Locally Product-like Statistical Manifolds and Their Hypersurfaces

Esra Erkan, Kazuhiko Takano and Mehmet Gülbahar

> Abstract. In this paper, almost product-like Riemannian manifolds are
> investigated. Basic properties on tangential hypersurfaces of almost
> product-like Riemannian manifolds are obtained. Some examples of tangential
> hypersurfaces are presented. Some relations involving the Riemannian
> curvature tensor of a tangential hypersurface are computed.

> AMS 2000 Mathematics Subject Classification: 53C05, 53C40, 53C42.
>
> Keywords: Riemann product manifold, hypersurface, statistical manifold

##  1 Introduction

The concept of statistical manifolds has important application areas and
various properties of statistical structures in geometric and physical terms
have been studied intensively recently. In physics, there exist various
application areas of statistical manifolds such as neural networks, machine
learning, artificial intelligence, black holes [8, 12, 19, 27]. Statistical
manifolds were firstly introduced by S. Amari [4] in 1985. Later, these
structures drew attention of several authors. Some basic properties of
hypersurfaces of statistical manifolds were presented by H. Furuhata in [13,
17], hypersurfaces of Sasakian statistical manifolds were investigated by F.
Wu, Y. Jiang, and L. Zhang in [14], hypersurfaces of almost Hermitian
statistical manifolds were investigated by H. Akbari and F. Malek in [3], etc.

As a new perspective on Hermitian statistical manifolds, the second author was
firstly introduced the notion of Hermite-like manifolds in [23, 24]. The most
important aspect of this point of view is that it reveals to us a more general
form of Hermitian geometry, which has a long history and on which many
mathematicians, physicists, and engineers have investigated. In fact, a
Hermite-like manifold becomes a Hermitian manifold when two almost complex
structures $\displaystyle J$ and $\displaystyle J^{\ast}$ are equal.

Inspired by the definition of Hermite-like manifolds, contact-like manifolds
were studied in [5, 15, 23, 24], para-Kaehler-like and quaternionic-like
Kaehler manifolds were investigated in [25, 26], the Wintgen-like inequality
on statistical warped product manifolds was computed in [21], etc.

The main purpose of this paper is to present the basic properties of almost
product-like Riemannian manifolds and their hypersurfaces. With these reviews
and using some obtained properties for hypersurfaces of almost product-like
Riemannian manifolds, we will be able to the opportunity to give some basic
definitions of para contact-like Riemannian manifolds.

##  2 Preliminaries

Let $\displaystyle(\widetilde{M},\widetilde{g})$ be a Riemannian manifold.
Denote a torsion-free affine connection by $\displaystyle\widetilde{\nabla}$.
The triple $\displaystyle(\widetilde{M},\widetilde{g},\widetilde{\nabla})$ is
called a statistical manifold if
$\displaystyle\widetilde{\nabla}\widetilde{g}$ is symmetric. For the
statistical manifold, we define another affine connection
$\displaystyle\widetilde{\nabla}^{\ast}$ by

$Z\widetilde{g}(X,Y)=\widetilde{g}(X,\widetilde{\nabla}_{Z}^{\ast}Y)+\widetilde{g}(\widetilde{\nabla}_{Z}X,Y),$
(2.1)

for any vector fields $\displaystyle X,Y$ and $\displaystyle Z$ on
$\displaystyle\widetilde{M}$. The affine connection
$\displaystyle\widetilde{\nabla}^{\ast}$ is called dual of
$\displaystyle\widetilde{\nabla}$ with respect to
$\displaystyle\widetilde{g}$. The affine connection
$\displaystyle\widetilde{\nabla}^{\ast}$ is torsion-free and satisfies
$\displaystyle(\widetilde{\nabla}^{\ast})^{\ast}=\widetilde{\nabla}$. It is
clear to see that
$\displaystyle\widetilde{\nabla}^{\circ}=\frac{1}{2}\left(\widetilde{\nabla}+\widetilde{\nabla}^{\ast}\right)$
is a metric connection. The pair
$\displaystyle(\widetilde{g},\widetilde{\nabla})$ is a statistical structure
on $\displaystyle\widetilde{M}$ is and only if so is
$\displaystyle(\widetilde{g},\widetilde{\nabla}^{\ast})$. Therefore the triple
$\displaystyle(\widetilde{M},\widetilde{g},\widetilde{\nabla}^{\ast})$ is a
statistical manifold, too. We denote by $\displaystyle\widetilde{R}$ and
$\displaystyle\widetilde{R}^{\ast}$ the curvature tensors on
$\displaystyle\widetilde{M}$ with respect to the affine connection
$\displaystyle\widetilde{\nabla}$ and its conjugate
$\displaystyle\widetilde{\nabla}^{\ast}$, respectively. Then we find

$\widetilde{g}(\widetilde{R}(X,Y)Z,W)=-\widetilde{g}(Z,\widetilde{R}^{\ast}(X,Y)W)$

where
$\displaystyle\widetilde{R}(X,Y)Z=\widetilde{\nabla}_{X}(\widetilde{\nabla}_{Y}Z)-\widetilde{\nabla}_{Y}(\widetilde{\nabla}_{X}Z)-\widetilde{\nabla}_{[X,Y]}Z$.
Therefore $\displaystyle\widetilde{R}$ vanishes identically if and only if so
is $\displaystyle\widetilde{R}^{\ast}$. We call flat if
$\displaystyle\widetilde{R}$ vanishes identically. If the curvature tensor
$\displaystyle\widetilde{R}$ with respect to the affine connection
$\displaystyle\widetilde{\nabla}$ satisfies

$\widetilde{R}(X,Y)Z=\widetilde{c}\left\\{\widetilde{g}(Y,Z)X-\widetilde{g}(X,Z)Y\right\\},$

then the statistical manifold
$\displaystyle(\widetilde{M},\widetilde{g},\widetilde{\nabla})$ is called a
space of constant curvature $\displaystyle\widetilde{c}$.

Suppose that $\displaystyle J$ and $\displaystyle J^{\ast}$ are almost complex
structures of type $\displaystyle(1,1)$ satisfying the condition

$\displaystyle\displaystyle\widetilde{g}\left(JX,Y\right)=-\widetilde{g}\left(X,J^{\ast}Y\right)$
(2.2)

for any tangent vector fields $\displaystyle X,Y\in\Gamma(T\widetilde{M})$.
Then $\displaystyle(\widetilde{M},\widetilde{g},J)$ is called an almost
Hermite-like manifold.

Let $\displaystyle F$ be a tensor field of type $\displaystyle(1,1)$ on
$\displaystyle\widetilde{M}$. If

$F^{2}=I,$ (2.3)

where $\displaystyle I$ denotes the identity map, then the pair
$\displaystyle(\widetilde{M},F)$ is called an almost product manifold with
almost product structure $\displaystyle F$. For any almost product manifold,
there exist two projections $\displaystyle P$ and $\displaystyle Q$ satisfying

$\displaystyle\displaystyle F=P-Q.$

If $\displaystyle F=I$ (resp. $\displaystyle F=-I$), we find $\displaystyle
P=I$, $\displaystyle Q=0$ (resp. $\displaystyle P=0$, $\displaystyle Q=I$). We
discuss the case of $\displaystyle F\neq\pm I$ in this paper. If an almost
product manifold admits a Riemannian metric $\displaystyle\widetilde{g}$ such
that the condition

$\displaystyle\displaystyle\widetilde{g}(FX,Y)=\widetilde{g}(X,FY)$

or, equivalently,

$\displaystyle\displaystyle\widetilde{g}(FX,FY)=\widetilde{g}(X,Y)$

is satisfied for any $\displaystyle X,Y\in\Gamma(T\widetilde{M})$, then
$\displaystyle(\widetilde{M},\widetilde{g},F)$ is called an almost product
Riemannian manifold [28].

Let $\displaystyle(\widetilde{M},\widetilde{g},F)$ be an almost product
Riemannian manifold and $\displaystyle\widetilde{\nabla}^{\circ}$ be the Levi-
Civita connection of $\displaystyle(\widetilde{M},\widetilde{g},F)$. In
particular, if $\displaystyle\widetilde{\nabla}^{{\circ}}F=0$, then
$\displaystyle(\widetilde{M},\widetilde{g},F)$ is called a locally product
Riemannian manifold [28].

For any hypersurface $\displaystyle(M,g)$ of
$\displaystyle(\widetilde{M},\widetilde{g},F)$, the Gauss and Weingarten
formulas are given as follows:

$\widetilde{\nabla}^{{\circ}}_{X}Y=\nabla^{{\circ}}_{X}Y+g(A^{{\circ}}_{N}X,Y)N,$
(2.4) $\widetilde{\nabla}^{{\circ}}_{X}N=-A^{{\circ}}_{N}X$ (2.5)

for any $\displaystyle X,Y\in\Gamma(TM)$, where $\displaystyle N$ is the unit
vector field, $\displaystyle\nabla^{{\circ}}$ is the induced linear connection
and $\displaystyle A^{{\circ}}_{N}$ is the shape operator on
$\displaystyle(M,g)$.

The hypersurface is called totally geodesic if $\displaystyle A^{{\circ}}_{N}$
vanishes identically, and it is called totally umbilical if there exists a
smooth function $\displaystyle\rho^{\circ}$ on $\displaystyle M$ such that

$A^{{\circ}}_{N}X=\rho^{\circ}X$ (2.6)

for any $\displaystyle X\in\Gamma(TM)$. Furthermore, $\displaystyle(M,g)$ is
called minimal if

$\textrm{Tr}\ A^{{\circ}}_{N}=0.$ (2.7)

For more details related to hypersurfaces and submanifolds, we refer to [9].

##  3 Almost Product-like Riemannian Manifolds

######  Definition 3.1.

If the Riemannian manifold $\displaystyle(\widetilde{M},\widetilde{g})$ with
the almost product structure $\displaystyle F$ which has another tensor field
$\displaystyle F^{\ast}$ of type $\displaystyle(1,1)$ satisfying

$\widetilde{g}(FX,Y)=\widetilde{g}(X,F^{\ast}Y)$ (3.1)

for any $\displaystyle X,Y\in\Gamma(T\widetilde{M})$, then the triple
$\displaystyle(\widetilde{M},\widetilde{g},F)$ is called an almost product-
like Riemannian manifold.

We see that $\displaystyle(F^{\ast})^{\ast}=F$,
$\displaystyle(F^{\ast})^{2}=I$ and

$\widetilde{g}(FX,F^{\ast}Y)=\widetilde{g}(X,Y).$ (3.2)

######  Lemma 3.2.

The triple $\displaystyle(\widetilde{M},\widetilde{g},F)$ is an almost
product-like Riemannian manifold if and only if so is
$\displaystyle(\widetilde{M},\widetilde{g},F^{\ast})$.

######  Example 3.3.

Let $\displaystyle\mathbb{R}^{4}$ be a smooth manifold with local coordinate
system $\displaystyle(x_{1},x_{2},x_{3},x_{4})$, which admits the following
almost product structure $\displaystyle F$

$F=\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\ 0&1&0&0\\\ \end{pmatrix}.$

Thus the pair $\displaystyle(\mathbb{R}^{4},F)$ is an almost product manifold.
If we put

$\widetilde{g}=\begin{pmatrix}1+e^{-x_{1}+x_{3}}&0&0&0\\\
0&e^{x_{1}-x_{3}}&0&0\\\ 0&0&e^{-x_{1}+x_{3}}&0\\\ 0&0&0&e^{x_{1}-x_{3}}\\\
\end{pmatrix}$

and

$F^{\ast}=\begin{pmatrix}0&0&(1+e^{x_{1}-x_{3}})^{-1}&0\\\ 0&0&0&1\\\
1+e^{x_{1}-x_{3}}&0&0&0\\\ 0&1&0&0\\\ \end{pmatrix},$

then the triple $\displaystyle(\mathbb{R}^{4},\widetilde{g},F)$ is an almost
product-like Riemannian manifold and so is
$\displaystyle(\mathbb{R}^{4},\widetilde{g},F^{\ast})$.

######  Definition 3.4.

If $\displaystyle F$ is parallel with respect to
$\displaystyle\widetilde{\nabla}$, then
$\displaystyle(\widetilde{M},\widetilde{g},\widetilde{\nabla},F)$ is called a
locally product-like statistical manifold.

From (3.1), we get

$\displaystyle\displaystyle\widetilde{g}\left((\widetilde{\nabla}_{X}F)Y,Z\right)=\widetilde{g}\left(Y,(\widetilde{\nabla}^{\ast}_{X}F^{\ast})Z\right).$

Hence we have

######  Lemma 3.5.

$\displaystyle(\widetilde{M},\widetilde{g},\widetilde{\nabla},F)$ is a locally
product-like statistical manifold if and only if so is
$\displaystyle(\widetilde{M},\widetilde{g},\widetilde{\nabla}^{\ast},F^{\ast})$.

