi y,\xi );\hfill \\ \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 \\ \hfill & F(x,y,\xi )=F(y,x,\xi )=-F(\varphi x,\varphi y,\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 \\ \hfill & F(x,y,\xi )=-F(y,x,\xi )=-F(\varphi x,\varphi y,\xi );\hfill \\ \hfill {\mathcal{F}}_{10}:& F(x,y,z)=-\eta \left(x\right)F(\xi ,\varphi y,\varphi z);\hfill \\ \hfill {\mathcal{F}}_{11}:& F(x,y,z)=\eta \left(x\right)\left\{\eta \left(y\right)\omega \left(z\right)+\eta \left(z\right)\omega \left(y\right)\right\}.\hfill \end{array}$$

(12)

Proposition 1. 

For the Lee forms on an ${\mathcal{F}}_s$-manifold following equalities are valid:

$$\begin{array}{cccc}\theta =\theta \circ {\varphi }^2,\hfill & {\theta }^{\ast }=-{\theta }^{\ast }\circ {\varphi }^2,\hfill & \omega =0\hfill & \mathit{for}s=1;\hfill \\ \theta =\theta \left(\xi \right)\eta ,\hfill & {\theta }^{\ast }=0,\hfill & \omega =0\hfill & \mathit{for}s=4;\hfill \end{array}$$

$$\begin{array}{cccc}\theta =0,\hfill & {\theta }^{\ast }={\theta }^{\ast }\left(\xi \right)\eta ,\hfill & \omega =0\hfill & \mathit{for}s=5;\hfill \\ \theta =0,\hfill & {\theta }^{\ast }=0,\hfill & \omega =\omega \circ {\varphi }^2\hfill & \mathit{for}s=11;\hfill \\ \theta =0,\hfill & {\theta }^{\ast }=0,\hfill & \omega =0\hfill & \mathit{for}\mathit{the}\mathit{other}\mathit{cases}\mathit{of}s.\hfill \end{array}$$

Proof. 

These properties follow by direct combination of (1), (2), (3), (6), (9), (10) and (12). Note that on an ${\mathcal{F}}_{11}$-manifold, the Lee form $\omega$ actually has no additional conditions. □

3. Classification of $\mathbf{\Pi }$-Manifolds Regarding the Pseudo-Riemannian Metric of the $\mathbf{\Pi }$-Structure

Let $\tilde{\nabla }$ be the Levi-Civita connection induced by the associated metric $\tilde{g}$ on $(\mathcal{M},\Pi )$. The tensor field $\tilde{F}$ of type $(0,3)$ is defined as follows:

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

(13)

Here and below, symbols with a tilde refer to the associated metric $\tilde{g}$ and the corresponding tensor $\tilde{F}$.

Due to the well-known Koszul formula for the metric $\tilde{g}$ and its Levi-Civita connection,

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

(14)

and the relation between g and $\tilde{g}$ in (4), the following expression of $\tilde{F}$ in terms of F, given in [26], is valid:

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

(15)

Lemma 2. 

On every Π-manifold, the corresponding Lee forms associated to F and $\tilde{F}$ coincide, i.e., the following equalities hold:

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

(16)

Proof. 

We contract (15) by ${\tilde{g}}^{ij}={\varphi }_k^j{g}^{ik}$ for $i,j,k\in \{1,2,\dots ,2n\}$, which is (4) in local coordinates. Meanwhile, we use the fact that $\eta \left(e_i\right)=g(e_i,\xi )$ is zero in the basis $\{e_i,\xi \}$ and the first property in (6) to obtain

$$\begin{array}{c}2{\tilde{g}}^{ij}\tilde{F}(e_i,e_j,z)={\varphi }_k^j{g}^{ik}\{F(\varphi e_j,e_i,z)+F(\varphi z,e_j,e_i)-F(e_j,e_i,\varphi z)-F(z,\varphi e_j,e_i)\hfill \\ \phantom{2{\tilde{g}}^{ij}\tilde{F}(e_i,e_j,z)={\varphi }_k^j{g}^{ik}\{}+\eta \left(z\right)\{F(e_i,e_j,\xi )-F(\varphi e_j,\varphi e_i,\xi )+F(e_i,\varphi e_j,\xi )\}\}.\hfill \end{array}$$

