# Invariant tensors related with natural connections for a class Riemannian
product manifolds

Dobrinka Gribacheva

###### Abstract.

Some invariant tensors in two Naveira classes of Riemannian product manifolds
are considered. These tensors are related with natural connections, i.e.
linear connections preserving the Riemannian metric and the product structure.

###### Key words and phrases:

Riemannian almost product manifold; Riemannian metric; product structure;
natural connection; curvature tensor; Riemannian P-tensor.

###### 2000 Mathematics Subject Classification:

53C15, 53C25.

## Introduction

A Riemannian almost product manifold $(M,P,g)$ is a differentiable manifold
$M$ for which almost product structure $P$ is compatible with the Riemannian
metric $g$ such that an isometry is induced in any tangent space of $M$.

The systematic development of the theory of Riemannian almost product
manifolds was started by K. Yano in [15].

In [11] A. M. Naveira gave a classification of Riemannian almost product
manifolds with respect to the covariant differentiation $\nabla P$, where
$\nabla$ is the Levi-Civita connection of $g$. This classification is very
similar to the Gray-Hervella classification in [1] of almost Hermitian
manifolds.

M. Staikova and K. Gribachev gave in [13] a classification of the Riemannian
almost product manifolds with ${\rm tr}P=0$. In this case the manifold $M$ is
even-dimensional.

For the class $\mathcal{W}_{1}$ of the Staikova-Gribachev classification is
valid
$\mathcal{W}_{1}=\overline{\mathcal{W}}_{3}\oplus\overline{\mathcal{W}}_{6}$,
where $\overline{\mathcal{W}}_{3}$ and $\overline{\mathcal{W}}_{6}$ are
classes of the Naveira classification. In some sense these manifolds have dual
geometries.

In [10], a connection $\nabla^{\prime}$ on a Riemannian almost product
manifold $(M,P,g)$ is called natural if $\nabla^{\prime}P=\nabla^{\prime}g=0$.
In [9], a tensor on such a manifold is called a Riemannian $P$-tensor if it
has properties similar to the properties of the Kähler tensor in Hermitian
geometry. In [4], a Riemannian $P$-tensor $K$ is defined on
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ by the
curvature tensor $R$ of $\nabla$ and the structure $P$.

In the present work111Partially supported by project NI11-FMI-004 of the
Scientific Research Fund, Paisii Hilendarski University of Plovdiv, Bulgaria,
we study manifolds $(M,P,g)$ from the class
$\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ for which the
curvature tensor of each natural connection is a Riemannian $P$-tensor.

We consider three tensors $B(L)$, $A(L)$ and $C(L)$ determined by arbitrary
Riemannian $P$-tensor $L$, where $B(L)$ is the Bochner tensor introduced in
[13]. We prove that $B(R^{\prime})=B(K)$ for arbitrary natural connection
$\nabla^{\prime}$ in Theorem 3.1. In Theorem 4.1 we prove that
$A(R^{\prime})=A(K)$ if $\nabla^{\prime}$ is the canonical connection
introduced in [10]. In Theorem 5.1 we prove that $C(R^{\prime})=C(K)$ if
$\nabla^{\prime}$ is a natural connection with parallel torsion. Moreover, we
consider a tensor $E(L)$ determined by a curvature-like tensor $L$. In Theorem
6.1 we prove that $E(R^{\prime})=E(R)$ for the natural connection
$\nabla^{\prime}=D$, considered in [3], in the case when $D$ has a parallel
torsion.

##  1\. Preliminaries

Let $(M,P,g)$ be a _Riemannian almost product manifold_ , i.e. a
differentiable manifold $M$ with a tensor field $P$ of type $(1,1)$ and a
Riemannian metric $g$ such that $P^{2}x=x$, $g(Px,Py)=g(x,y)$ for any $x$, $y$
of the algebra $\mathfrak{X}(M)$ of the smooth vector fields on $M$. Further
$x,y,z,w$ will stand for arbitrary elements of $\mathfrak{X}(M)$ or vectors in
the tangent space $T_{c}M$ at $c\in M$.

In [11] A.M. Naveira gives a classification of Riemannian almost product
manifolds with respect to the tensor $F$ of type (0,3), defined by
$F(x,y,z)=g\left(\left(\nabla_{x}P\right)y,z\right),$ where $\nabla$ is the
Levi-Civita connection of $g$.

In this work we consider manifolds $(M,P,g)$ with ${\rm tr}{P}=0$. In this
case $M$ is an even-dimensional manifold. We assume that $\dim{M}=2n$.

Using the Naveira classification, in [13] M. Staikova and K. Gribachev give a
classification of Riemannian almost product manifolds $(M,P,g)$ with ${\rm
tr}P=0$. The basic classes of this classification are $\mathcal{W}_{1}$,
$\mathcal{W}_{2}$ and $\mathcal{W}_{3}$. Their intersection is the class
$\mathcal{W}_{0}$ of the _Riemannian $P$-manifolds_ ([12]), determined by the
condition $F=0$. This class is an analogue of the class of Kähler manifolds in
the geometry of almost Hermitian manifolds.