######  Lemma 3.6.

Let $\displaystyle(\widetilde{M},\widetilde{g},\widetilde{\nabla},F)$ be a
locally product-like statistical manifold. If $\displaystyle\widetilde{M}$ is
of constant curvature $\displaystyle\widetilde{c}$, then
$\displaystyle\widetilde{c}=0$, that is, $\displaystyle\widetilde{M}$ is flat.

###### Proof.

From $\displaystyle\widetilde{R}(X,Y)FZ=F\widetilde{R}(X,Y)Z$, we find

$\widetilde{c}\left\\{\widetilde{g}(Y,FZ)X-\widetilde{g}(X,FZ)Y\right\\}=\widetilde{c}\left\\{\widetilde{g}(Y,Z)FX-\widetilde{g}(X,Z)FY\right\\}.$

If $\displaystyle\widetilde{c}\neq 0$, then we get $\displaystyle
mFX-(\text{tr}F)X=0$, where $\displaystyle m=\textrm{dim}\widetilde{M}$ and
$\displaystyle\textrm{tr}F=\sum\limits_{\alpha=1}^{m}\widetilde{g}(Fe_{\alpha},e_{\alpha})$
for local orthonormal basis $\displaystyle\\{e_{1},\ldots,e_{m}\\}$ of
$\displaystyle\widetilde{M}$. It is clear that $\displaystyle\text{tr}F=\mp
m$, which yields that $\displaystyle F=\mp I$ holds. Thus we obtain
$\displaystyle\widetilde{c}=0$. ∎

######  Example 3.7.

Let $\displaystyle(\mathbb{R}^{4},\widetilde{g},F)$ be an almost product-like
Riemannian manifold of Example 3.3. We put the affine connection
$\displaystyle\widetilde{\nabla}$ as follows:

$\displaystyle\displaystyle{\widetilde{\nabla}_{\partial_{1}}}{\partial_{1}}$
$\displaystyle\displaystyle=\widetilde{\nabla}_{\partial_{1}}\partial_{3}=\widetilde{\nabla}_{\partial_{3}}\partial_{1}=\widetilde{\nabla}_{\partial_{3}}\partial_{3}=e^{-x_{1}+x_{3}}(\partial_{1}+\partial_{3}),$
$\displaystyle\displaystyle{\widetilde{\nabla}_{\partial_{1}}}{\partial_{2}}$
$\displaystyle\displaystyle={\widetilde{\nabla}_{\partial_{2}}}{\partial_{1}}={\widetilde{\nabla}_{\partial_{3}}}{\partial_{4}}={\widetilde{\nabla}_{\partial_{4}}}{\partial_{3}}=-e^{-x_{1}+x_{3}}\partial_{2}-(1+e^{-x_{1}+x_{3}})\partial_{4},$
$\displaystyle\displaystyle{\widetilde{\nabla}_{\partial_{1}}}{\partial_{4}}$
$\displaystyle\displaystyle={\widetilde{\nabla}_{\partial_{4}}}{\partial_{1}}={\widetilde{\nabla}_{\partial_{2}}}{\partial_{3}}={\widetilde{\nabla}_{\partial_{3}}}{\partial_{2}}=-(1+e^{-x_{1}+x_{3}})\partial_{2}-e^{-x_{1}+x_{3}}\partial_{4},$
$\displaystyle\displaystyle{\widetilde{\nabla}_{\partial_{2}}}{\partial_{2}}$
$\displaystyle\displaystyle={\widetilde{\nabla}_{\partial_{4}}}{\partial_{4}}=-e^{x_{1}-x_{3}}\partial_{1}+\partial_{2}-e^{x_{1}-x_{3}}\partial_{3}-\partial_{4},$
$\displaystyle\displaystyle{\widetilde{\nabla}_{\partial_{2}}}{\partial_{4}}$
$\displaystyle\displaystyle={\widetilde{\nabla}_{\partial_{4}}}{\partial_{2}}=-e^{x_{1}-x_{3}}\partial_{1}-\partial_{2}-e^{x_{1}-x_{3}}\partial_{3}+\partial_{4}.$

Then we find

$\displaystyle\displaystyle{\widetilde{\nabla}^{\ast}_{\partial_{1}}}{\partial_{1}}$
$\displaystyle\displaystyle=-\frac{e^{-x_{1}+x_{3}}(2+e^{-x_{1}+x_{3}})}{1+e^{-x_{1}+x_{3}}}{\partial_{1}}-(1+e^{-x_{1}+x_{3}}){\partial_{3}},$
$\displaystyle\displaystyle{\widetilde{\nabla}^{\ast}_{\partial_{1}}}{\partial_{2}}$
$\displaystyle\displaystyle={\widetilde{\nabla}^{\ast}_{\partial_{2}}}{\partial_{1}}={\widetilde{\nabla}^{\ast}_{\partial_{1}}}{\partial_{4}}={\widetilde{\nabla}^{\ast}_{\partial_{4}}}{\partial_{1}}=(1+e^{-x_{1}+x_{3}})(\partial_{2}+\partial_{4}),$
$\displaystyle\displaystyle{\widetilde{\nabla}^{\ast}_{\partial_{1}}}{\partial_{3}}$
$\displaystyle\displaystyle={\widetilde{\nabla}^{\ast}_{\partial_{3}}}{\partial_{1}}=-\frac{e^{-2(x_{1}-x_{3})}}{1+e^{-x_{1}+x_{3}}}{\partial_{1}}-(1+e^{-x_{1}+x_{3}}){\partial_{3}},$
$\displaystyle\displaystyle{\widetilde{\nabla}^{\ast}_{\partial_{2}}}{\partial_{2}}$
$\displaystyle\displaystyle={\widetilde{\nabla}^{\ast}_{\partial_{4}}}{\partial_{4}}=\frac{1}{1+e^{-x_{1}+x_{3}}}{\partial_{1}}-{\partial_{2}}+\frac{1+e^{-x_{1}+x_{3}}}{e^{-2(x_{1}-x_{3})}}{\partial_{3}}+{\partial_{4}},$
$\displaystyle\displaystyle{\widetilde{\nabla}^{\ast}_{\partial_{2}}}{\partial_{3}}$
$\displaystyle\displaystyle={\widetilde{\nabla}^{\ast}_{\partial_{3}}}{\partial_{2}}={\widetilde{\nabla}^{\ast}_{\partial_{3}}}{\partial_{4}}={\widetilde{\nabla}^{\ast}_{\partial_{4}}}{\partial_{3}}=e^{-x_{1}+x_{3}}({\partial_{2}+{\partial_{4}}}),$
$\displaystyle\displaystyle{\widetilde{\nabla}^{\ast}_{\partial_{2}}}{\partial_{4}}$
$\displaystyle\displaystyle={\widetilde{\nabla}^{\ast}_{\partial_{4}}}{\partial_{2}}={e^{x_{1}-x_{3}}}{\partial_{1}}+{\partial_{2}}+{e^{x_{1}-x_{3}}}{\partial_{3}}-{\partial_{4}},$
$\displaystyle\displaystyle{\widetilde{\nabla}^{\ast}_{\partial_{3}}}{\partial_{3}}$
$\displaystyle\displaystyle=-\frac{e^{-2(x_{1}-x_{3})}}{1+e^{-x_{1}+x_{3}}}{\partial_{1}}+(1-e^{-x_{1}+x_{3}}){\partial_{3}}.$

Therefore $\displaystyle(\mathbb{R}^{4},\widetilde{g},F,\widetilde{\nabla})$
is a locally product-like statistical manifold and so is
$\displaystyle(\mathbb{R}^{4},\widetilde{g},F^{\ast},\widetilde{\nabla}^{\ast})$.

Next, let $\displaystyle(M,g)$ be a submanifold of the statistical manifold
$\displaystyle\widetilde{M}$. The Gauss and Weingarten formulas with respect
to affine connections $\displaystyle\widetilde{\nabla}$ and
$\displaystyle\widetilde{\nabla}^{\ast}$ are denoted by

$\displaystyle\displaystyle\widetilde{\nabla}_{X}Y$
$\displaystyle\displaystyle=\nabla_{X}Y+\sigma(X,Y),$
$\displaystyle\displaystyle\widetilde{\nabla}_{X}V$
$\displaystyle\displaystyle=-A_{V}X+D_{X}V,$
$\displaystyle\displaystyle\widetilde{\nabla}_{X}^{\ast}Y$
$\displaystyle\displaystyle=\nabla_{X}^{\ast}Y+\sigma^{\ast}(X,Y),$
$\displaystyle\displaystyle\widetilde{\nabla}_{X}^{\ast}V$
$\displaystyle\displaystyle=-A_{V}^{\ast}X+D_{X}^{\ast}V$

for any vector fields $\displaystyle X,Y$ tangent to $\displaystyle M$ and
$\displaystyle V$ normal to $\displaystyle M$, respectively. Because of
$\displaystyle\widetilde{\nabla}$ (resp.
$\displaystyle\widetilde{\nabla}^{\ast}$) is torsion-free, the affine
connection $\displaystyle\nabla$ (resp. $\displaystyle\nabla^{\ast}$) is
torsion-free and the second fundamental form $\displaystyle\sigma$ (resp.
$\displaystyle\sigma^{\ast}$) is symmetric. Affine connections
$\displaystyle\nabla$ and $\displaystyle\nabla^{\ast}$ are dual each other.
The triple $\displaystyle(M,g,\nabla)$ (resp.
$\displaystyle(M,g,\nabla^{\ast})$) is a statistical submanifold of
$\displaystyle\widetilde{M}$. Also, $\displaystyle D$ and $\displaystyle
D^{\ast}$ are dual affine connections each other. From
$\displaystyle(\widetilde{\nabla}^{\ast})^{\ast}=\widetilde{\nabla}$, we find
$\displaystyle(\nabla^{\ast})^{\ast}=\nabla$,
$\displaystyle(\sigma^{\ast})^{\ast}=\sigma$,
$\displaystyle(A^{\ast})^{\ast}=A$ and $\displaystyle(D^{\ast})^{\ast}=D$. If
$\displaystyle\sigma=0$ (resp. $\displaystyle\sigma^{\ast}=0$), then
$\displaystyle M$ is said to be totally geodesic with respect to
$\displaystyle\widetilde{\nabla}$ (resp.
$\displaystyle\widetilde{\nabla}^{\ast}$). H. Furuhata and I. Hasegawa [18]
studied statistical submanifolds.

######  Lemma 3.8.

For any vector fields $\displaystyle X,Y$ tangent to $\displaystyle M$ and
$\displaystyle V$ normal to $\displaystyle M$, we get

$\displaystyle\displaystyle\widetilde{g}(\sigma(X,Y),V)$
$\displaystyle\displaystyle=g(Y,A^{\ast}_{V}X),$
$\displaystyle\displaystyle\widetilde{g}(\sigma^{\ast}(X,Y),V)$
$\displaystyle\displaystyle=g(Y,A_{V}X).$

The second fundamental form $\displaystyle\sigma$ (resp.
$\displaystyle\sigma^{\ast}$) vanishes zero if and only if the shape operator
$\displaystyle A^{\ast}$ (resp. $\displaystyle A$) so is.