Due to Lemma 1 and (9) with respect to $\tilde{g}$, the last equality takes the following form:

$$\begin{array}{c}2\tilde{\theta }\left(z\right)={\varphi }_k^j{\varphi }_j^l{g}^{ik}F(e_l,e_i,z)+{\varphi }_k^j{g}^{ik}F(\varphi z,e_j,e_i)-{\varphi }_k^j{g}^{ik}F(e_j,e_i,\varphi z)-{\varphi }_k^j{g}^{ik}F(z,\varphi e_j,e_i)\hfill \\ \phantom{2\tilde{\theta }\left(z\right)=}+\eta \left(z\right)\{{\varphi }_k^j{g}^{ik}F(e_i,e_j,\xi )-{\varphi }_k^j{g}^{ik}F(\varphi e_j,\varphi e_i,\xi )+{\varphi }_k^j{g}^{ik}F(e_i,\varphi e_j,\xi )\}.\hfill \end{array}$$

(17)

In the last expression, we perform the following calculations on the local coordinates of the particular terms, using the last equalities of (1) and (3), as well as the first two definitions in (9), the first property in (10), and the traces in Lemma 1:

$$\begin{array}{c}{\varphi }_k^j{g}^{ik}F(\varphi e_j,e_i,z)={\varphi }_k^j{\varphi }_j^l{g}^{ik}F(e_l,e_i,z)={\delta }_k^l{g}^{ik}F(e_l,e_i,z)=g^{ik}F(e_k,e_i,z)=\theta \left(z\right),\hfill \\ {\varphi }_k^j{g}^{ik}F(\varphi z,e_j,e_i)=g^{ik}F(\varphi z,\varphi e_k,e_i)=0,\hfill \\ {\varphi }_k^j{g}^{ik}F(e_j,e_i,\varphi z)={\varphi }_k^i{g}^{jk}F(e_j,e_i,\varphi z)=g^{jk}F(e_j,\varphi e_k,\varphi z)={\theta }^{\ast }\left(\varphi z\right)=-\theta \left({\varphi }^{2}z\right),\hfill \\ {\varphi }_k^j{g}^{ik}F(z,\varphi e_j,e_i)=g^{ik}F(z,{\varphi }^2{e}_k,e_i)=-g^{ik}F(z,e_k,e_i)=0,\hfill \\ {\varphi }_k^j{g}^{ik}F(e_i,e_j,\xi )=g^{ik}F(e_i,\varphi e_k,\xi )={\theta }^{\ast }\left(\xi \right),\hfill \\ {\varphi }_k^j{g}^{ik}F(\varphi e_j,\varphi e_i,\xi )={\varphi }_j^l{\varphi }_k^j{g}^{ik}F(e_l,\varphi e_i,\xi )=g^{il}F(e_l,\varphi e_i,\xi )={\theta }^{\ast }\left(\xi \right),\hfill \\ {\varphi }_k^j{g}^{ik}F(e_i,\varphi e_j,\xi )={\varphi }_j^l{\varphi }_k^j{g}^{ik}F(e_i,e_l,\xi )=g^{il}F(e_i,e_l,\xi )=\theta \left(\xi \right).\hfill \end{array}$$

Thus, (17) takes the following form:

$$2\tilde{\theta }\left(z\right)=\theta \left(z\right)+\theta \left({\varphi }^{2}z\right)+\eta \left(z\right)\theta \left(\xi \right),$$

which is simplified as $\tilde{\theta }\left(z\right)=\theta \left(z\right)$ because of the second equality in (1). Therefore, we prove the first equality in (16). In this way we derive the other two dependencies in (16). □

Using (15), as well as (6) and (7), we verify directly that $\tilde{F}$ has the same properties as for F given in (6) and (8)—i.e., the following is true:

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

(18)

$$\left({\tilde{\nabla }}_x\eta \right)\left(y\right)=\tilde{g}\left({\tilde{\nabla }}_x\xi ,y\right)=-\tilde{F}(x,\varphi y,\xi ),\eta \left({\tilde{\nabla }}_x\xi \right)=0.$$