The class $\mathcal{W}_{1}$ from the Staikova-Gribachev classification
consists of the Riemannian product manifolds which are locally conformal
equivalent to Riemannian $P$-manifolds. This class plays a similar role of the
role of the class of the conformal Kähler manifolds in almost Hermitian
geometry. We will say that a manifold from the class $\mathcal{W}_{1}$ is a
_$\mathcal{W}_{1}$ -manifold_.

The characteristic condition for the class $\mathcal{W}_{1}$ is the following

$\begin{array}[]{l}\mathcal{W}_{1}:F(x,y,z)=\frac{1}{2n}\big{\\{}g(x,y)\theta(z)-g(x,Py)\theta(Pz)\big{.}\\\\[4.0pt]
\phantom{\mathcal{W}_{1}:F(x,y,z)=\frac{1}{2n}}+g(x,z)\theta(y)-g(x,Pz)\theta(Py)\big{\\}},\end{array}$

where the associated 1-form $\theta$ is determined by
$\theta(x)=g^{ij}F(e_{i},e_{j},x).$ Here $g^{ij}$ will stand for the
components of the inverse matrix of $g$ with respect to a basis $\\{e_{i}\\}$
of $T_{c}M$ at $c\in M$. The 1-form $\theta$ is _closed_ , i.e. ${\rm
d}\theta=0$, if and only if
$\left(\nabla_{x}\theta\right)y=\left(\nabla_{y}\theta\right)x$. Moreover,
$\theta\circ P$ is a closed 1-form if and only if
$\left(\nabla_{x}\theta\right)Py=\left(\nabla_{y}\theta\right)Px$.

In [13] it is proved that
$\mathcal{W}_{1}=\overline{\mathcal{W}}_{3}\oplus\overline{\mathcal{W}}_{6}$,
where $\overline{\mathcal{W}}_{3}$ and $\overline{\mathcal{W}}_{6}$ are the
classes from the Naveira classification determined by the following
conditions:

$\begin{array}[]{rl}\overline{\mathcal{W}}_{3}:&\quad
F(A,B,\xi)=\frac{1}{n}g(A,B)\theta^{v}(\xi),\quad F(\xi,\eta,A)=0,\\\\[4.0pt]
\overline{\mathcal{W}}_{6}:&\quad
F(\xi,\eta,A)=\frac{1}{n}g(\xi,\eta)\theta^{h}(A),\quad
F(A,B,\xi)=0,\end{array}$

where $A,B,\xi,\eta\in\mathfrak{X}(M)$, $PA=A$, $PB=B$, $P\xi=-\xi$,
$P\eta=-\eta$, $\theta^{v}(x)=\frac{1}{2}\left(\theta(x)-\theta(Px)\right)$,
$\theta^{h}(x)=\frac{1}{2}\left(\theta(x)+\theta(Px)\right)$. In the case when
${\rm tr}P=0$, the above conditions for $\overline{\mathcal{W}}_{3}$ and
$\overline{\mathcal{W}}_{6}$ can be written for any $x,y,z$ in the following
form:

$\begin{array}[]{rl}\overline{\mathcal{W}}_{3}:&F(x,y,z)=\frac{1}{2n}\bigl{\\{}\left[g(x,y)+g(x,Py)\right]\theta(z)\\\\[4.0pt]
&+\left[g(x,z)+g(x,Pz)\right]\theta(y)\bigr{\\}},\quad\theta(Px)=-\theta(x),\\\\[4.0pt]
\overline{\mathcal{W}}_{6}:&F(x,y,z)=\frac{1}{2n}\bigl{\\{}\left[g(x,y)-g(x,Py)\right]\theta(z)\\\\[4.0pt]
&+\left[g(x,z)-g(x,Pz)\right]\theta(y)\bigr{\\}},\quad\theta(Px)=\theta(x).\end{array}$

In [13], a tensor $L$ of type (0,4) with properties

$L(x,y,z,w)=-L(y,x,z,w)=-L(x,y,w,z),$ $L(x,y,z,w)+L(y,z,x,w)+L(z,x,y,w)=0$

is called a _curvature-like tensor_. Such a tensor on a Riemannian almost
product manifold $(M,P,g)$ with the property

$L(x,y,Pz,Pw)=L(x,y,z,w)$

is called a _Riemannian $P$-tensor_ in [9]. This notion is an analogue of the
notion of a Kähler tensor in Hermitian geometry.