######  Corollary 3.9.

We find

$\displaystyle\displaystyle g(A^{\ast}_{V}X,Y)$
$\displaystyle\displaystyle=g(X,A^{\ast}_{V}Y),$ $\displaystyle\displaystyle
g(A_{V}X,Y)$ $\displaystyle\displaystyle=g(X,A_{V}Y).$

For any vector field $\displaystyle X$ tangent to $\displaystyle M$, we put
([28])

$\displaystyle\displaystyle FX=fX+hX,\quad\quad
F^{\ast}X=f^{\ast}X+h^{\ast}X,$

where $\displaystyle fX$ (resp. $\displaystyle f^{\ast}X$) is the tangent part
of $\displaystyle FX$ (resp. $\displaystyle F^{\ast}X$) and $\displaystyle hX$
(resp. $\displaystyle h^{\ast}X$) the normal part of $\displaystyle FX$ (resp.
$\displaystyle F^{\ast}X$). For any vector field $\displaystyle V$ normal to
$\displaystyle M$ we set

$\displaystyle\displaystyle FV=tV+sV,\quad\quad
F^{\ast}V=t^{\ast}V+s^{\ast}V,$

where $\displaystyle tV$ (resp. $\displaystyle t^{\ast}V$) is the tangent part
of $\displaystyle FV$ (resp. $\displaystyle F^{\ast}V$) and $\displaystyle sV$
(resp. $\displaystyle s^{\ast}V$) the normal part of $\displaystyle FV$ (resp.
$\displaystyle F^{\ast}V$). From $\displaystyle(F^{\ast})^{\ast}=F$,
$\displaystyle F^{2}=I$ and $\displaystyle(F^{\ast})^{2}=I$, we get

$(f^{\ast})^{\ast}=f,\quad\quad\quad(h^{\ast})^{\ast}=h,\quad\quad\quad(t^{\ast})^{\ast}=t,\quad\quad\quad(s^{\ast})^{\ast}=s,$
$f^{2}X=X-thX,\quad hfX+shX=0,\quad ftV+tsV=0,\quad s^{2}V=V-htV$

and

$\displaystyle\displaystyle(f^{\ast})^{2}X$
$\displaystyle\displaystyle=X-t^{\ast}h^{\ast}X,$ $\displaystyle\displaystyle
h^{\ast}f^{\ast}X+s^{\ast}h^{\ast}X$ $\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle f^{\ast}t^{\ast}V+t^{\ast}s^{\ast}V$
$\displaystyle\displaystyle=0,$ $\displaystyle\displaystyle(s^{\ast})^{2}V$
$\displaystyle\displaystyle=V-h^{\ast}t^{\ast}V.$

Because of $\displaystyle\widetilde{g}(FE,G)=\widetilde{g}(E,F^{\ast}G)$ for
any vector fields $\displaystyle E$ and $\displaystyle G$ on
$\displaystyle\widetilde{M}$, we find

$\displaystyle\displaystyle g(fX,Y)$
$\displaystyle\displaystyle=g(X,f^{\ast}Y),$ $\displaystyle\displaystyle
g(fX,f^{\ast}Y)$
$\displaystyle\displaystyle=g(X,Y)-\widetilde{g}(hX,h^{\ast}Y),$
$\displaystyle\displaystyle\widetilde{g}(hX,V)$
$\displaystyle\displaystyle=g(X,t^{\ast}V),$
$\displaystyle\displaystyle\widetilde{g}(h^{\ast}X,V)$
$\displaystyle\displaystyle=g(X,tV),$
$\displaystyle\displaystyle\widetilde{g}(sU,V)$
$\displaystyle\displaystyle=g(U,s^{\ast}V),$
$\displaystyle\displaystyle\widetilde{g}(sU,s^{\ast}V)$
$\displaystyle\displaystyle=\widetilde{g}(U,V)-g(tU,t^{\ast}V).$

Thus $\displaystyle f$ (resp. $\displaystyle s$) vanishes identically if and
only if $\displaystyle f^{\ast}$ (resp. $\displaystyle s^{\ast}$) so is, and
$\displaystyle h$ (resp. $\displaystyle t$) vanishes identically is and only
if $\displaystyle t^{\ast}$ (resp. $\displaystyle h^{\ast}$) so is. Hence we
have

######  Lemma 3.10.

We find for each $\displaystyle p\in M$

  1. (1)

$\displaystyle F\left(T_{p}M\right)\subset T_{p}M$ if and only if
$\displaystyle
F^{\ast}\left(T_{p}M\right)^{\bot}\subset\left(T_{p}M\right)^{\bot}$.

  2. (2)

$\displaystyle F\left(T_{p}M\right)\subset\left(T_{p}M\right)^{\bot}$ if and
only if $\displaystyle
F^{\ast}\left(T_{p}M\right)\subset\left(T_{p}M\right)^{\bot}$.

  3. (3)

$\displaystyle F\left(T_{p}M\right)^{\bot}\subset T_{p}M$ if and only if
$\displaystyle F^{\ast}\left(T_{p}M\right)^{\bot}\subset T_{p}M$.

  4. (4)

$\displaystyle F\left(T_{p}M\right)^{\bot}\subset\left(T_{p}M\right)^{\bot}$
if and only if $\displaystyle F^{\ast}\left(T_{p}M\right)\subset T_{p}M$.

If $\displaystyle F\left(T_{p}M\right)\subset T_{p}M$ (resp. $\displaystyle
F^{\ast}\left(T_{p}M\right)\subset T_{p}M$) for each $\displaystyle p\in M$,
then $\displaystyle M$ is said to be $\displaystyle F$-invariant (resp.
$\displaystyle F^{\ast}$-invariant) in $\displaystyle\widetilde{M}$. Then
$\displaystyle h$ and $\displaystyle t^{\ast}$ (resp. $\displaystyle h^{\ast}$
and $\displaystyle t$) vanish identically. because of $\displaystyle f^{2}=I$
and $\displaystyle g(fX,f^{\ast}Y)=g(X,Y)$, the triple $\displaystyle(M,g,f)$
is an almost product-like Riemannian manifold. Hence we have

######  Theorem 3.11.

Let $\displaystyle M$ be a submanifold of an almost product-like Riemannian
manifold $\displaystyle\widetilde{M}$. A necessary and sufficient condition
for $\displaystyle M$ to be $\displaystyle F$-invariant or $\displaystyle
F^{\ast}$-invariant is that the triple $\displaystyle(M,g,f)$ is an almost
product-like Riemannian manifold.

######  Example 3.12.

Let $\displaystyle(\mathbb{R}^{4},\widetilde{g},F)$ be an almost product-like
Riemannian manifold of Example 3.3. We define the immersion
$\displaystyle\widetilde{f}:M\rightarrow\mathbb{R}^{4}$ by
$\displaystyle\widetilde{f}(x_{1},x_{3})=(x_{1},x_{2},x_{3},x_{4})$. Then the
induced metric $\displaystyle g$ is given by

$\displaystyle\displaystyle g=\left(\begin{array}[]{cc}1+e^{-x_{1}+x_{3}}&0\\\
0&e^{-x_{1}+x_{3}}\\\ \end{array}\right).$

We find

$\displaystyle\displaystyle f=\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&0\\\
1&0&0&0\\\ 0&0&0&0\\\ \end{array}\right),\ \ h=O,\ \ \ t=O,\ \ \
s=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&1\\\ 0&0&0&0\\\ 0&1&0&0\\\
\end{array}\right)$

and

$\displaystyle\displaystyle
f^{\ast}=\left(\begin{array}[]{cccc}0&0&\left(1+e^{x_{1}-x_{3}}\right)^{-1}&0\\\
0&0&0&0\\\ 1+e^{x_{1}-x_{3}}&0&0&0\\\ 0&0&0&0\\\ \end{array}\right),\ \
h^{\ast}=O,\ $ $\displaystyle\displaystyle t^{\ast}=O,\
s^{\ast}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&1\\\ 0&0&0&0\\\
0&1&0&0\\\ \end{array}\right).$

Therefore the triple $\displaystyle(M,g,f)$ is $\displaystyle F$-invariant and
$\displaystyle F^{\ast}$-invariant of $\displaystyle\mathbb{R}^{4}$.

Moreover, we get

$\displaystyle\displaystyle g((\nabla_{Z}f)X,Y)$
$\displaystyle\displaystyle=g(X,(\nabla^{\ast}_{Z}f^{\ast})Y),$
$\displaystyle\displaystyle\widetilde{g}((D_{X}s)U,V)$
$\displaystyle\displaystyle=\widetilde{g}(U,(D^{\ast}_{X}s^{\ast})V).$

Thus we have

######  Lemma 3.13.

The structure $\displaystyle f$ (resp. $\displaystyle s$) is parallel with
respect to $\displaystyle\nabla$ (resp. $\displaystyle D$) if and only if
$\displaystyle f^{\ast}$ (resp. $\displaystyle s^{\ast}$) is parallel with
respect to $\displaystyle\nabla^{\ast}$ (resp. $\displaystyle D^{\ast}$).

######  Lemma 3.14.

In the locally product-like statistical manifold, we get

$\displaystyle\displaystyle(\nabla_{X}f)Y-A_{hY}X-t(\sigma(X,Y))$
$\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle(\overline{D}_{X}h)Y+\sigma(X,fY)-s(\sigma(X,Y))$
$\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle(\overline{\nabla}_{X}t)V-A_{sV}X+f(A_{V}X)$
$\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle(D_{X}s)V+\sigma(X,tV)+h(A_{V}X)$
$\displaystyle\displaystyle=0,$

where we put

$\displaystyle\displaystyle(\nabla_{X}f)Y$
$\displaystyle\displaystyle=\nabla_{X}(fY)-f(\nabla_{X}Y),$
$\displaystyle\displaystyle(\overline{D}_{X}h)Y$
$\displaystyle\displaystyle=D_{X}(hY)-h(\nabla_{X}Y),$
$\displaystyle\displaystyle(\overline{\nabla}_{X}t)V$
$\displaystyle\displaystyle=\nabla_{X}(tV)-t(D_{X}V),$
$\displaystyle\displaystyle(D_{X}s)V$
$\displaystyle\displaystyle=D_{X}(sV)-s(D_{X}V).$

###### Proof.

We find

$\displaystyle\displaystyle\left(\widetilde{\nabla}_{X}F\right)Y$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\left(\nabla_{X}f\right)Y-A_{hY}X-t\left(\sigma(X,Y)\right)+\left(\overline{D}_{X}h\right)Y$
$\displaystyle\displaystyle+\sigma(X,fY)-s(\sigma(X,Y)),$
$\displaystyle\displaystyle\left(\widetilde{\nabla}_{X}F\right)V$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\left(\overline{\nabla}_{X}t\right)V-A_{sV}X-f\left(A_{V}X\right)+\left(D_{X}s\right)V$
$\displaystyle\displaystyle+\sigma(X,tV)+h(A_{V}X).$

From $\displaystyle\widetilde{\nabla}F=0$, we have the results. ∎

######  Corollary 3.15.

In the locally product-like statistical manifold, we get

$\displaystyle\displaystyle\left(\nabla^{\ast}_{X}f^{\ast}\right)Y-A^{\ast}_{h^{\ast}Y}X-t^{\ast}\left(\sigma^{\ast}(X,Y)\right)$
$\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle\left(\overline{D}^{\ast}_{X}h^{\ast}\right)Y+\sigma^{\ast}(X,f^{\ast}Y)-s^{\ast}\left(\sigma^{\ast}(X,Y)\right)$
$\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle\left(\overline{\nabla}^{\ast}_{X}t^{\ast}\right)V-A^{\ast}_{s^{\ast}V}X+f^{\ast}(A^{\ast}_{V}X)$
$\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle\left(D^{\ast}_{X}s^{\ast}\right)V+\sigma^{\ast}(X,t^{\ast}V)+h^{\ast}(A^{\ast}_{V}X)$
$\displaystyle\displaystyle=0,$

where we put

$\displaystyle\displaystyle\left(\nabla^{\ast}_{X}f^{\ast}\right)Y$
$\displaystyle\displaystyle=\nabla^{\ast}_{X}(f^{\ast}Y)-f^{\ast}(\nabla^{\ast}_{X}Y),$
$\displaystyle\displaystyle\left(\overline{D}^{\ast}_{X}h^{\ast}\right)Y$
$\displaystyle\displaystyle=D^{\ast}_{X}(h^{\ast}Y)-h^{\ast}(\nabla^{\ast}_{X}Y),$
$\displaystyle\displaystyle\left(\overline{\nabla}^{\ast}_{X}t^{\ast}\right)V$
$\displaystyle\displaystyle=\nabla^{\ast}_{X}(t^{\ast}V)-t^{\ast}(D^{\ast}_{X}V),$
$\displaystyle\displaystyle\left(D^{\ast}_{X}s^{\ast}\right)V$
$\displaystyle\displaystyle=D^{\ast}_{X}(s^{\ast}V)-s^{\ast}(D^{\ast}_{X}V).$

######  Example 3.16.

$\displaystyle(\mathbb{R}^{4},\widetilde{g},F,\widetilde{\nabla})$ be a
locally product-like statistical manifold of Example 3.7 and
$\displaystyle(M,g)$ be a statistical submanifold of Example 3.12. If
$\displaystyle\partial_{1}$, $\displaystyle\partial_{3}$ is tangent to
$\displaystyle M$ and $\displaystyle\partial_{2}$, $\displaystyle\partial_{4}$
is normal to $\displaystyle M$, then we find