(19)

We obtain the following consequences of the expression in (15) using the properties of the $\Pi$-structure, given in (1) and (3), as well as the properties of F in (6) and (7):

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

(20)

We aim to classify $\Pi$-manifolds with respect to the associated metric $\tilde{g}$. Since the fundamental tensor $\tilde{F}$ generated by it belongs to the same vector space $\mathcal{F}$ as F, we can use its decomposition into 11 subspaces ${\mathcal{F}}_s$ as in [22] and thus obtain the corresponding 11 basic classes ${\tilde{\mathcal{F}}}_s$.

Proposition 2. 

The following definition conditions of $\tilde{F}$ for a Π-manifold in every basic class ${\tilde{\mathcal{F}}}_s$ with respect to the associated compatible metric $\tilde{g}$ are valid:

$$\begin{array}{cc}\hfill {\tilde{\mathcal{F}}}_1:& \tilde{F}(x,y,z)={\displaystyle \frac{1}{2n}}\{\tilde{g}(\varphi x,\varphi y)\tilde{\theta }\left({\varphi }^{2}z\right)+\tilde{g}(\varphi x,\varphi z)\tilde{\theta }\left({\varphi }^{2}y\right)\hfill \\ \hfill & \phantom{\tilde{F}(x,y,z)={\displaystyle \frac{1}{2n}}\{}-\tilde{g}(x,\varphi y)\tilde{\theta }\left(\varphi z\right)-\tilde{g}(x,\varphi z)\tilde{\theta }\left(\varphi y\right)\};\hfill \\ \hfill {\tilde{\mathcal{F}}}_2:& \tilde{F}(\xi ,y,z)=\tilde{F}(x,y,\xi )=0,\hfill \\ \hfill & \tilde{F}(x,y,\varphi z)+\tilde{F}(y,z,\varphi x)+\tilde{F}(z,x,\varphi y)=0,\tilde{\theta }=0;\hfill \\ \hfill {\tilde{\mathcal{F}}}_3:& \tilde{F}(\xi ,y,z)=\tilde{F}(x,y,\xi )=0,\hfill \\ \hfill & \tilde{F}(x,y,z)+\tilde{F}(y,z,x)+\tilde{F}(z,x,y)=0;\hfill \\ \hfill {\tilde{\mathcal{F}}}_4:& \tilde{F}(x,y,z)={\displaystyle \frac{\tilde{\theta }\left(\xi \right)}{2n}}\left\{\tilde{g}(\varphi x,\varphi y)\tilde{\eta }\left(z\right)+\tilde{g}(\varphi x,\varphi z)\tilde{\eta }\left(y\right)\right\};\hfill \\ \hfill {\tilde{\mathcal{F}}}_5:& \tilde{F}(x,y,z)={\displaystyle \frac{{\tilde{\theta }}^{\ast }\left(\xi \right)}{2n}}\left\{\tilde{g}(x,\varphi y)\tilde{\eta }\left(z\right)+\tilde{g}(x,\varphi z)\tilde{\eta }\left(y\right)\right\};\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\tilde{\mathcal{F}}}_6:& \tilde{F}(x,y,z)=\tilde{\eta }\left(z\right)\tilde{F}(x,y,\xi )+\tilde{\eta }\left(y\right)\tilde{F}(x,z,\xi ),\hfill \\ \hfill & \tilde{F}(x,y,\xi )=\tilde{F}(y,x,\xi )=\tilde{F}(\varphi x,\varphi y,\xi );\hfill \\ \hfill {\tilde{\mathcal{F}}}_7:& \tilde{F}(x,y,z)=\tilde{\eta }\left(z\right)\tilde{F}(x,y,\xi )+\tilde{\eta }\left(y\right)\tilde{F}(x,z,\xi ),\hfill \\ \hfill & \tilde{F}(x,y,\xi )=-\tilde{F}(y,x,\xi )=\tilde{F}(\varphi x,\varphi y,\xi );\hfill \\ \hfill {\tilde{\mathcal{F}}}_8:& \tilde{F}(x,y,z)=\tilde{\eta }\left(z\right)\tilde{F}(x,y,\xi )+\tilde{\eta }\left(y\right)\tilde{F}(x,z,\xi ),\hfill \\ \hfill & \tilde{F}(x,y,\xi )=\tilde{F}(y,x,\xi )=-\tilde{F}(\varphi x,\varphi y,\xi );\hfill \\ \hfill {\tilde{\mathcal{F}}}_9:& \tilde{F}(x,y,z)=\tilde{\eta }\left(z\right)\tilde{F}(x,y,\xi )+\tilde{\eta }\left(y\right)\tilde{F}(x,z,\xi ),\hfill \\ \hfill & \tilde{F}(x,y,\xi )=-\tilde{F}(y,x,\xi )=-\tilde{F}(\varphi x,\varphi y,\xi );\hfill \\ \hfill {\tilde{\mathcal{F}}}_{10}:& \tilde{F}(x,y,z)=-\tilde{\eta }\left(x\right)\tilde{F}(\xi ,\varphi y,\varphi z);\hfill \\ \hfill {\tilde{\mathcal{F}}}_{11}:& \tilde{F}(x,y,z)=\tilde{\eta }\left(x\right)\left\{\tilde{\eta }\left(y\right)\tilde{\omega }\left(z\right)+\tilde{\eta }\left(z\right)\tilde{\omega }\left(y\right)\right\}.\hfill \end{array}$$