Let $S$ be a (0,2)-tensor on a Riemannian almost product manifold. In [13] it
is proved that

(1.1) $\begin{split}\psi_{1}(S)(x,y,z,w)&=g(y,z)S(x,w)-g(x,z)S(y,w)\\\\[4.0pt]
&+S(y,z)g(x,w)-S(x,z)g(y,w)\end{split}$

is a curvature-like tensor if and only if $S(x,y)=S(y,x)$, and the tensor

(1.2) $\psi_{2}(S)(x,y,z,w)=\psi_{1}(S)(x,y,Pz,Pw)$

is curvature-like if and only if $S(x,Py)=S(y,Px)$. Obviously

$\psi_{2}(S)(x,y,Pz,Pw)=\psi_{1}(S)(x,y,z,w).$

If $\psi_{1}(S)$ and $\psi_{2}(S)$ are curvature-like tensors, then
$\left(\psi_{1}+\psi_{2}\right)(S)$ is a Riemannian $P$-tensor. The tensors

(1.3)
$\pi_{1}=\frac{1}{2}\psi_{1}(g),\qquad\pi_{2}=\frac{1}{2}\psi_{2}(g),\qquad\pi_{3}=\psi_{1}(\widetilde{g})=\psi_{2}(\widetilde{g})$

are curvature-like, where $\widetilde{g}(x,y)=g(x,Py)$, and the tensors
$\pi_{1}+\pi_{2}$, $\pi_{3}$ are Riemannian $P$-tensors.

The curvature tensor $R$ of $\nabla$ is determined by
$R(x,y)z=\nabla_{x}\nabla_{y}z-\nabla_{y}\nabla_{x}z-\nabla_{[x,y]}z$ and the
corresponding tensor of type (0,4) is defined as follows
$R(x,y,z,w)=g(R(x,y)z,w)$. We denote the Ricci tensor and the scalar curvature
of $R$ by $\rho$ and $\tau$, respectively, i.e.
$\rho(y,z)=g^{ij}R(e_{i},y,z,e_{j})$ and $\tau=g^{ij}\rho(e_{i},e_{j})$. The
associated Ricci tensor $\rho^{*}$ and the associated scalar curvature
$\tau^{*}$ of $R$ are determined by $\rho^{*}(y,z)=g^{ij}R(e_{i},y,z,Pe_{j})$
and $\tau^{*}=g^{ij}\rho^{*}(e_{i},e_{j})$. In a similar way there are
determined the Ricci tensor $\rho(L)$ and the scalar curvature $\tau(L)$ for
any curvature-like tensor $L$ as well as the associated quantities
$\rho^{*}(L)$ and $\tau^{*}(L)$.

In [10], a linear connection $\nabla^{\prime}$ on a Riemannian almost product
manifold $(M,P,g)$ is called a _natural connection_ if
$\nabla^{\prime}P=\nabla^{\prime}g=0$.

In [2], it is established that the natural connections $\nabla^{\prime}$ on a
$\mathcal{W}_{1}$-manifold $(M,P,g)$ form a 2-parametric family, where the
torsion $T$ of $\nabla^{\prime}$ is determined by

(1.4)
$\begin{split}T(x,y,z)&=\frac{1}{2n}\left\\{g(y,z)\theta(Px)-g(x,z)\theta(Py)\right\\}\\\\[4.0pt]
&\phantom{=\
}+\lambda\left\\{g(y,z)\theta(x)-g(x,z)\theta(y)\right.\\\\[4.0pt]
&\phantom{=\
+\lambda\left\\{\right.}\left.+g(y,Pz)\theta(Px)-g(x,Pz)\theta(Py)\right\\}\\\\[4.0pt]
&\phantom{=\ }+\mu\left\\{g(y,Pz)\theta(x)-g(x,Pz)\theta(y)\right.\\\\[4.0pt]
&\phantom{=\
+\mu\left\\{\right.}\left.+g(y,z)\theta(Px)-g(x,z)\theta(Py)\right\\},\end{split}$

where $\lambda,\mu\in\mathbb{R}$.

Let $Q$ be the tensor determined by

(1.5) $\nabla^{\prime}_{x}y=\nabla_{x}y+Q(x,y).$

The corresponding tensor of type (0,3), according to [5], satisfies

(1.6) $Q(x,y,z)=T(z,x,y).$

Let us recall the following statement.

######  Theorem 1.1 ([5]).

Let $R^{\prime}$ is the curvature tensor of a natural connection
$\nabla^{\prime}$ on a $\mathcal{W}_{1}$-manifold $(M,P,g)$. Then the
following relation is valid:

(1.7)
$R=R^{\prime}-g(p,p)\pi_{1}-g(q,q)\pi_{2}-g(p,q)\pi_{3}-\psi_{1}(S^{\prime})-\psi_{2}(S^{\prime\prime}),$

where

$\begin{array}[]{l}p=\lambda\Omega+\left(\mu+\frac{1}{2n}\right)P\Omega,\quad
q=\lambda P\Omega+\mu\Omega,\quad g(\Omega,x)=\theta(x),\end{array}$
$\begin{array}[]{rl}&S^{\prime}(y,z)=\lambda\left(\nabla^{\prime}_{y}\theta\right)z+\left(\mu+\frac{1}{2n}\right)\left(\nabla^{\prime}_{y}\theta\right)Pz\\\\[4.0pt]
&\phantom{S^{\prime}(y,z)=}-\frac{1}{2n}\left\\{\lambda\theta(y)\theta(Pz)+\mu\theta(y)\theta(z)\right\\},\\\\[4.0pt]
&S^{\prime\prime}(y,z)=\lambda\left(\nabla^{\prime}_{y}\theta\right)z+\mu\left(\nabla^{\prime}_{y}\theta\right)Pz\\\\[4.0pt]
&\phantom{S^{\prime\prime}(y,z)=}+\frac{1}{2n}\left\\{\lambda\theta(Py)\theta(z)+\mu\theta(Py)\theta(Pz)\right\\}.\end{array}$

##  2\. Some properties of the natural connections on the manifolds of the
class $\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$

Let $(M,P,g)$ is a Riemannian product manifold of the class
$\overline{\mathcal{W}}_{3}$ or the class $\overline{\mathcal{W}}_{6}$, i.e.
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$. Then for
the 1-form $\theta$ and the vector $\Omega$ we have

(2.1) $\theta(Pz)=\varepsilon\theta(z),\qquad P\Omega=\varepsilon\Omega,$

where $\varepsilon=1$ for $(M,P,g)\in\overline{\mathcal{W}}_{3}$ and
$\varepsilon=-1$ for $(M,P,g)\in\overline{\mathcal{W}}_{6}$.

Let $\nabla^{\prime}$ be a natural connection on
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$.

Using (1.4), (1.6) and (2.1), we obtain for the tensor $Q$ determined by (1.5)
the following

$\begin{split}Q(x,y)&=\left[\lambda+\varepsilon\left(\mu+\frac{1}{2n}\right)\right]\left[g(x,y)-\theta(y)x\right]\\\\[4.0pt]
&\phantom{=\
}\left(\mu+\varepsilon\lambda\right)\left[g(x,Py)-\theta(y)Px\right].\end{split}$

Now, for the curvature tensors $R$ and $R^{\prime}$ of $\nabla$ and
$\nabla^{\prime}$, it is valid (1.7), where

(2.2) $\displaystyle
p=\left(\lambda+\varepsilon\mu+\frac{\varepsilon}{2n}\right)\Omega,\qquad
q=(\mu+\varepsilon\lambda)\Omega,$ (2.3) $\displaystyle
S^{\prime}(y,z)=\left(\lambda+\varepsilon\mu+\frac{\varepsilon}{2n}\right)\left(\nabla^{\prime}_{y}\theta\right)z-\frac{\mu+\varepsilon\lambda}{2n}\theta(y)\theta(z),$
(2.4) $\displaystyle
S^{\prime\prime}(y,z)=\left(\lambda+\varepsilon\mu\right)\left(\nabla^{\prime}_{y}\theta\right)z+\frac{\mu+\varepsilon\lambda}{2n}\theta(y)\theta(z).$

Further we consider manifolds
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with
closed 1-form $\theta$. In this case, the tensor $K$, determined by

(2.5) $K(x,y,z,w)=\frac{1}{2}\left[R(x,y,z,w)+R(x,y,Pz,Pw)\right],$

is a Riemannian $P$-tensor ([4]).

If $(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ has a
closed 1-form $\theta$, then the curvature tensor $R^{\prime}$ of a natural
connection $\nabla^{\prime}$ is also a Riemannian $P$-tensor. Indeed, from
(1.7) it is clear, that $R^{\prime}$ is a Riemannian $P$-tensor if and only if
$\psi_{1}(S^{\prime})$ and $\psi_{2}(S^{\prime\prime})$ are curvature-like
tensors, i.e. if and only if $S^{\prime}(y,z)=S^{\prime}(z,y)$ and
$S^{\prime\prime}(y,Pz)=S^{\prime\prime}(z,Py)$. According to (2.3) and (2.4),
the latter conditions are valid if and only if

(2.6)
$\left(\nabla^{\prime}_{y}\theta\right)z=\left(\nabla^{\prime}_{z}\theta\right)y.$

In [5], it is proved that for any $\mathcal{W}_{1}$-manifold the following
equality is valid:

$\begin{split}\left(\nabla^{\prime}_{y}\theta\right)z-\left(\nabla^{\prime}_{z}\theta\right)y&=\left(\nabla_{y}\theta\right)z-\left(\nabla_{z}\theta\right)y\\\\[4.0pt]
&-\frac{1}{2n}\left\\{\theta(Py)\theta(z)-\theta(y)\theta(Pz)\right\\}.\end{split}$

Bearing in mind (2.1), the latter equality implies that equality (2.6) is
valid on $(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$
if and only if
$\left(\nabla_{y}\theta\right)z=\left(\nabla_{z}\theta\right)y$, i.e. if and
only if the 1-form $\theta$ is closed.