$\displaystyle\displaystyle\nabla_{\partial_{i}}\partial_{j}$
$\displaystyle\displaystyle=e^{-x_{1}+x_{3}}(\partial_{1}+\partial_{3}),\ \ \
\sigma(\partial_{i},\partial_{j})=0,\ \ (i,j\in\\{1,3\\}),$
$\displaystyle\displaystyle A_{\partial_{2}}\partial_{1}$
$\displaystyle\displaystyle=A_{\partial_{4}}\partial_{3}=0,\ \ \ \
D_{\partial_{1}}\partial_{2}=D_{\partial_{3}}\partial_{4}=-e^{-x_{1}+x_{3}}\partial_{2}-(1+e^{-x_{1}+x_{3}})\partial_{4},$
$\displaystyle\displaystyle A_{\partial_{2}}\partial_{3}$
$\displaystyle\displaystyle=A_{\partial_{4}}\partial_{1}=0,\ \ \ \
D_{\partial_{1}}\partial_{4}=D_{\partial_{3}}\partial_{2}=-(1+e^{-x_{1}+x_{3}})\partial_{2}-e^{-x_{1}+x_{3}}\partial_{4},$

and

$\displaystyle\displaystyle\nabla^{\ast}_{\partial_{1}}\partial_{1}$
$\displaystyle\displaystyle=-\tfrac{e^{-x_{1}+x_{3}}(2+e^{-x_{1}+x_{3}})}{1+e^{-x_{1}+x_{3}}}\partial_{1}-(1+e^{-x_{1}+x_{3}})\partial_{3},\
\sigma^{\ast}(\partial_{1},\partial_{1})=0,$
$\displaystyle\displaystyle\nabla^{\ast}_{\partial_{1}}\partial_{3}$
$\displaystyle\displaystyle=\nabla^{\ast}_{\partial_{3}}\partial_{1}=-\tfrac{e^{-2(x_{1}-x_{3})}}{1+e^{-x_{1}+x_{3}}}\partial_{1}-(1+e^{-x_{1}+x_{3}})\partial_{3},\
\sigma^{\ast}(\partial_{1},\partial_{3})=0,$
$\displaystyle\displaystyle\nabla^{\ast}_{\partial_{3}}\partial_{3}$
$\displaystyle\displaystyle=-\tfrac{e^{-2(x_{1}-x_{3})}}{1+e^{-x_{1}+x_{3}}}\partial_{1}+(1-e^{-x_{1}+x_{3}})\partial_{3},\
\ \ \ \ \ \ \ \ \ \ \ \ \sigma^{\ast}(\partial_{3},\partial_{3})=0,$
$\displaystyle\displaystyle A^{\ast}_{\partial_{2}}\partial_{1}$
$\displaystyle\displaystyle=A^{\ast}_{\partial_{4}}\partial_{1}=0,\ \ \ \
D^{\ast}_{\partial_{1}}\partial_{2}=D^{\ast}_{\partial_{1}}\partial_{4}=(1+e^{-x_{1}+x_{3}})(\partial_{2}+\partial_{4}),$
$\displaystyle\displaystyle A^{\ast}_{\partial_{2}}\partial_{3}$
$\displaystyle\displaystyle=A^{\ast}_{\partial_{4}}\partial_{3}=0,\ \ \ \
D^{\ast}_{\partial_{3}}\partial_{2}=D^{\ast}_{\partial_{3}}\partial_{4}=e^{-x_{1}+x_{3}}(\partial_{2}+\partial_{4}).$

Thus the triple $\displaystyle(M,g,\nabla)$ (resp.
$\displaystyle(M,g,\nabla^{\ast})$) is totally geodesic relative to
$\displaystyle\widetilde{\nabla}$ (resp.
$\displaystyle\widetilde{\nabla}^{\ast}$). Also, the structure $\displaystyle
f$ (resp. $\displaystyle f^{\ast}$) is parallel with respect to
$\displaystyle\nabla$ (resp. $\displaystyle\nabla^{\ast}$). Therefore
$\displaystyle(M,g,f,\nabla)$ is a locally product-like statistical manifold
and so is $\displaystyle(M,g,f^{\ast},\nabla^{\ast})$.

######  Example 3.17.

Let $\displaystyle(\mathbb{R}^{4},\widetilde{g},F,\widetilde{\nabla})$ be a
locally product-like statistical manifold of Example 3.7. We define the
immersion $\displaystyle\widetilde{f}:M\rightarrow\mathbb{R}^{4}$ by
$\displaystyle\widetilde{f}(x_{1},x_{2},x_{3})=(x_{1},x_{2},x_{3},x_{4})$.
Then the induced metric $\displaystyle g$ is given by

$g=\begin{pmatrix}1+e^{-x_{1}+x_{3}}&0&0\\\ 0&e^{x_{1}-x_{3}}&0\\\
0&0&e^{-x_{1}+x_{3}}\\\ \end{pmatrix}.$

If $\displaystyle\partial_{1},\partial_{2},\partial_{3}$ is tangent to
$\displaystyle M$ and $\displaystyle\partial_{4}$ is normal to $\displaystyle
M$, then we find

$\displaystyle\displaystyle{\nabla}_{\partial_{i}}{\partial_{j}}$
$\displaystyle\displaystyle=e^{-x_{1}+x_{3}}(\partial_{1}+\partial_{3}),$
$\displaystyle\displaystyle\sigma(\partial_{i},\partial_{j})$
$\displaystyle\displaystyle=0,\ \ \ i,j\in\left\\{1,3\right\\},$
$\displaystyle\displaystyle{\nabla}_{\partial_{1}}{\partial_{2}}$
$\displaystyle\displaystyle={\nabla}_{\partial_{2}}{\partial_{1}}=-e^{-x_{1}+x_{3}}\partial_{2},$
$\displaystyle\displaystyle\sigma(\partial_{1},\partial_{2})$
$\displaystyle\displaystyle=-(1+e^{-x_{1}+x_{3}})\partial_{4},$
$\displaystyle\displaystyle{\nabla}_{\partial_{2}}{\partial_{2}}$
$\displaystyle\displaystyle=-e^{x_{1}-x_{3}}\partial_{1}+\partial_{2}-e^{x_{1}-x_{3}}\partial_{3},$
$\displaystyle\displaystyle\sigma(\partial_{2},\partial_{2})$
$\displaystyle\displaystyle=-\partial_{4},$
$\displaystyle\displaystyle{\nabla}_{\partial_{2}}{\partial_{3}}$
$\displaystyle\displaystyle={\nabla}_{\partial_{3}}{\partial_{2}}=-(1+e^{-x_{1}+x_{3}})\partial_{2},$
$\displaystyle\displaystyle\sigma(\partial_{2},\partial_{3})$
$\displaystyle\displaystyle=-e^{-x_{1}+x_{3}}\partial_{4},$
$\displaystyle\displaystyle{A_{\partial_{4}}}\partial_{1}$
$\displaystyle\displaystyle=(1+e^{-x_{1}+x_{3}})\partial_{2},$
$\displaystyle\displaystyle{D_{\partial_{1}}}\partial_{4}$
$\displaystyle\displaystyle=e^{-x_{1}+x_{3}}\partial_{4},$
$\displaystyle\displaystyle{A_{\partial_{4}}}\partial_{2}$
$\displaystyle\displaystyle=e^{x_{1}-x_{3}}\partial_{1}+\partial_{2}+e^{x_{1}-x_{3}}\partial_{3},$
$\displaystyle\displaystyle{D_{\partial_{2}}}\partial_{4}$
$\displaystyle\displaystyle=\partial_{4},$
$\displaystyle\displaystyle{A_{\partial_{4}}}\partial_{3}$
$\displaystyle\displaystyle=e^{-x_{1}+x_{3}}\partial_{2},$
$\displaystyle\displaystyle{D_{\partial_{3}}}\partial_{4}$
$\displaystyle\displaystyle=-(1+e^{-x_{1}+x_{3}})\partial_{4}$

and

$\displaystyle\displaystyle{\nabla}^{\ast}_{\partial_{1}}{\partial_{1}}$
$\displaystyle\displaystyle=-\dfrac{e^{-x_{1}+x_{3}}(2+e^{-x_{1}+x_{3}})}{1+e^{-x_{1}+x_{3}}}{\partial_{1}}-(1+e^{-x_{1}+x_{3}}){\partial_{3}},\quad\sigma^{\ast}(\partial_{1},\partial_{1})=0,$
$\displaystyle\displaystyle{\nabla}^{\ast}_{\partial_{1}}{\partial_{2}}$
$\displaystyle\displaystyle={\nabla}^{\ast}_{\partial_{2}}{\partial_{1}}=(1+e^{-x_{1}+x_{3}})\partial_{2},\quad\sigma^{\ast}(\partial_{1},\partial_{2})=(1+e^{-x_{1}+x_{3}})\partial_{4},$
$\displaystyle\displaystyle{\nabla}^{\ast}_{\partial_{1}}{\partial_{3}}$
$\displaystyle\displaystyle={\nabla}^{\ast}_{\partial_{3}}{\partial_{1}}=-\frac{e^{-2(x_{1}-x_{3})}}{1+e^{-x_{1}+x_{3}}}\partial_{1}-(1+e^{-x_{1}+x_{3}})\partial_{3},\sigma^{\ast}(\partial_{1},\partial_{3})=0,$
$\displaystyle\displaystyle{\nabla}^{\ast}_{\partial_{2}}{\partial_{2}}$
$\displaystyle\displaystyle=\frac{1}{1+e^{-x_{1}+x_{3}}}\partial_{1}-\partial_{2}+\frac{1+e^{-x_{1}+x_{3}}}{e^{-2(x_{1}-x_{3})}}\partial_{3},\quad\sigma^{\ast}(\partial_{2},\partial_{2})=\partial_{4},$

$\displaystyle\displaystyle{\nabla}^{\ast}_{\partial_{2}}\partial_{3}$
$\displaystyle\displaystyle=\nabla^{\ast}_{\partial_{3}}\partial_{2}=e^{-x_{1}+x_{3}}\partial_{2},\quad\sigma^{\ast}(\partial_{2},\partial_{3})=e^{-x_{1}+x_{3}}\partial_{4},$
$\displaystyle\displaystyle{\nabla}^{\ast}_{\partial_{3}}{\partial_{3}}$
$\displaystyle\displaystyle=-\frac{e^{-2(x_{1}-x_{3})}}{1+e^{-x_{1}+x_{3}}}\partial_{1}+(1-e^{-x_{1}+x_{3}}){\partial_{3}},\quad\sigma^{\ast}(\partial_{3},\partial_{3})=0,$
$\displaystyle\displaystyle A^{\ast}_{\partial_{4}}\partial_{1}$
$\displaystyle\displaystyle=-(1+e^{-x_{1}+x_{3}})\partial_{2},\quad
D^{\ast}_{\partial_{1}}\partial_{4}=(1+e^{-x_{1}+x_{3}})\partial_{4},$
$\displaystyle\displaystyle A^{\ast}_{\partial_{4}}\partial_{2}$
$\displaystyle\displaystyle=-e^{x_{1}-x_{3}}\partial_{1}-\partial_{2}-e^{x_{1}-x_{3}}\partial_{3},\quad
D^{\ast}_{\partial_{2}}\partial_{4}=-\partial_{4},$
$\displaystyle\displaystyle A^{\ast}_{\partial_{4}}\partial_{3}$
$\displaystyle\displaystyle=-e^{-x_{1}+x_{3}}\partial_{2},\quad
D^{\ast}_{\partial_{3}}\partial_{4}=e^{-x_{1}+x_{3}}\partial_{4}.$

Thus the triple $\displaystyle(M,g,\nabla)$ is a statistical hypersurface of
$\displaystyle\mathbb{R}^{4}$ and so is $\displaystyle(M,g,\nabla^{\ast})$.

For any vector fields $\displaystyle X,Y,Z$ tangent to $\displaystyle M$, we
get

$\widetilde{R}(X,Y)Z=R(X,Y)Z-A_{\sigma(Y,Z)}X+A_{\sigma(X,Z)}Y+(D_{X}\sigma)(Y,Z)-(D_{Y}\sigma)(X,Z),$
$\displaystyle\displaystyle\widetilde{R}^{\ast}(X,Y)Z$
$\displaystyle\displaystyle=R^{\ast}(X,Y)Z-A^{\ast}_{\sigma^{\ast}(Y,Z)}X+A^{\ast}_{\sigma^{\ast}(X,Z)}Y+(D^{\ast}_{X}\sigma^{\ast})(Y,Z)$
$\displaystyle\displaystyle\quad-(D^{\ast}_{Y}\sigma^{\ast})(X,Z),$

where we put

$(D_{X}\sigma)(Y,Z)=D_{X}\\{\sigma(Y,Z)\\}-\sigma(\nabla_{X}Y,Z)-\sigma(Y,\nabla_{X}Z),$
$(D^{\ast}_{X}\sigma^{\ast})(Y,Z)=D^{\ast}_{X}\\{\sigma^{\ast}(Y,Z)\\}-\sigma^{\ast}(\nabla^{\ast}_{X}Y,Z)-\sigma^{\ast}(Y,\nabla^{\ast}_{X}Z).$

Hence we have the equation of Gauss relative to
$\displaystyle\widetilde{\nabla}$ and the equation of Codazzi relative to
$\displaystyle\widetilde{\nabla}$ [18].