(22)

Proof. 

Recall that $\tilde{\eta }=\eta$, which is argued immediately after (4). We then use the identities in (1), (2), (3), and (6), along with the relations specified in Lemma 2 and (20). After that, we substitute F according to (12) in (15) and obtain $\tilde{F}$ for each class. □

Having the formula of $\tilde{F}$ with respect to F in (15), we also need the inverse formula given in the following.

Proposition 3. 

The expression of F in terms of $\tilde{F}$ on an arbitrary Π-manifold has the following form:

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

(23)

The above general relation has the following consequences:

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

(24)

Proof. 

Using (4), (13), (14), (18) and (19), we obtain the equalities in (23). From there, due to the properties of the $\Pi$-structure, the specific expressions in (24) for F in terms of $\tilde{F}$ follow. □

Note that Formula (15), combined with its inverse form (23), effectively defines an involution in the vector space $\mathcal{F}$ of classification tensors. This shows the equality of the two compatible metrics g and $\tilde{g}$. The relations (15) and (23) show that the passage from F to $\tilde{F}$ is governed by a linear invertible transformation of the vector space $\mathcal{F}$ of classification tensors. Therefore, the correspondence between the basic classes can be interpreted as the action of this transformation on the invariant decomposition $\mathcal{F}={\mathcal{F}}_1\oplus \cdots \oplus {\mathcal{F}}_{11}$. Theorem 1 describes explicitly how the components of this decomposition behave under the replacement of the metric g by its associated metric $\tilde{g}$.

We ask ourselves whether the classification of $\Pi$-manifolds with respect to one of the metrics g and $\tilde{g}$ differs from that of the other. The answer is given in the following.

Theorem 1. 

If a manifold $(\mathcal{M},\Pi )$ belongs to a basic class ${\mathcal{F}}_s$ or ${\tilde{\mathcal{F}}}_s$, $s\in \{0,1,\dots ,11\}$, 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 implications:

(i)	${\mathcal{F}}_s\iff {\tilde{\mathcal{F}}}_s$,       $s\in \{0,1,\dots ,7,11\}$;
(ii)	${\mathcal{F}}_9\iff {\tilde{\mathcal{F}}}_{10}$;
(iii)	${\mathcal{F}}_{10}\iff {\tilde{\mathcal{F}}}_9$;
(iv)	${\mathcal{F}}_8\Rightarrow {\tilde{\mathcal{F}}}_8\oplus {\tilde{\mathcal{F}}}_9\oplus {\tilde{\mathcal{F}}}_{10}$,      ${\tilde{\mathcal{F}}}_8\Rightarrow {\mathcal{F}}_8\oplus {\mathcal{F}}_9\oplus {\mathcal{F}}_{10}$,

as each of the two direct sums cannot be reduced to any of its components except ${\mathcal{F}}_0\equiv {\tilde{\mathcal{F}}}_0$.