######  Theorem 2.1.

Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ be with a
closed 1-form $\theta$. Then the following equality is valid

(2.7) $K=R^{\prime}-\left(\psi_{1}+\psi_{2}\right)(S),$

where

(2.8)
$\begin{split}S(y,z)&=\left(\lambda+\varepsilon\mu+\frac{\varepsilon}{4n}\right)\left(\nabla^{\prime}_{y}\theta\right)z\\\\[4.0pt]
&+\frac{g(p,p)+g(q,q)}{4}g(y,z)+\frac{g(p,q)}{2}g(y,Pz).\end{split}$

###### Proof.

According to Theorem 1.1, for $(M,P,g)$ it is valid the equality

(2.9)
$\begin{split}R(x,y,z,w)&=\left\\{R^{\prime}-g(p,p)\pi_{1}-g(q,q)\pi_{2}-g(p,q)\pi_{3}\right.\\\\[4.0pt]
&\phantom{=\left\\{\right.}\left.-\psi_{1}(S^{\prime})-\psi_{2}(S^{\prime\prime})\right\\}(x,y,z,w).\end{split}$

In (2.9), we substitute $Pz$ and $Pw$ for $z$ and $w$, respectively. We add
the obtained equality to (2.9). Then, taking into account (1.1), (1.2), (1.3),
(2.3), (2.4), (2.5), (2.8) and the properties of the curvature-like tensors
$\psi_{1}(S^{\prime})$ and $\psi_{2}(S^{\prime\prime})$, we get (2.7). ∎

In Section 3, Section 4 and Section 5, we find some Riemannian $P$-tensors
determined by $K$ on a manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$. We establish that the found tensors coincide with the
corresponding tensors determined by the curvature tensor $R^{\prime}$ of a
natural connection $\nabla^{\prime}$. In Section 6, we find a curvature-like
tensor determined by $R$ on such a manifold and establish that this tensor
coincides with the corresponding tensor determined by the curvature tensor
$R^{\prime}$ of the special natural connection $D$ investigated in [5], in the
case when $D$ has a parallel torsion.

##  3\. An arbitrary natural connection on a manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$

In [13], it is defined a Bochner tensor $B(L)$ for an arbitrary Riemannian
$P$-tensor $L$ on a $\mathcal{W}_{1}$-manifold $(M,P,g)$ ($\dim M\geq 6$) as
follows:

(3.1)
$\begin{split}B(L)&=L-\frac{1}{2(n-2)}\left\\{(\psi_{1}+\psi_{2})(\rho(L))\phantom{\frac{1}{2(n-1)}}\right.\\\\[4.0pt]
&\left.\phantom{=L}-\frac{1}{2(n-1)}\left[\tau(L)(\pi_{1}+\pi_{2})+\tau^{*}(L)\pi_{3}\right]\right\\}.\end{split}$

Let us remark that $B(L)$ is also a Riemannian $P$-tensor.

######  Theorem 3.1.

Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ ($\dim
M\geq 6$) be with a closed 1-form $\theta$. If $R^{\prime}$ is the curvature
tensor of a natural connection $\nabla^{\prime}$, then $B(R^{\prime})=B(K)$.

###### Proof.

Relation (2.7) implies the following equality for the Ricci tensors $\rho(K)$
and $\rho^{\prime}$ of $K$ and $R^{\prime}$, respectively:

(3.2) $\rho(K)=\rho^{\prime}-{\rm tr}S\ g-{\rm tr}\widetilde{S}\
\widetilde{g}-2(n-2)S,$

where $\widetilde{S}(y,z)=S(y,Pz)$. Then we get the following equalities for
the scalar curvatures:

(3.3) ${\rm tr}S=\frac{\tau^{\prime}-\tau(K)}{4(n-1)},\qquad{\rm
tr}\widetilde{S}=\frac{\tau^{\prime*}-\tau^{*}(K)}{4(n-1)}.$

Equalities (3.2) and (3.3) imply

(3.4)
$\begin{split}S=\frac{1}{2(n-2)}\left\\{\rho^{\prime}-\rho(K)-\frac{(\tau^{\prime}-\tau(K))g+(\tau^{\prime*}-\tau^{*}(K))\widetilde{g}}{4(n-1)}\right\\}.\end{split}$

From (1.1), (1.2), (1.3) and (3.4), we have

(3.5) $\begin{split}&(\psi_{1}+\psi_{2})(S)=\\\\[4.0pt]
&=\frac{1}{2(n-2)}\left\\{(\psi_{1}+\psi_{2})(\rho^{\prime})-(\psi_{1}+\psi_{2})(\rho(K))\phantom{\frac{1}{2(n-2)}\left\\{\right.=}\right.\\\\[4.0pt]
&\left.\phantom{\frac{1}{2(n-2)}\left\\{\right.=}-\frac{(\tau^{\prime}-\tau(K))(\pi_{1}+\pi_{2})+(\tau^{\prime*}-\tau^{*}(K))\pi_{3}}{2(n-1)}\right\\}.\end{split}$

Using (3.5), (2.7) and the definition (3.1) of the Bochner tensor, we obtain
$B(K)=B(R^{\prime})$. ∎

##  4\. The canonical connection on a manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$

The canonical connection on a Riemannian almost product manifold is a natural
connection introduced in [10] as an analogue of the Hermitian connection on
almost Hermitian manifold. A connection of such a type on almost contact
B-metric manifolds is considered in [7], [8].