######  Proposition 3.18.

For any vector field $\displaystyle W$ tangent to $\displaystyle M$, we get

$\displaystyle\displaystyle\widetilde{g}(\widetilde{R}(X,Y)Z,W)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
g(R(X,Y)Z,W)-\widetilde{g}(\sigma(Y,Z),\sigma^{\ast}(X,W))$
$\displaystyle\displaystyle+\widetilde{g}(\sigma(X,Z),\sigma^{\ast}(Y,W)),$
$\displaystyle\displaystyle(\widetilde{R}(X,Y)Z)^{\perp}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle(D_{X}\sigma)(Y,Z)-(D_{Y}\sigma)(X,Z).$

Also, we have the equation of Gauss relative to
$\displaystyle\widetilde{\nabla}^{\ast}$ and the equation of Codazzi relative
to $\displaystyle\widetilde{\nabla}^{\ast}$.

######  Proposition 3.19.

For any vector field $\displaystyle W$ tangent to $\displaystyle M$, we get

$\displaystyle\displaystyle\widetilde{g}(\widetilde{R}^{\ast}(X,Y)Z,W)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
g(R^{\ast}(X,Y)Z,W)-\widetilde{g}(\sigma^{\ast}(Y,Z),\sigma(X,W))$
$\displaystyle\displaystyle+\widetilde{g}(\sigma^{\ast}(X,Z),\sigma(Y,W)),$
$\displaystyle\displaystyle(\widetilde{R}^{\ast}(X,Y)Z)^{\perp}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle(D^{\ast}_{X}\sigma^{\ast})(Y,Z)-(D^{\ast}_{Y}\sigma^{\ast})(X,Z).$

For any vector fields $\displaystyle X,Y$ tangent to $\displaystyle M$ and
$\displaystyle V$ normal to $\displaystyle M$, we obtain

$\displaystyle\displaystyle\widetilde{R}(X,Y)V$
$\displaystyle\displaystyle=-(\nabla_{X}A)_{V}Y+(\nabla_{Y}A)_{V}X+R^{\perp}(X,Y)V-\sigma(X,A_{V}Y)$
$\displaystyle\displaystyle\quad+\sigma(Y,A_{V}X),$
$\displaystyle\displaystyle\widetilde{R}^{\ast}(X,Y)V$
$\displaystyle\displaystyle=-(\nabla^{\ast}_{X}A^{\ast})_{V}Y+(\nabla^{\ast}_{Y}A^{\ast})_{V}X+(R^{\ast})^{\perp}(X,Y)V-\sigma^{\ast}(X,A^{\ast}_{V}Y)$
$\displaystyle\displaystyle\quad+\sigma^{\ast}(Y,A^{\ast}_{V}X),$

where we put

$\displaystyle\displaystyle(\nabla_{X}A)_{V}Y$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\nabla_{X}(A_{V}Y)-A_{D_{X}V}Y-A_{V}(\nabla_{X}Y),$
$\displaystyle\displaystyle R^{\perp}(X,Y)V$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle D_{X}(D_{Y}V)-D_{Y}(D_{X}V)-D_{[X,Y]}V,$
$\displaystyle\displaystyle(\nabla^{\ast}_{X}A^{\ast})_{V}Y$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\nabla^{\ast}_{X}(A^{\ast}_{V}Y)-A^{\ast}_{D^{\ast}_{X}V}Y-A^{\ast}_{V}(\nabla^{\ast}_{X}Y),$
$\displaystyle\displaystyle(R^{\ast})^{\perp}(X,Y)V$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
D^{\ast}_{X}(D^{\ast}_{Y}V)-D^{\ast}_{Y}(D^{\ast}_{X}V)-D^{\ast}_{[X,Y]}V.$

Hence we have equation of Ricci relative to $\displaystyle\widetilde{\nabla}$.

######  Proposition 3.20.

For any vector field $\displaystyle U$ normal to $\displaystyle M$, we get

$\widetilde{g}(\widetilde{R}(X,Y)V,U)=\widetilde{g}(R^{\perp}(X,Y)V,U)+g([A^{\ast}_{U},A_{V}]X,Y).$

Also we have the equation Ricci relative to
$\displaystyle\widetilde{\nabla}^{\ast}$.

######  Proposition 3.21.

For any vector field $\displaystyle U$ normal to $\displaystyle M$, we get

$\widetilde{g}(\widetilde{R}^{\ast}(X,Y)V,U)=\widetilde{g}((R^{\ast})^{\perp}(X,Y)V,U)+g([A_{U},A^{\ast}_{V}]X,Y).$

From $\displaystyle\widetilde{R}(X,Y)FZ=F\widetilde{R}(X,Y)Z$ and
$\displaystyle\widetilde{R}(X,Y)FV=F\widetilde{R}(X,Y)V$, we find

######  Proposition 3.22.

Let $\displaystyle(M,g)$ be the submanifold of the locally product-like
statistical manifold $\displaystyle\widetilde{M}$. For any vector fields
$\displaystyle X,Y,Z$ tangent to $\displaystyle M$ and any vector field
$\displaystyle V$ normal to $\displaystyle M$, we have

$\displaystyle\displaystyle R(X,Y)fZ-
A_{\sigma(Y,fZ)}X+A_{\sigma(X,fZ)}Y-(\nabla_{X}A)_{hZ}Y+(\nabla_{Y}A)_{hZ}X$
$\displaystyle\displaystyle=f(R(X,Y)Z)-f(A_{\sigma(Y,Z)}X)+f(A_{\sigma(X,Z)}Y)+t\left\\{(D_{X}\sigma)(Y,Z)\right.$
$\displaystyle\displaystyle\left.\quad-(D_{Y}\sigma)(X,Z)\right\\},$
$\displaystyle\displaystyle
R^{\perp}(X,Y)hZ-\sigma(X,A_{hZ}Y)+\sigma(Y,A_{hZ}X)+(D_{X}\sigma)(Y,fZ)$
$\displaystyle\displaystyle\quad-(D_{Y}\sigma)(X,fZ)=h(R(X,Y)Z)-h(A_{\sigma(Y,Z)}X)+h(A_{\sigma(X,Z)}Y)$
$\displaystyle\displaystyle\quad+s\\{(D_{X}\sigma)(Y,Z)-(D_{Y}\sigma)(X,Z)\\},$
$\displaystyle\displaystyle R(X,Y)tV-
A_{\sigma(Y,tV)}X+A_{\sigma(X,tV)}Y-(\nabla_{X}A)_{sV}Y+(\nabla_{Y}A)_{sV}X$
$\displaystyle\displaystyle=-f((\nabla_{X}A)_{V}Y)+f((\nabla_{Y}A)_{V}X)+t(R^{\perp}(X,Y)V)-t(\sigma(X,A_{V}Y))$
$\displaystyle\displaystyle\quad+t(\sigma(Y,A_{V}X)),$
$\displaystyle\displaystyle
R^{\perp}(X,Y)sV-\sigma(X,A_{sV}Y)+\sigma(Y,A_{sV}X)+(D_{X}\sigma)(Y,tV)$
$\displaystyle\displaystyle-(D_{Y}\sigma)(X,tV)=s(R^{\perp}(X,Y)V)-s(\sigma(X,A_{V}Y))+s(\sigma(Y,A_{V}X))$
$\displaystyle\displaystyle-h((\nabla_{X}A)_{V}Y)+h((\nabla_{Y}A)_{V}X).$

######  Corollary 3.23.

Let $\displaystyle(M,g)$ be the submanifold of the locally product-like
statistical manifold $\displaystyle\widetilde{M}$. For any vector fields
$\displaystyle X,Y,Z$ tangent to $\displaystyle M$ and any vector field
$\displaystyle V$ normal to $\displaystyle M$, we have

$\displaystyle\displaystyle
R^{\ast}(X,Y)f^{\ast}Z-A^{\ast}_{\sigma^{\ast}(Y,f^{\ast}Z)}X+A^{\ast}_{\sigma^{\ast}(X,f^{\ast}Z)}Y-(\nabla^{\ast}_{X}A^{\ast})_{h^{\ast}Z}Y$
$\displaystyle\displaystyle+(\nabla^{\ast}_{Y}A^{\ast})_{h^{\ast}Z}X=f^{\ast}(R^{\ast}(X,Y)Z)-f^{\ast}(A^{\ast}_{\sigma^{\ast}(Y,Z)}X)+f^{\ast}(A^{\ast}_{\sigma^{\ast}(X,Z)}Y)$
$\displaystyle\displaystyle+t^{\ast}\\{(D^{\ast}_{X}\sigma^{\ast})(Y,Z)-(D^{\ast}_{Y}\sigma^{\ast})(X,Z)\\},$
$\displaystyle\displaystyle(R^{\ast})^{\perp}(X,Y)h^{\ast}Z-\sigma^{\ast}(X,A^{\ast}_{h^{\ast}Z}Y)+\sigma^{\ast}(Y,A^{\ast}_{h^{\ast}Z}X)+(D^{\ast}_{X}\sigma^{\ast})(Y,f^{\ast}Z)$
$\displaystyle\displaystyle-(D^{\ast}_{Y}\sigma^{\ast})(X,f^{\ast}Z)=h^{\ast}(R^{\ast}(X,Y)Z)-h^{\ast}(A^{\ast}_{\sigma^{\ast}(Y,Z)}X)+h^{\ast}(A^{\ast}_{\sigma^{\ast}(X,Z)}Y)$
$\displaystyle\displaystyle+s^{\ast}\\{(D^{\ast}_{X}\sigma^{\ast})(Y,Z)-(D^{\ast}_{Y}\sigma^{\ast})(X,Z)\\},$
$\displaystyle\displaystyle
R^{\ast}(X,Y)t^{\ast}V-A^{\ast}_{\sigma^{\ast}(Y,t^{\ast}V)}X+A^{\ast}_{\sigma^{\ast}(X,t^{\ast}V)}Y-(\nabla^{\ast}_{X}A^{\ast})_{s^{\ast}V}Y$
$\displaystyle\displaystyle+(\nabla^{\ast}_{Y}A^{\ast})_{s^{\ast}V}X=-f^{\ast}((\nabla^{\ast}_{X}A^{\ast})_{V}Y)+f^{\ast}((\nabla^{\ast}_{Y}A^{\ast})_{V}X)$
$\displaystyle\displaystyle+t^{\ast}((R^{\perp})^{\ast}(X,Y)V)-t^{\ast}(\sigma^{\ast}(X,A^{\ast}_{V}Y))+t^{\ast}(\sigma^{\ast}(Y,A^{\ast}_{V}X)),$
$\displaystyle\displaystyle(R^{\ast})^{\perp}(X,Y)s^{\ast}V-\sigma^{\ast}(X,A^{\ast}_{s^{\ast}V}Y)+\sigma^{\ast}(Y,A^{\ast}_{s^{\ast}V}X)+(D^{\ast}_{X}\sigma^{\ast})(Y,t^{\ast}V)$
$\displaystyle\displaystyle-(D^{\ast}_{Y}\sigma^{\ast})(X,t^{\ast}V)=s^{\ast}((R^{\ast})^{\perp}(X,Y),V)-s^{\ast}(\sigma^{\ast}(X,A^{\ast}_{V}Y))$
$\displaystyle\displaystyle+s^{\ast}(\sigma^{\ast}(Y,A^{\ast}_{V}X))-h^{\ast}((\nabla^{\ast}_{X}A^{\ast})_{V}Y)+h^{\ast}((\nabla^{\ast}_{Y}A^{\ast})_{V}X).$

Next, in the almost product-like statistical manifold
$\displaystyle(\widetilde{M},\widetilde{g},\widetilde{\nabla},F)$, we put for
any vector fields $\displaystyle X,Y,Z$ on $\displaystyle\widetilde{M}$