Proof. 

(i) We show the reasoning for only some of the cases of s in the first statement. The rest are obtained in a similar way.

Let us consider the case $s=1$ as an example of the main classes, i.e., those of the basic classes in which F or $\tilde{F}$, respectively, is expressed explicitly through the corresponding metric and Lee forms. Then from the expression of F for an ${\mathcal{F}}_1$-manifold in (12) we obtain that $F(x,y,\xi )$ and $F(\xi ,y,z)$ are zero and the following is valid:

$$\begin{array}{c}F(\varphi y,z,x)+F(\varphi z,y,x)-F(y,\varphi z,x)-F(z,\varphi y,x)\hfill \\ ={\displaystyle \frac{1}{n}}\left\{g(x,\varphi y)\theta \left({\varphi }^{2}z\right)+g(x,\varphi z)\theta \left({\varphi }^{2}y\right)-g(\varphi x,\varphi y)\theta \left(\varphi z\right)-g(\varphi x,\varphi z)\theta \left(\varphi y\right)\right\}.\hfill \end{array}$$

We substitute these expressions into (15) and apply (5) and (16). Then the result takes the form given in (21) for $s=1$. The reverse implication is obtained in a similar way, now using the formula in (23).

Let us consider the case $s=7$ as an example of the basic classes in (i) that are not main. Taking into account the properties of ${\mathcal{F}}_7$ in (12) and the second formula in (20), we deduce by (15) as follows:

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

This means that $(\mathcal{M},\Pi )$ belongs to ${\tilde{\mathcal{F}}}_7$ according to (22). The reasoning for the reverse direction is analogous using (23).

(ii) Let an ${\mathcal{F}}_9$-manifold be given. Then from (12) we get

$$\begin{array}{cc}\hfill & F(x,y,\xi )-F(\varphi y,\varphi x,\xi )=0,\hfill \\ \hfill & F(\varphi y,z,x)+F(\varphi z,y,x)-F(y,\varphi z,x)-F(z,\varphi y,x)\hfill \\ \hfill & =-4F(y,\varphi z,\xi )\eta \left(x\right)-F(x,\varphi y,\xi )\eta \left(z\right)-F(x,\varphi z,\xi )\eta \left(y\right).\hfill \end{array}$$

Hence, (15) takes the following form $\tilde{F}(x,y,z)=-2\eta \left(x\right)F(y,\varphi z,\xi ),$ which implies

$$\tilde{F}(x,y,\xi )=\eta \left(x\right)\tilde{F}(\xi ,y,z),\tilde{F}(\xi ,y,z)=-2F(y,\varphi z,\xi ).$$

The last equality and the properties (12) of F in ${\mathcal{F}}_9$ lead to the following:

$$\tilde{F}(\xi ,\varphi y,\varphi z)=-2F(\varphi y,z,\xi )=2F(y,\varphi z,\xi )=\tilde{F}(\xi ,y,z).$$

Therefore, the expression $\tilde{F}(x,y,z)=-\eta \left(x\right)\tilde{F}(\xi ,\varphi y,\varphi z)$ is valid, i.e., the considered manifold $(\mathcal{M},\Pi )$ belongs to ${\tilde{\mathcal{F}}}_{10}$.

The reverse implication in (ii) is obtained in a similar way, now using the formula in (23) and the corresponding conditions in (22).

(iii) The results in this case are obtained in a similar way as in the previous one.