We define the tensor $A(L)$ for an arbitrary Riemannian $P$-tensor $L$ by the
equality

(4.1) $A(L)=L-\frac{\tau(L)(\pi_{1}+\pi_{2}-\varepsilon\pi_{3})}{4n(n-1)}.$

Obviously, $A(L)$ is also a Riemannian $P$-tensor.

######  Theorem 4.1.

Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ be with a
closed 1-form $\theta$. If $R^{\prime}$ is the curvature tensor of the
canonical connection, then $A(R^{\prime})=A(K)$.

###### Proof.

In [5], it is shown the canonical connection on a $\mathcal{W}_{1}$-manifold
is determined by $\lambda=0$ and $\mu=-\frac{1}{4n}$. Then, (2.2) implies
$p=\frac{\varepsilon\Omega}{4n}$, $q=-\frac{\Omega}{4n}$ and therefore

(4.2) $g(p,p)=g(q,q)=-\varepsilon g(p,q)=\frac{\theta(\Omega)}{16n^{2}}.$

From (2.8) and (4.2) it is follows
$S=\frac{\theta(\Omega)}{32n^{2}}(g-\varepsilon\widetilde{g})$. Then, because
of (1.3), we have
$(\psi_{1}+\psi_{2})(S)=\frac{\theta(\Omega)}{16n^{2}}(\pi_{1}+\pi_{2}-\varepsilon\pi_{3})$.
Thus, (2.7) takes the form

(4.3)
$K=R^{\prime}-\frac{\theta(\Omega)(\pi_{1}+\pi_{2}-\varepsilon\pi_{3})}{16n^{2}}.$

By virtue of (4.3), we obtain the following equalities

(4.4)
$\begin{split}&\rho(K)=\rho^{\prime}-\frac{(n-1)\theta(\Omega)(g-\varepsilon\widetilde{g})}{8n^{2}},\\\\[4.0pt]
&\theta(\Omega)=\frac{4n(\tau^{\prime}-\tau(K))}{n-1}=-\frac{4n\varepsilon(\tau^{\prime*}-\tau^{*}(K))}{n-1}.\end{split}$

Bearing in mind (4.3) and (4.4), by suitable calculations we get

$R^{\prime}-\frac{\tau^{\prime}(\pi_{1}+\pi_{2}-\varepsilon\pi_{3})}{4n(n-1)}=K-\frac{\tau(K)(\pi_{1}+\pi_{2}-\varepsilon\pi_{3})}{4n(n-1)}.$

Then, according to (4.1), we have $A(R^{\prime})=A(K)$. ∎

In [12], a 2-plane $\alpha=(x,y)$ in $T_{c}M$ is called a totally real 2-plane
if $\alpha$ is orthogonal to $P\alpha$. Its sectional curvatures with respect
to $R^{\prime}$

$\nu^{\prime}=\frac{R^{\prime}(x,y,y,x)}{\pi_{1}(x,y,y,x)},\qquad\nu^{\prime*}=\frac{R^{\prime}(x,y,y,Px)}{\pi_{1}(x,y,y,x)}$

are called totally real sectional curvatures with respect to $R^{\prime}$.

######  Theorem 4.2.

A manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$ has point-wise constant totally real sectional
curvatures

$\nu^{\prime}=\frac{\tau^{\prime}}{4n(n-1)},\qquad\nu^{\prime*}=-\frac{\varepsilon\tau^{\prime}}{4n(n-1)}$

with respect to the curvature tensor $R^{\prime}$ of the canonical connection
if and only if $A(R^{\prime})=0$ (or equivalently $A(K)=0$).

###### Proof.

According to (4.1), the condition for annulment of $A(R^{\prime})$ is the
condition

$R^{\prime}=\frac{\tau^{\prime}(\pi_{1}+\pi_{2}-\varepsilon\pi_{3})}{4n(n-1)}.$

Then, bearing in mind [12], we establish the truthfulness of the statement. ∎

##  5\. An natural connection with parallel torsion on a manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$

We define the tensor $C(L)$ for an arbitrary Riemannian $P$-tensor $L$ by the
equality

(5.1) $C(L)=L-\frac{\tau(L)(\pi_{1}+\pi_{2})+\tau^{*}(L)\pi_{3}}{4n(n-1)}.$

Obviously, $C(L)$ is also a Riemannian $P$-tensor.