$\displaystyle\displaystyle\widetilde{R}(X,Y)Z$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\widetilde{c}[\widetilde{g}(Y,Z)X-\widetilde{g}(X,Z)Y+\widetilde{g}(Y,FZ)FX-\widetilde{g}(X,FZ)FY$
(3.7)
$\displaystyle\displaystyle+\left\\{\widetilde{g}(FX,Y)-\widetilde{g}(X,FY)\right\\}FZ],$

where $\displaystyle\widetilde{c}$ is a constant. Then the tensor
$\displaystyle\widetilde{R}$ is satisfied the Bianchi’s 1st and 2nd
identities, and $\displaystyle F\widetilde{R}(X,Y)=\widetilde{R}(X,Y)F$.
Moreover, we have

$\displaystyle\displaystyle\widetilde{R}^{\ast}(X,Y)Z$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\widetilde{c}[\widetilde{g}(Y,Z)X-\widetilde{g}(X,Z)Y+\widetilde{g}(Y,F^{\ast}Z)F^{\ast}X-\widetilde{g}(X,F^{\ast}Z)F^{\ast}Y$
(3.8)
$\displaystyle\displaystyle+\left\\{\widetilde{g}(F^{\ast}X,Y)-\widetilde{g}(X,F^{\ast}Y)\right\\}F^{\ast}Z].$

We discuss the statistical submanifold $\displaystyle M$ of the almost
product-like statistical manifold
$\displaystyle(\widetilde{M},\widetilde{g},\widetilde{\nabla},F)$ satisfying
the condition (3.7). For any vector fields $\displaystyle X,Y,Z$ tangent to
$\displaystyle M$ and $\displaystyle V$ normal to $\displaystyle M$, we get

$\displaystyle\displaystyle
R(X,Y)Z=\widetilde{c}[g(Y,Z)X-g(X,Z)Y+g(Y,fZ)fX-g(X,fZ)fY$
$\displaystyle\displaystyle+\left\\{\
g(fX,Y)-g(X,fY)\right\\}fZ]+A_{\sigma(Y,Z)}X-A_{\sigma(X,Z)}Y,$
$\displaystyle\displaystyle(D_{X}\sigma)(Y,Z)-(D_{Y}\sigma)(X,Z)=\widetilde{c}[g(Y,fZ)hX-g(X,fZ)hY$
$\displaystyle\displaystyle+\left\\{\ g(fX,Y)-g(X,fY)\right\\}hZ],$
$\displaystyle\displaystyle(\nabla_{X}A)_{V}Y-(\nabla_{Y}A)_{V}X=\widetilde{c}[-g(Y,tV)fX+g(X,tV)fY-$
$\displaystyle\displaystyle\left\\{\ g(fX,Y)-g(X,fY)\right\\}tV],$
$\displaystyle\displaystyle
R^{\perp}(X,Y)V\sigma(X,A_{V}Y)+\sigma(Y,A_{V}X)=\widetilde{c}[g(Y,tV)hX-g(X,tV)hY$
$\displaystyle\displaystyle+\left\\{\ g(fX,Y)-g(X,fY)\right\\}sV].$

If the second fundamental form $\displaystyle\sigma$ is parallel with respect
to $\displaystyle D$, then we find from (3.7)

$\displaystyle\displaystyle\widetilde{c}[g(Y,fZ)hX-g(X,fZ)hY+\left\\{\
g(fX,Y)-g(X,fY)\right\\}hZ]=0.$

From this equation, we get $\displaystyle\widetilde{c}=0$ or

$\displaystyle\displaystyle h[g(Y,fZ)X-g(X,fZ)Y+\left\\{\
g(fX,Y)-g(X,fY)\right\\}Z]=0,$

We assume $\displaystyle g(Y,fZ)X-g(X,fZ)Y+\left\\{\
g(fX,Y)-g(X,fY)\right\\}Z=0$, which implies that $\displaystyle f=0$ if
$\displaystyle\text{dim}M>2$.

######  Theorem 3.24.

Let $\displaystyle(M,g)$ be the statistical submanifold of the locally
product-like statistical manifold $\displaystyle\widetilde{M}$ satisfying the
condition (3.7). If the second fundamental form $\displaystyle\sigma$ is
parallel with respect to $\displaystyle D$, then we get

  * (1)

$\displaystyle\widetilde{c}=0$, that is, $\displaystyle\widetilde{M}$ is flat,
or

  * (2)

$\displaystyle M$ is $\displaystyle F$-invariant of
$\displaystyle\widetilde{M}$, or

  * (3)

$\displaystyle M$ is $\displaystyle F$-anti-invariant.

######  Corollary 3.25.

Let $\displaystyle(M,g)$ be the statistical submanifold of the locally
product-like statistical manifold $\displaystyle\widetilde{M}$ satisfying the
condition (3.7). If $\displaystyle M$ is totally geodesic with respect to
$\displaystyle\widetilde{\nabla}$, then we get

  * (1)

$\displaystyle\widetilde{c}=0$, that is, $\displaystyle\widetilde{M}$ is flat,
or

  * (2)

$\displaystyle M$ is $\displaystyle F$-invariant of
$\displaystyle\widetilde{M}$, or

  * (3)

$\displaystyle M$ is $\displaystyle F$-anti-invariant, which is of constant
curvature $\displaystyle\widetilde{c}$.

######  Corollary 3.26.

Let $\displaystyle(M,g)$ be the statistical submanifold of the locally
product-like statistical manifold $\displaystyle\widetilde{M}$ satisfying the
condition (3.7). If the second fundamental form $\displaystyle\sigma^{\ast}$
is parallel with respect to $\displaystyle D^{\ast}$, then we get

  * (1)

$\displaystyle\widetilde{c}=0$, that is, $\displaystyle\widetilde{M}$ is flat,
or

  * (2)

$\displaystyle M$ is $\displaystyle F^{\ast}$-invariant of
$\displaystyle\widetilde{M}$, or

  * (3)

$\displaystyle M$ is $\displaystyle F^{\ast}$-anti-invariant.

######  Corollary 3.27.

Let $\displaystyle(M,g)$ be the statistical submanifold of the locally
product-like statistical manifold $\displaystyle\widetilde{M}$ satisfying the
condition (3.7). If $\displaystyle M$ is totally geodesic with respect to
$\displaystyle\widetilde{\nabla}^{\ast}$, then we get

  * (1)

$\displaystyle\widetilde{c}=0$, that is, $\displaystyle\widetilde{M}$ is flat,
or

  * (2)

$\displaystyle M$ is $\displaystyle F^{\ast}$-invariant of
$\displaystyle\widetilde{M}$, or

  * (3)

$\displaystyle M$ is $\displaystyle F^{\ast}$-anti-invariant, which is of
constant curvature $\displaystyle\widetilde{c}$.

##  4 Hypersurfaces of almost product-like Riemannian manifolds

Let $\displaystyle(\widetilde{M},\widetilde{g},F)$ be an almost product-like
Riemannian manifold and $\displaystyle(M,g)$ be a hypersurface of
$\displaystyle\widetilde{M}$. If $\displaystyle N$ is the unit normal vector
field of $\displaystyle M$, then we put

$\displaystyle\displaystyle FN=\xi+\mu_{1}N$

and

$\displaystyle\displaystyle F^{\ast}N=\xi^{\ast}+\mu_{2}N,$

where $\displaystyle\xi$ and $\displaystyle\xi^{\ast}$ are tangential parts of
$\displaystyle FN$ and $\displaystyle F^{\ast}N$, respectively. Here,
$\displaystyle\mu_{1}$ and $\displaystyle\mu_{2}$ are smooth functions on
$\displaystyle\widetilde{M}$. Considering (3.1) in the last two equations, we
obtain $\displaystyle\mu_{1}=\mu_{2}$. Thus, we can write

$FN=\xi+\mu N$ (4.1)

and

$F^{\ast}N=\xi^{\ast}+\mu N,$ (4.2)

where $\displaystyle\mu$ is a smooth function on $\displaystyle\widetilde{M}$.
Using (4.1) and (4.2) in (3.2), we derive

$1-\mu^{2}=g(\xi,\xi^{\ast}).$ (4.3)

Based on (4.3), we obtain the following lemma:

######  Lemma 4.1.

Let $\displaystyle(M,g)$ be a hypersurface of $\displaystyle\widetilde{M}$.
Then we have the following statements:

  * i)

If $\displaystyle\mu=1$ then $\displaystyle\xi$ and $\displaystyle\xi^{\ast}$
are orthogonal.

  * ii)

If $\displaystyle\mu=0$ then $\displaystyle g(\xi,\xi^{\ast})=1$.

######  Definition 4.2.

Let $\displaystyle(M,g)$ be a hypersurface of $\displaystyle\widetilde{M}$. If
$\displaystyle FN$ and $\displaystyle F^{\ast}N$ lie on
$\displaystyle\Gamma(TM)$, then $\displaystyle(M,g)$ is called a tangential
hypersurface.

Now, let $\displaystyle(M,g)$ be a tangential hypersurface of
$\displaystyle\widetilde{M}$. Then we find

$FN=\xi\quad and\quad F^{\ast}N=\xi^{\ast}.$ (4.4)

From Lemma 4.1, we note that $\displaystyle\xi$ is not perpendicular to
$\displaystyle\xi^{\ast}$ and $\displaystyle(\xi^{\ast})^{\ast}=\xi$. For any
$\displaystyle X\in\Gamma(TM)$, we write

$FX=\varphi X+\eta^{\ast}(X)N$ (4.5)

and

$F^{\ast}X=\varphi^{\ast}X+\eta(X)N,$ (4.6)

where $\displaystyle\varphi X$, $\displaystyle\varphi^{\ast}X\in\Gamma(TM)$,
$\displaystyle\eta$ and $\displaystyle\eta^{\ast}$ are $\displaystyle 1-$
forms on $\displaystyle M$. Then we get

$\eta^{\ast}(X)=g(X,\xi^{\ast}),\quad\eta(X)=g(X,\xi)$ (4.7)

and $\displaystyle(\eta^{\ast})^{\ast}=\eta$. Moreover, it is easy to see from
Lemma 4.1 that $\displaystyle\eta(\xi^{\ast})=1$ and
$\displaystyle\eta^{\ast}(\xi)=1$ hold.

######  Lemma 4.3.

For any tangential hypersurface, we have the following relations:

$\varphi^{2}X=X-\eta^{\ast}(X)\xi,$ (4.8) $\eta^{\ast}(\varphi X)=0,$ (4.9)
$(\varphi^{\ast})^{2}X=X-\eta(X)\xi^{\ast},$ (4.10) $\eta(\varphi^{\ast}X)=0,$
(4.11) $\varphi\xi=0\quad and\quad\varphi^{\ast}\xi^{\ast}=0.$ (4.12)

######  Lemma 4.4.

Let $\displaystyle(M,g)$ be tangential hypersurface of
$\displaystyle\widetilde{M}$. Then we have the following relations for any
$\displaystyle X,Y\in\Gamma(TM)$:

$\displaystyle\displaystyle g(\varphi X,Y)$
$\displaystyle\displaystyle=g(X,\varphi^{\ast}Y),$ $\displaystyle\displaystyle
g(\varphi X,\varphi^{\ast}Y)$
$\displaystyle\displaystyle=g(X,Y)-\eta^{\ast}(X)\eta(Y).$

Moreover, $\displaystyle(\varphi^{\ast})^{\ast}=\varphi$ holds.

######  Example 4.5.

Let $\displaystyle(\mathbb{R}^{4},\widetilde{g},F)$ be an almost product-like
Riemannian manifold of Example 3.3 and $\displaystyle\widetilde{f}$ be an
immersion of Example 3.17. If we set

$N=e^{-\frac{1}{2}(x_{1}-x_{3})}\partial_{4},$

then $\displaystyle N$ is an unit normal to $\displaystyle\mathbb{R}^{4}$ and
$\displaystyle\xi=e^{-\frac{1}{2}(x_{1}-x_{3})}\partial_{2}$. We obtain

$\displaystyle\displaystyle f=\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&0\\\
1&0&0&0\\\ 0&0&0&0\\\ \end{array}\right),\
h=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&1&0&0\\\
\end{array}\right),\ $ $\displaystyle\displaystyle
t=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&1\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{array}\right),\ s=O$

and

$\displaystyle\displaystyle
f^{\ast}=\left(\begin{array}[]{cccc}0&0&(1+e^{x_{1}-x_{3}})^{-1}&0\\\
0&0&0&0\\\ 1+e^{x_{1}-x_{3}}&0&0&0\\\ 0&0&0&0\\\ \end{array}\right),\ \
h^{\ast}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&1&0&0\\\ \end{array}\right),$ $\displaystyle\displaystyle
t^{\ast}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&1\\\ 0&0&0&0\\\
0&0&0&0\\\ \end{array}\right),\ \ s^{\ast}=O,$

which implies from $\displaystyle s=O$ that the pair $\displaystyle(M,g)$ is a
tangential hypersurface. Moreover, we find $\displaystyle\varphi=f$,
$\displaystyle\eta^{\ast}(\partial_{1})=0$,
$\displaystyle\eta^{\ast}(\partial_{2})=e^{\frac{1}{2}(x_{1}-x_{3})}$ and
$\displaystyle\eta^{\ast}(\partial_{3})=0$.