(iv) Now we consider $(\mathcal{M},\Pi )\in {\mathcal{F}}_8$, the most interesting case among the hypotheses in the statement. From (12) for $s=8$ and the second equality in (20), we get

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

which implies

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

(25)

where

$$\tilde{F}(\xi ,y,z)=2F(y,z,\xi ),\tilde{F}(x,y,\xi )=F(x,y,\xi )+F(x,\varphi y,\xi ).$$

(26)

The first equality in (26) shows that $\tilde{F}(\xi ,y,z)$ is symmetric regarding y and z as well as hybrid with respect to $\varphi$, since $F(y,z,\xi )$ is such in ${\mathcal{F}}_8$—i.e.,

$$\tilde{F}(\xi ,y,z)=\tilde{F}(\xi ,z,y),\tilde{F}(\xi ,\varphi y,\varphi z)=-\tilde{F}(\xi ,y,z).$$

Therefore, the first addend in (25) is equal to ${\tilde{F}}_{10}(x,y,z)$ according to (22) for $s=10$.

On the other hand, the second equality in (26) means that $\tilde{F}(x,y,\xi )$ has a symmetric component $F(x,y,\xi )$ and an antisymmetric component $F(x,\varphi y,\xi )$ with respect to x and y according to the properties (12) of F in ${\mathcal{F}}_8$. Moreover, this form of $\tilde{F}(x,y,\xi )$ is hybrid with respect to $\varphi$, i.e., we have $\tilde{F}(\varphi x,\varphi y,\xi )=-\tilde{F}(x,y,\xi )$. Thus, the sum of the last two addends in (25) is equal to ${\tilde{F}}_8(x,y,z)+{\tilde{F}}_9(x,y,z)$, taking into account (12) and (22).

In conclusion, (25) and the arguments above show that $(\mathcal{M},\Pi )$ belongs to the direct sum ${\tilde{\mathcal{F}}}_8\oplus {\tilde{\mathcal{F}}}_9\oplus {\tilde{\mathcal{F}}}_{10}$ in this case. Moreover, if we want $(\mathcal{M},\Pi )$ to belong only to some of the components of this direct sum, i.e., the corresponding tensor component of $\tilde{F}$ for $s\in \{8,9,10\}$ becomes equal to zero, considering (22), then it is easy to see in (26) that the other components also vanish and $(\mathcal{M},\Pi )$ becomes an ${\tilde{\mathcal{F}}}_0$-manifold.

In a similar way, we motivate the opposite direction of the assertion (iv) by using (22) and (23) instead of (12) and (15), respectively. □

Note that the correspondences in statements (ii) and (iii) of Theorem 1 are between two basic classes characterized by different symmetry properties of the classification tensors in their respective classifications. They are induced by the transformation rule connecting F and $\tilde{F}$ and arise naturally from the interchange of the two compatible metrics.

Theorem 1 shows that the passage from g to $\tilde{g}$ preserves most basic classes, interchanges the classes ${\mathcal{F}}_9$ and ${\mathcal{F}}_{10}$, and produces a non-trivial splitting phenomenon for ${\mathcal{F}}_8$. Therefore, the classification with respect to the associated metric is not merely a reformulation of the original one but reveals additional structural information concerning the metric pair of the $\Pi$-structure.

The exceptional role of the class ${\mathcal{F}}_8$ follows from the fact that the correspondence between F and $\tilde{F}$ does not preserve this component. Instead, its image simultaneously contains components of the classes ${\tilde{\mathcal{F}}}_8$, ${\tilde{\mathcal{F}}}_9$, and ${\tilde{\mathcal{F}}}_{10}$, which explains the splitting phenomenon described in Theorem 1 (iv).

Theorem 1 describes how the basic components of the decomposition of the tensor space $\mathcal{F}$ are affected by the replacement of the compatible metric g with its associated metric $\tilde{g}$. In particular, some components are preserved and some are interchanged, whereas the component ${\mathcal{F}}_8$ gives rise to a mixture of the components ${\tilde{\mathcal{F}}}_8$, ${\tilde{\mathcal{F}}}_9$, and ${\tilde{\mathcal{F}}}_{10}$.

Corollary 1. 

The following direct sums of basic classes of Π-manifolds are invariant under the replacement of one metric of the pair $(g,\tilde{g})$ with the other, considering the corresponding classifications with respect to F and $\tilde{F}$:

(i)

${\mathcal{F}}_9\oplus {\mathcal{F}}_{10}\iff {\tilde{\mathcal{F}}}_9\oplus {\tilde{\mathcal{F}}}_{10}$;

(ii)

${\mathcal{F}}_8\oplus {\mathcal{F}}_9\oplus {\mathcal{F}}_{10}\iff {\tilde{\mathcal{F}}}_8\oplus {\tilde{\mathcal{F}}}_9\oplus {\tilde{\mathcal{F}}}_{10}$.