######  Theorem 5.1.

Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ be with a
closed 1-form $\theta$. If $R^{\prime}$ is the curvature tensor of a natural
connection with a parallel torsion, then $C(R^{\prime})=C(K)$.

###### Proof.

In [5], it is proved that a natural connection $\nabla^{\prime}$ on a
$\mathcal{W}_{1}$-manifold has a parallel torsion if and only if the 1-form
$\theta$ is also parallel, i.e. $\nabla^{\prime}\theta=0$. Then, (2.8) implies

$S=\frac{g(p,p)+g(q,q)}{4}g+\frac{g(p,q)}{2}\widetilde{g}.$

Then, because of (1.3), we have

$(\psi_{1}+\psi_{2})(S)=\frac{g(p,p)+g(q,q)}{2}(\pi_{1}+\pi_{2})+g(p,q)\pi_{3}.$

Thus, (2.7) takes the form

(5.2) $K=R^{\prime}-\frac{g(p,p)+g(q,q)}{2}(\pi_{1}+\pi_{2})+g(p,q)\pi_{3}.$

By virtue of (5.2), we obtain

$\rho(K)=\rho^{\prime}-(n-1)[g(p,p)+g(q,q)]g-2(n-1)g(p,q)\widetilde{g}),$

which implies

(5.3) $\begin{split}&\tau(K)=\tau^{\prime}-2n(n-1)[g(p,p)+g(q,q)],\\\\[4.0pt]
&\tau^{*}(K)=\tau^{\prime*}-4n(n-1)g(p,q).\end{split}$

Bearing in mind (5.2) and (5.3), by suitable calculations we get

$R^{\prime}-\frac{\tau^{\prime}(\pi_{1}+\pi_{2})+\tau^{\prime*}\pi_{3}}{4n(n-1)}=K-\frac{\tau(K)(\pi_{1}+\pi_{2})+\tau^{*}(K)\pi_{3}}{4n(n-1)}.$

Then, according to (5.1), we have $C(R^{\prime})=C(K)$. ∎

######  Theorem 5.2.

A manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$ has point-wise constant totally real sectional
curvatures

$\nu^{\prime}=\frac{\tau^{\prime}}{4n(n-1)},\qquad\nu^{\prime*}=\frac{\tau^{\prime*}}{4n(n-1)}$

with respect to the curvature tensor $R^{\prime}$ of an arbitrary natural
connection with parallel torsion if and only if $C(R^{\prime})=0$ (or
equivalently $C(K)=0$).

###### Proof.

According to (5.1), the condition for annulment of $C(R^{\prime})$ is the
condition

$R^{\prime}=\frac{\tau^{\prime}(\pi_{1}+\pi_{2})+\tau^{\prime*}\pi_{3}}{4n(n-1)}.$

Then, bearing in mind [12], we establish the truthfulness of the statement. ∎

##  6\. The natural connection $D$ ($\lambda=\mu=0$) with parallel torsion on
a manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ with a
closed 1-form $\theta$

In [3], it is studied the natural connection $D$ determined by $\lambda=\mu=0$
on a $\mathcal{W}_{1}$-manifold $(M,P,g)$.

Now we consider the case when
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ is with a
closed 1-form $\theta$ and the connection $\nabla^{\prime}=D$ has a parallel
torsion. Then, from (2.2), (2.3) and (2.4) we have
$p=\frac{\varepsilon\Omega}{2n}$, $q=S^{\prime}=S^{\prime\prime}=0$ and
therefore (1.7) takes the form

(6.1) $R=R^{\prime}-\frac{\theta(\Omega)\pi_{1}}{4n^{2}}.$

The latter equality implies
$\rho=\rho^{\prime}-\frac{(2n-1)\theta(\Omega)}{4n^{2}}g,$ which gives us

(6.2)
$\tau=\tau^{\prime}-\frac{(2n-1)\theta(\Omega)}{2n},\qquad\tau^{*}=\tau^{\prime*}.$

We define the tensor $E(L)$ for an arbitrary curvature-like tensor $L$ by the
equality

(6.3) $E(L)=L-\frac{\tau(L)\pi_{1}}{2n(2n-1)}.$

Obviously, $E(L)$ is also a curvature-like tensor.

######  Theorem 6.1.

Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ be with a
closed 1-form $\theta$. If $R^{\prime}$ is the curvature tensor of the
connection $D$ with parallel torsion, then $E(R^{\prime})=E(R)$.

###### Proof.

Equalities (6.1) and (6.2) imply

$R-\frac{\tau\pi_{1}}{2n(2n-1)}=R^{\prime}-\frac{\tau^{\prime}\pi_{1}}{2n(2n-1)}.$

Then, according to (6.3), we have $E(R^{\prime})=E(R)$. ∎

######  Theorem 6.2.

Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ be with a
closed 1-form $\theta$ and $D$ be with a parallel torsion. Then $D$ is flat if
and only if $E(R^{\prime})=0$ (or equivalently $E(R)=0$).

###### Proof.

Let $E(R^{\prime})=0$ be valid, i.e.