##  5 Tangential hypersurfaces of locally product-like statistical manifolds

Let $\displaystyle(M,g)$ be a tangential hypersurface of an almost product-
like statistical manifold $\displaystyle\widetilde{M}$. The Gauss and
Weingarten formulas with respect to $\displaystyle\widetilde{\nabla}$ and
$\displaystyle\widetilde{\nabla}^{\ast}$ are given by

$\displaystyle\displaystyle\widetilde{\nabla}_{X}Y$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\nabla_{X}Y+\
^{{}^{\prime}}\sigma(X,Y)N,$
$\displaystyle\displaystyle\widetilde{\nabla}_{X}N$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle-A_{N}X+\kappa(X)N,$
$\displaystyle\displaystyle\widetilde{\nabla}^{\ast}_{X}Y$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\nabla^{\ast}_{X}Y+\
^{{}^{\prime}}\sigma^{\ast}(X,Y)N,$
$\displaystyle\displaystyle\widetilde{\nabla}^{\ast}_{X}N$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-A^{\ast}_{N}X+\kappa^{\ast}(X)N,$

where $\displaystyle\kappa$ and $\displaystyle\kappa^{\ast}$ are
$\displaystyle 1-$forms. Then we find
${}^{{}^{\prime}}\sigma(X,Y)=g(A_{N}^{\ast}X,Y)$, ${}^{{}^{\prime}}\\\
\sigma^{\ast}(X,Y)=g(A_{N}X,Y)$, $\displaystyle\kappa(X)+\kappa^{\ast}(X)=0$.
Thus, we get

$\widetilde{\nabla}_{X}Y=\nabla_{X}Y+g(A^{\ast}_{N}X,Y)N,$ (5.1)
$\widetilde{\nabla}_{X}N=-A_{N}X+\kappa(X)N,$ (5.2)
$\widetilde{\nabla}^{\ast}_{X}Y=\nabla^{\ast}_{X}Y+g(A_{N}X,Y)N,$ (5.3)
$\widetilde{\nabla}^{\ast}_{X}N=-A^{\ast}_{N}X-\kappa(X)N.$ (5.4)

######  Proposition 5.1.

Let $\displaystyle(M,g)$ be a tangential hypersurface of a locally product-
like statistical manifold $\displaystyle\widetilde{M}$. Then we have the
following relations:

$\nabla_{X}\xi=-\varphi(A_{N}X)+\kappa(X)\xi,$ (5.5)
$\nabla^{\ast}_{X}\xi^{\ast}=-\varphi^{\ast}(A^{\ast}_{N}X)-\kappa(X)\xi^{\ast},$
(5.6)
$\eta\left(A_{N}^{\ast}X\right)+\eta^{\ast}\left(A_{N}X\right)=0,\quad\quad
A_{N}\xi^{\ast}+A_{N}^{\ast}\xi=0,$ (5.7)
$\kappa(X)=\eta^{\ast}(\nabla_{X}\xi)=-\eta(\nabla^{\ast}_{X}\xi^{\ast}).$
(5.8)

###### Proof.

Using $\displaystyle FN=\xi$, $\displaystyle F^{\ast}N=\xi^{\ast}$, (4.5) and
(5.3), we can write

$\displaystyle\displaystyle(\widetilde{\nabla}_{X}F)N$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\nabla_{X}\xi+\varphi(A_{N}X)-\kappa(X)\xi+\\{\eta(A_{N}^{\ast}X)+\eta^{\ast}(A_{N}X)\\}N,$
$\displaystyle\displaystyle(\widetilde{\nabla}^{\ast}_{X}F^{\ast})N$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\nabla^{\ast}_{X}\xi^{\ast}+\varphi^{\ast}(A^{\ast}_{N}X)-\kappa(X)\xi^{\ast}+\\{\eta(A_{N}^{\ast}X)+\eta^{\ast}(A_{N}X)\\}N,$

which implies from
$\displaystyle\widetilde{\nabla}F=\widetilde{\nabla}^{\ast}F^{\ast}=0$ that
the proof is completed. ∎

######  Proposition 5.2.

Let $\displaystyle(M,g)$ be a tangential hypersurface of a locally product-
like statistical manifold $\displaystyle\widetilde{M}$. Then we have

$(\nabla_{X}\varphi)Y=g(A^{\ast}_{N}X,Y)\xi+\eta^{\ast}(Y)A_{N}X,$ (5.9)
$(\nabla^{\ast}_{X}\varphi^{\ast})Y=g(A_{N}X,Y)\xi^{\ast}+\eta(Y)A^{\ast}_{N}X.$
(5.10)

###### Proof.

Using (5.1) and (5.3), we get

$\displaystyle\displaystyle(\widetilde{\nabla}_{X}F)Y=(\nabla_{X}\varphi)Y-g(A_{N}^{\ast}X,Y)\xi-\eta^{\ast}(Y)A_{N}X,$
$\displaystyle\displaystyle(\widetilde{\nabla}^{\ast}_{X}F^{\ast})Y=(\nabla^{\ast}_{X}\varphi^{\ast})Y-g(A_{N}X,Y)\xi^{\ast}-\eta(Y)A^{\ast}_{N}X,$

which yields from
$\displaystyle\widetilde{\nabla}F=\widetilde{\nabla}^{\ast}F^{\ast}=0$ that
$\displaystyle(\ref{eq40})$ and $\displaystyle(\ref{eq42})$ hold. ∎

######  Example 5.3.

Let $\displaystyle(\mathbb{R}^{4},\widetilde{g},F)$ be a locally product-like
statistical manifold of Example 3.7 and $\displaystyle(M,g)$ be a tangential
hypersurface of Example 4.5. By straightforward calculate we find

$\displaystyle\displaystyle\nabla_{\partial_{1}}\xi=\nabla_{\partial_{3}}\xi=-\frac{1}{2}(1+2e^{-x_{1}+x_{3}})\xi,\
\
\nabla_{\partial_{2}}\xi=-e^{\frac{1}{2}(x_{1}-x_{3})}(\partial_{2}+\partial_{3})+\xi$

and

$\displaystyle\displaystyle(\nabla_{\partial_{1}}\varphi)\partial_{1}$
$\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle(\nabla_{\partial_{2}}\varphi)\partial_{1}$
$\displaystyle\displaystyle=-2\cosh\frac{1}{2}(x_{1}-x_{3})\xi,$
$\displaystyle\displaystyle(\nabla_{\partial_{3}}\varphi)\partial_{1}$
$\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle(\nabla_{\partial_{1}}\varphi)\partial_{2}$
$\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle(\nabla_{\partial_{2}}\varphi)\partial_{2}$
$\displaystyle\displaystyle=e^{x_{1}-x_{3}}(\partial_{1}+\partial_{3}),$
$\displaystyle\displaystyle(\nabla_{\partial_{3}}\varphi)\partial_{2}$
$\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle(\nabla_{\partial_{1}}\varphi)\partial_{3}$
$\displaystyle\displaystyle=0,$
$\displaystyle\displaystyle(\nabla_{\partial_{2}}\varphi)\partial_{3}$
$\displaystyle\displaystyle=-e^{-\frac{1}{2}(x_{1}-x_{3})}\xi,$
$\displaystyle\displaystyle(\nabla_{\partial_{3}}\varphi)\partial_{3}$
$\displaystyle\displaystyle=0.$

Also, we get

$\displaystyle\displaystyle A_{N}\partial_{1}$
$\displaystyle\displaystyle=-A^{\ast}_{N}\partial_{1}=(1+e^{-x_{1}+x_{3}})\xi,$
$\displaystyle\displaystyle\kappa(\partial_{1})$
$\displaystyle\displaystyle=-\frac{1}{2}(1+2e^{-x_{1}+x_{3}}),$
$\displaystyle\displaystyle A_{N}\partial_{2}$
$\displaystyle\displaystyle=-A^{\ast}_{N}\partial_{2}=e^{\frac{1}{2}(x_{1}-x_{3})}(\partial_{1}+\partial_{3})+\xi,$
$\displaystyle\displaystyle\kappa(\partial_{2})$
$\displaystyle\displaystyle=1,$ $\displaystyle\displaystyle A_{N}\partial_{3}$
$\displaystyle\displaystyle=-A^{\ast}_{N}\partial_{3}=e^{-x_{1}+x_{3}}\xi,$
$\displaystyle\displaystyle\kappa(\partial_{3})$
$\displaystyle\displaystyle=-\frac{1}{2}(1+2e^{x_{1}-x_{3}}),$

which yields that Proposition 5.1 and Proposition 5.2 hold.

From Proposition 3.22 and Proposition 5.1, we have

######  Proposition 5.4.

Let $\displaystyle(M,g)$ be a tangential hypersurface of a locally product-
like statistical manifold $\displaystyle\widetilde{M}$. Then we obtain

$\displaystyle\displaystyle R(X,Y)\xi$
$\displaystyle\displaystyle=-\varphi\left((\overline{\nabla}_{X}A)_{N}Y\right)+\varphi\left((\overline{\nabla}_{Y}A)_{N}X\right)-\eta^{\ast}(A_{N}Y)A_{N}X$
$\displaystyle\displaystyle\quad+\eta^{\ast}(A_{N}X)A_{N}Y-\left\\{g([A_{N},A^{\ast}_{N}]X,Y)-(d\kappa)(X,Y)\right\\}\xi,$
$\displaystyle\displaystyle R^{\ast}(X,Y)\xi^{\ast}$
$\displaystyle\displaystyle=-\varphi^{\ast}\left((\overline{\nabla}^{\ast}_{X}A^{\ast})_{N}Y\right)+\varphi^{\ast}\left((\overline{\nabla}^{\ast}_{Y}A^{\ast})_{N}X\right)-\eta(A^{\ast}_{N}Y)A^{\ast}_{N}X$
$\displaystyle\displaystyle\quad+\eta(A^{\ast}_{N}X)A^{\ast}_{N}Y+\left\\{g([A_{N},A^{\ast}_{N}]X,Y)-(d\kappa)(X,Y)\right\\}\xi^{\ast},$

where we put

$\displaystyle\displaystyle(\overline{\nabla}_{X}A)_{N}Y$
$\displaystyle\displaystyle=\nabla_{X}(A_{N}Y)-A_{\widetilde{\nabla}_{X}N}Y-A_{N}(\nabla_{X}Y),$
$\displaystyle\displaystyle(\nabla_{X}\kappa)(Y)$
$\displaystyle\displaystyle=X\left\\{\kappa(Y)\right\\}-\kappa(\nabla_{X}Y).$

######  Proposition 5.5.

Let $\displaystyle(M,g)$ be a tangential hypersurface of a locally product-
like statistical manifold $\displaystyle\widetilde{M}$. Then we obtain

$\displaystyle\displaystyle\widetilde{R}(X,Y)Z$
$\displaystyle\displaystyle=R(X,Y)Z-g(A^{\ast}_{N}Y,Z)A_{N}X+g(A^{\ast}_{N}X,Z)A_{N}Y$
$\displaystyle\displaystyle\quad+\\{g((\overline{\nabla}^{\ast}_{X}A^{\ast})_{N}Y,Z)-g((\overline{\nabla}^{\ast}_{Y}A^{\ast})_{N}X,Z)\\}N,$
$\displaystyle\displaystyle\widetilde{R}(X,Y)N$
$\displaystyle\displaystyle=-(\overline{\nabla}_{X}A)_{N}Y+(\overline{\nabla}_{Y}A)_{N}X-\left\\{g([A_{N},A^{\ast}_{N}]X,Y)\right.$
$\displaystyle\displaystyle\quad\left.-(d\kappa)(X,Y)\right\\}N,$
$\displaystyle\displaystyle\widetilde{R}^{\ast}(X,Y)Z$
$\displaystyle\displaystyle=R^{\ast}(X,Y)Z-g(A_{N}Y,Z)A^{\ast}_{N}X+g(A_{N}X,Z)A^{\ast}_{N}Y$
$\displaystyle\displaystyle\quad+\\{g((\overline{\nabla}_{X}A)_{N}Y,Z)-g((\overline{\nabla}_{Y}A)_{N}X,Z)\\}N,$
$\displaystyle\displaystyle\widetilde{R}^{\ast}(X,Y)N$
$\displaystyle\displaystyle=-(\overline{\nabla}^{\ast}_{X}A^{\ast})_{N}Y+(\overline{\nabla}^{\ast}_{Y}A^{\ast})_{N}X$
$\displaystyle\displaystyle\quad+\left\\{g([A_{N},A^{\ast}_{N}]X,Y)-(d\kappa)(X,Y)\right\\}N.$