Proof. 

Given Theorem 1 for cases (ii) and (iii), we immediately conclude the equivalence in (i) above.

Similarly, based on Theorem 1 (iv), we derive the statement in (ii) of the present corollary. □

Note that, as in Theorem 1 (iv), the direct sums occurring as images of the implications in Corollary 1 cannot, in general, be reduced to any of their individual components, except for the trivial class ${\mathcal{F}}_0\equiv {\tilde{\mathcal{F}}}_0$.

4. Example

Let us consider a three-dimensional real connected Lie group $\mathcal{L}$ and its associated Lie algebra $\mathfrak{l}$. Suppose that $\{E_0,E_1,E_2\}$ is 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:

$$\begin{array}{c}\varphi E_0=0,\varphi E_1=E_2,\varphi E_2=E_1,\xi =E_0,\eta \left(E_0\right)=1,\eta \left(E_1\right)=\eta \left(E_2\right)=0,\end{array}$$

$$g(E_i,E_j)={\delta }_{ij},\tilde{g}(E_i,E_j)=g(E_i,\varphi E_j)+\eta \left(E_i\right)\eta \left(E_j\right),i,j\in \{0,1,2\}.$$

In [27] it is proved that the manifold $(\mathcal{L},\varphi ,\xi ,\eta ,g,\tilde{g})$, 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 example, for the basic class ${\mathcal{F}}_8$ we have the following:

$$[E_0,E_1]=\lambda E_2,[E_0,E_2]=-\lambda E_1,[E_1,E_2]=2\lambda E_0,$$

where $\lambda$ is an arbitrary real parameter. It is also shown that

$$\lambda =F(E_1,E_0,E_1)=F(E_1,E_1,E_0)=-F(E_2,E_0,E_2)=-F(E_2,E_2,E_0)$$

(27)

are the only non-zero components of F with respect to the basis $\{E_0,E_1,E_2\}$ on a three-dimensional ${\mathcal{F}}_8$-manifold that is not an ${\mathcal{F}}_0$-manifold.

Recall from [23] that the basic classes of three-dimensional $\Pi$-manifolds are only ${\mathcal{F}}_1$, ${\mathcal{F}}_4$, ${\mathcal{F}}_5$, ${\mathcal{F}}_8$, ${\mathcal{F}}_9$, ${\mathcal{F}}_{10}$, and ${\mathcal{F}}_{11}$. For the present example, we are only interested in the components $F_s$$(s=8,9,10)$ of F in the corresponding basic classes ${\mathcal{F}}_s$, which are expressed as follows:

$$\begin{array}{c}{F}_8(x,y,z)=\lambda \left\{x^1\left(y^0{z}^1+y^1{z}^0\right)-x^2\left(y^0{z}^2+y^2{z}^0\right)\right\},\lambda =F_{110}=-F_{220};\hfill \\ F_9(x,y,z)=\mu \left\{x^1\left(y^0{z}^2+y^2{z}^0\right)-x^2\left(y^0{z}^1+y^1{z}^0\right)\right\},\mu =F_{120}=-F_{210};\hfill \\ F_{10}(x,y,z)=\nu x^0\left(y^1{z}^1-y^2{z}^2\right),\nu =F_{011}=-F_{022},\hfill \end{array}$$

(28)

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 $F(E_i,E_j,E_k)$.