(6.4) $R^{\prime}(x,y,z,w)=\frac{\tau^{\prime}}{2n(2n-1)}\pi_{1}(x,y,z,w).$

In (6.4), we substitute $Pz$ and $Pw$ for $z$ and $w$, respectively. Taking
into account that $R^{\prime}$ is a Riemannian $P$-tensor and
$\pi_{1}(x,y,Pz,Pw)=\pi_{2}(x,y,z,w)$, we obtain

(6.5) $R^{\prime}=\frac{\tau^{\prime}}{2n(2n-1)}\pi_{2}.$

From (6.4) and (6.5) it is follows $\tau^{\prime}(\pi_{1}-\pi_{2})=0$ and
because of $\pi_{1}\neq\pi_{2}$ we have $\tau^{\prime}=0$. Then
$R^{\prime}=0$, according to (6.4), i.e. $D$ is a flat connection.

Vice versa, let $D$ be flat, i.e. $R^{\prime}=0$. Then $\tau^{\prime}=0$ and
bearing in mind the definition of $E(R^{\prime})$ we obtain $E(R^{\prime}=0$.
∎

######  Corollary 6.3.

Let the manifold
$(M,P,g)\in\overline{\mathcal{W}}_{3}\cup\overline{\mathcal{W}}_{6}$ be with a
closed 1-form $\theta$ and $D$ be flat with a parallel torsion. Then $(M,P,g)$
is a space form with a negative scalar curvature $\tau$.

###### Proof.

If $D$ is flat, then by Theorem 6.2 we have $E(R)=0$, i.e.

$R=\frac{\tau}{2n(2n-1)}\pi_{1}.$

This means that the manifold is a space form. Moreover, $\tau^{\prime}=0$ for
a flat connection $D$ and therefore $\tau=-\frac{2n-1}{2n}\theta(\Omega)$,
because of (6.2). Thus, since $\theta(\Omega)=g(\Omega,\Omega)>0$, we obtain
$\tau<0$. ∎

## References

  * [1] A. Gray, L. Hervella, _The sixteen classes of almost Hermitian manifolds and their linear invariants._ Ann. Mat. Pura Appl. 123 (1980), 35–58.
  * [2] D. Gribacheva, _Natural connections on Riemannian product manifolds._ Compt. rend. Acad. bulg. Sci. 64 (2011), no. 6, 799–806.
  * [3] D. Gribacheva, _A natural connection on a basic class of Riemannian product manifolds._ Int. J. Geom. Methods Mod. Phys., 9 (2012), no. 7, 1250057 (14 pages).
  * [4] D. Gribacheva, _Curvature properties of two Naveira classes of Riemannian product manifolds._ Plovdiv Univ. Sci. Works – Math. (In Press), arXiv:1204.5838.
  * [5] D. Gribacheva, D. Mekerov, _Natural connections on conformal Riemannian $P$-manifolds._ Compt. rend. Acad. bulg. Sci. 65 (2012), no. 5, 581–590.
  * [6] H. Hayden, _Subspaces of a space with torsion._ Proc. London Math. Soc. 34 (1934), 27–50.
  * [7] M. Manev, M. Ivanova. _Canonical-type connection on almost contact manifolds with B-metric_ ; arXiv:1203.0137.
  * [8] M. Manev, M. Ivanova. _Almost contact B-metric manifolds with curvature tensor of Kähler type_. Plovdiv Univ. Sci. Works – Math., vol. 39, no. 3 (2012) (In Press); arXiv:1203.3290.
  * [9] D. Mekerov. _On Riemannian almost product manifolds with nonintegrable structure._ J. Geom. 89 (2008), no. 1-2, 119–129.
  * [10] V. Mihova, _Canonical connections and the canonical conformal group on a Riemannian almost product manifold._ Serdica Math. P., 15 (1989), 351–358.
  * [11] A. M. Naveira, _A classification of Riemannian almost product manifolds._ Rend. Math. 3 (1983), 577–592.
  * [12] M. Staikova, _Curvature properties of Riemannian $P$-manifolds._ Plovdiv Univ. Sci. Works - Math. 32 (1987), no. 3, 241–251.
  * [13] M. Staikova, K. Gribachev, _Canonical connections and their conformal invariants on Riemannian $P$-manifolds._ Serdica Math. P. 18 (1992), 150–161.
  * [14] M. Staikova, K. Gribachev, D. Mekerov, _Riemannian $P$-manifolds of constant sectional curvatures._ Serdica Math. J. 17 (1991), 212–219.
  * [15] K. Yano, _Differential geometry on complex and almost complex spaces._ Pure and Applied Math. 49, New York, Pergamon Press Book, 1965.

D. Gribacheva
Department of Algebra and Geometry
Faculty of Mathematics and Informatics
University of Plovdiv
236 Bulgaria Blvd
4003 Plovdiv, Bulgaria
<EMAIL_ADDRESS>