From $\displaystyle\widetilde{R}(X,Y)FZ=F(\widetilde{R}(X,Y)Z)$ and
$\displaystyle\widetilde{R}^{\ast}(X,Y)F^{\ast}Z=F^{\ast}(\widetilde{R}^{\ast}(X,Y)Z)$,
we get

######  Proposition 5.6.

Let $\displaystyle(M,g)$ be a tangential hypersurface of a locally product-
like statistical manifold $\displaystyle\widetilde{M}$. Then we obtain

$\displaystyle\displaystyle R(X,Y)\varphi Z+g(A^{\ast}_{N}X,\varphi
Z)A_{N}Y-g(A^{\ast}_{N}Y,\varphi Z)A_{N}X$
$\displaystyle\displaystyle-\eta^{\ast}(Z)\left\\{(\overline{\nabla}_{X}A)_{N}Y-(\overline{\nabla}_{Y}A)_{N}X\right\\}=\varphi(R(X,Y)Z)-g(A^{\ast}_{N}Y,Z)\varphi(A_{N}X)$
$\displaystyle\displaystyle+g(A^{\ast}_{N}X,Z)\varphi(A_{N}Y)+\\{g((\overline{\nabla}^{\ast}_{X}A^{\ast})_{N}Y,Z)-g((\overline{\nabla}^{\ast}_{Y}A^{\ast})_{N}X,Z)\\}\xi,$

and

$\displaystyle\displaystyle
R^{\ast}(X,Y)\varphi^{\ast}Z+g(A_{N}X,\varphi^{\ast}Z)A^{\ast}_{N}Y-g(A_{N}Y,\varphi^{\ast}Z)A^{\ast}_{N}X$
$\displaystyle\displaystyle-\eta(Z)\left\\{(\overline{\nabla}^{\ast}_{X}A^{\ast})_{N}Y-(\overline{\nabla}^{\ast}_{Y}A^{\ast})_{N}X\right\\}=\varphi^{\ast}(R^{\ast}(X,Y)Z)+\left\\{g((\overline{\nabla}_{X}A)_{N}Y,Z)\right.$
$\displaystyle\displaystyle\left.-g((\overline{\nabla}_{Y}A)_{N}X,Z)\right\\}\xi^{\ast}-g(A_{N}Y,Z)\varphi^{\ast}(A^{\ast}_{N}X)+g(A_{N}X,Z)\varphi^{\ast}(A^{\ast}_{N}Y).$

From $\displaystyle\widetilde{R}(X,Y)FN=F(\widetilde{R}(X,Y)N)$,
$\displaystyle\widetilde{R}^{\ast}(X,Y)F^{\ast}N=F^{\ast}(\widetilde{R}^{\ast}(X,Y)N)$,
we get

######  Proposition 5.7.

Let $\displaystyle(M,g)$ be a tangential hypersurface of a locally product-
like statistical manifold $\displaystyle\widetilde{M}$. Then we obtain

$\displaystyle\displaystyle\eta((\overline{\nabla}^{\ast}_{X}A^{\ast})_{N}Y)-\eta((\overline{\nabla}^{\ast}_{Y}A^{\ast})_{N}X)=-\eta^{\ast}((\overline{\nabla}_{X}A)_{N}Y)+\eta^{\ast}((\overline{\nabla}_{Y}A)_{N}X).$

Next, we discuss the tangential hypersurface $\displaystyle(M,g)$ of an almost
product-like statistical manifold satisfying (3.7). Then we find from
Proposition 5.6

$\displaystyle\displaystyle R(X,Y)\varphi Z$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\widetilde{c}\left[g(Y,Z)X-g(X,Z)Y+g(Y,\varphi
Z)\varphi X-g(X,\varphi Z)\varphi Y\right.$ (5.11)
$\displaystyle\displaystyle\left.+\\{g(\varphi X,Y)-g(X,\varphi Y)\\}\varphi
Z\right]+g(A^{\ast}_{N}Y,Z)A_{N}X$
$\displaystyle\displaystyle-g(A^{\ast}_{N}X,Z)A_{N}Y,$
$\displaystyle\displaystyle
g((\overline{\nabla}^{\ast}_{X}A^{\ast})_{N}Y,Z)-g((\overline{\nabla}^{\ast}_{Y}A^{\ast})_{N}X,Z)$
$\displaystyle\displaystyle=\widetilde{c}[\eta^{\ast}(X)g(Y,\varphi
Z)-\eta^{\ast}(Y)g(X,\varphi Z)$
$\displaystyle\displaystyle\quad+\eta^{\ast}(Z)\\{g(\varphi X,Y)-g(X,\varphi
Y)\\}],$ (5.12)
$\displaystyle\displaystyle(\overline{\nabla}_{X}A)_{N}Y-(\overline{\nabla}_{Y}A)_{N}X$
$\displaystyle\displaystyle=\widetilde{c}\left[\eta(X)\varphi Y-\eta(Y)\varphi
X\right.$ $\displaystyle\displaystyle\quad\left.+\\{g(X,\varphi Y)-g(\varphi
X,Y)\\}\xi\right],$ (5.13) $\displaystyle\displaystyle
g([A_{N},A_{N}^{\ast}]X,Y)-(d\kappa)(X,Y)=\widetilde{c}\\{\eta(X)\eta^{\ast}(Y)-\eta(Y)\eta^{\ast}(X)\\}.$
(5.14)

If $\displaystyle M$ is totally geodesic with respect to
$\displaystyle\widetilde{\nabla}$, that is, ${}^{\prime}\sigma=0$, then we get
from (5.13)

$\displaystyle\displaystyle\widetilde{c}[\eta^{\ast}(X)g(Y,\varphi
Z)-\eta^{\ast}(Y)g(X,\varphi Z)+\eta^{\ast}(Z)\\{g(\varphi X,Y)-g(X,\varphi
Y)\\}]=0,$

which denotes that $\displaystyle\widetilde{c}=0$ or

$\eta^{\ast}(X)g(Y,\varphi Z)-\eta^{\ast}(Y)g(X,\varphi
Z)+\eta^{\ast}(Z)\\{g(\varphi X,Y)-g(X,\varphi Y)\\}=0.$ (5.15)

Therefore we get $\displaystyle g(\varphi X,Y)=g(X,\varphi Y)$, namely,
$\displaystyle\varphi^{\ast}=\varphi$. It is east to see from (5.15) that
$\displaystyle\eta^{\ast}(X)\varphi Z=g(X,\varphi Z)\xi^{\ast}$ holds. Thus
$\displaystyle X=\eta^{\ast}(X)\xi=\eta(X)\xi^{\ast}$ holds. Because of any
vector field $\displaystyle X$ tangent to $\displaystyle M$ is parallel to
$\displaystyle\xi$ and $\displaystyle\xi^{\ast}$, we can not select to the
$\displaystyle n$th linearly independence vector fields, where $\displaystyle
n=\textrm{dim}M(>2)$. Hence we have

######  Theorem 5.8.

Let $\displaystyle(M,g)$ be the tangential hypersurface of the locally
product-like statistical manifold $\displaystyle\widetilde{M}$ satisfying the
condition (3.7). If $\displaystyle M$ is totally geodesic with respect to
$\displaystyle\widetilde{\nabla}$, then $\displaystyle\widetilde{M}$ is flat.

##  6 Conclusions and future works

The theory of hypersurfaces of Riemannian manifolds admitting various
differentiable structures includes comprehensive geometric properties. For
example, every hypersurface of an almost complex space form possesses a
contact structure (cf. [1, 10, 20]), every hypersurface of an almost contact
space form possesses a natural $\displaystyle f$-structure (cf.[7, 16]) and
every hypersurface of almost product Riemannian manifolds admits a para
contact structure under some conditions (cf. [2, 11]). With the help of these
features, each hypersurface of an almost complex space form becomes a contact
metric manifold, each hypersurface of an almost contact space form becomes a
metric $\displaystyle f$-manifold and some special hypersurfaces become a para
contact manifold. Thus, the basic relations and properties of contact metric
manifolds, metric $\displaystyle f$-manifolds, and para contact metric
manifolds can be constructed by examining the basic properties of the geometry
of these hypersurfaces. In this study, as a result of examining tangential
hypersurfaces of almost product-like Riemannian manifolds, para contact-like
structures have coincided.

Para contact manifolds firstly defined by K. Sato and K. K. Matsumoto in [22]
as follows:

Let $\displaystyle(\overline{M},\overline{g})$ be a $\displaystyle
m$-dimensional Riemannian manifold, $\displaystyle\overline{\varphi}$ be a
$\displaystyle(1,1)$ tensor field, $\displaystyle\overline{\xi}$ be a tangent
vector field and $\displaystyle\overline{\eta}$ be a $\displaystyle 1$-form on
$\displaystyle M$. Then
$\displaystyle(\overline{M},\overline{g},\overline{\varphi},\overline{\xi},\overline{\eta})$
is called an almost para contact Riemannian manifold if the following
relations are satisfied for any $\displaystyle X,Y\in\Gamma(T\overline{M})$:

$\displaystyle\displaystyle\overline{\varphi}^{2}X=X-\overline{\eta}(X)\overline{\xi},\
\ \ \overline{\eta}(\overline{\xi})=1,\ \ \
\overline{\varphi}\overline{\xi}=0,\ \ \
\overline{\eta}(\overline{\varphi}X)=0$

and

$\displaystyle\displaystyle
g(\overline{\varphi}X,\overline{\varphi}Y)=\overline{g}(X,Y)-\overline{\eta}(X)\overline{\eta}(Y).$

Considering the definition of almost para contact manifolds, it is possible to
give the following definitions:

######  Definition 6.1.

A Riemannian manifold $\displaystyle(M,g)$ is called an almost para contact-
like manifold if there exists a differentiable structure
$\displaystyle(\varphi,\varphi^{\ast},\xi,\xi^{\ast},\eta,\eta^{\ast})$
consisting of tensor fields $\displaystyle\varphi$ and
$\displaystyle\varphi^{\ast}$ of type $\displaystyle(1,1)$, vector fields
$\displaystyle\xi$ and $\displaystyle\xi^{\ast}$, $\displaystyle 1$-forms
$\displaystyle\eta$ and $\displaystyle\eta^{\ast}$ such that the following
relations are satisfied

$\displaystyle\displaystyle\varphi^{2}=I-\eta^{\ast}\otimes\xi,\ \ \
\varphi\xi=0,\ \ \ \eta(\xi^{\ast})=1,\ \ \ \eta\circ\varphi^{\ast}=0,$
$\displaystyle\displaystyle(\varphi^{\ast})^{2}=I-\eta\otimes\xi^{\ast},\ \ \
\varphi^{\ast}\xi^{\ast}=0,\ \ \ \eta^{\ast}(\xi)=1\ \ \
\eta^{\ast}\circ\varphi=0,$

where $\displaystyle I$ denotes the identity map.

######  Definition 6.2.

An almost para contact-like manifold
$\displaystyle(M,g,\varphi,\varphi^{\ast},\xi,\xi^{\ast},\eta,\eta^{\ast})$ is
called a para contact-like metric manifold if the following relation is
satisfied for any $\displaystyle X,Y\in\Gamma(TM)$:

$\displaystyle\displaystyle g(\varphi
X,\varphi^{\ast}Y)=g(X,Y)-\eta^{\ast}(X)\eta(Y).$

From Lemma 4.3, Lemma 4.4, Definition 6.1 and Definition 6.2, we get

######  Corollary 6.3.

Every tangential hypersurface of an almost product-like manifold is a para
contact-like metric manifold.

Considering Definition 6.1 and Definition 6.2, the problem of examining the
geometric and physical properties naturally arises in para contact-like
manifolds and their submanifolds.

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Esra Erkan
Address: Department of Mathematics, Faculty of Science and Art, Harran
University,
Sanliurfa, TURKEY.
e-mail<EMAIL_ADDRESS>
Kazuhiko Takano
Address: Department of Mathematics, School of General Education, Shinshu
University, Nagano 390-8621, Japan.
e-mail<EMAIL_ADDRESS>
Mehmet Gülbahar
Address: Department of Mathematics, Faculty of Science and Art, Harran
University,
Sanliurfa, TURKEY.
<EMAIL_ADDRESS>