Now we use the formula in (15) to express the components ${\tilde{F}}_{ijk}=\tilde{F}(E_i,E_j,E_k)$ in terms of $F_{ijk}$ for all $i,j,k\in \{0,1,2\}$, and we get

$$\begin{array}{cc}\hfill \lambda & ={\tilde{F}}_{101}={\tilde{F}}_{110}=-{\tilde{F}}_{202}=-{\tilde{F}}_{220}={\tilde{F}}_{102}={\tilde{F}}_{120}=-{\tilde{F}}_{201}=-{\tilde{F}}_{210}\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\tilde{F}}_{011}=-{\displaystyle \frac{1}{2}}{\tilde{F}}_{022}.\hfill \end{array}$$

(29)

Obviously, taking into account (27) and (29), we conclude that the corresponding parameter $\tilde{\lambda }$ of an ${\tilde{\mathcal{F}}}_8$-manifold coincides with $\lambda$, i.e., $\tilde{\lambda }=\lambda$. But in (29) other non-zero components of $\tilde{F}$ are also present. Using [27] again, we deduce from (28) and (22) that these components are the basic components of an ${\tilde{\mathcal{F}}}_9$-manifold and an ${\tilde{\mathcal{F}}}_{10}$-manifold:

$$\tilde{\mu }={\tilde{F}}_{102}={\tilde{F}}_{120}=-{\tilde{F}}_{201}=-{\tilde{F}}_{210},\tilde{\nu }={\tilde{F}}_{011}=-{\tilde{F}}_{022}.$$

(30)

The same results can also be obtained directly from Formula (15) by evaluating the corresponding components of ${\tilde{F}}_{ijk}$.

Then (27), (29) and (30) imply the relation

$$\tilde{\lambda }=\tilde{\mu }={\displaystyle \frac{1}{2}}\tilde{\nu }=\lambda$$

and we therefore obtain that the manifold $(\mathcal{L},\varphi ,\xi ,\eta ,g,\tilde{g})$ belongs to ${\tilde{\mathcal{F}}}_8\oplus {\tilde{\mathcal{F}}}_9\oplus {\tilde{\mathcal{F}}}_{10}$ provided that $\lambda \ne 0$.

Furthermore, the constructed $\Pi$-manifold does not belong to any of the basic classes ${\tilde{\mathcal{F}}}_8$, ${\tilde{\mathcal{F}}}_9$ or ${\tilde{\mathcal{F}}}_{10}$, nor to their double direct sums ${\tilde{\mathcal{F}}}_8\oplus {\tilde{\mathcal{F}}}_9$, ${\tilde{\mathcal{F}}}_8\oplus {\tilde{\mathcal{F}}}_{10}$ or ${\tilde{\mathcal{F}}}_9\oplus {\tilde{\mathcal{F}}}_{10}$ with respect to $\tilde{F}$.

The present example supports case (iv) of Theorem 1. The remaining cases can be easily illustrated in a similar way.

Formulae (28) show that the classes ${\mathcal{F}}_9$ and ${\mathcal{F}}_{10}$ are determined by the parameters $\mu$ and $\nu$, respectively. According to Theorem 1 (ii) and (iii), under the passage from F to $\tilde{F}$, the component corresponding to $\mu$ becomes a ${\tilde{\mathcal{F}}}_{10}$-component. In contrast, the component corresponding to $\nu$ becomes a ${\tilde{\mathcal{F}}}_9$-component. Therefore, the classes ${\mathcal{F}}_9$ and ${\mathcal{F}}_{10}$ are exchanged under the replacement of the compatible metric.

5. Conclusions

In this paper, we studied the classification of almost paracontact almost paracomplex Riemannian manifolds with respect to the pseudo-Riemannian metric associated through the structure. Using the relation between the corresponding fundamental tensors F and $\tilde{F}$, we established explicit correspondences between the basic classes of the original classification and those determined by the associated metric. Several invariance properties under the interchange of the compatible metrics were obtained, and the exceptional classes were analyzed in detail. The theoretical results were illustrated by examples of three-dimensional Lie groups belonging to the relevant classes.

The results reveal how the geometric characteristics of these manifolds change under the passage from the Riemannian metric to the associated pseudo-Riemannian metric and provide a unified description of the corresponding classifications. Further research may be directed towards the study of curvature properties, natural connections, and geometric invariants of these manifolds from the viewpoint of the associated metric. Another interesting direction is the investigation of analogous correspondences for subclasses determined by additional geometric conditions and for manifolds of higher dimensions.

It would also be interesting to study whether similar correspondences hold for other tensorial characteristics associated with this pair of compatible metrics.