**2. Preliminaries**

Let (*N*˜ , ∇˜ , *g*˜) be a statistical manifold and *N* be a submanifold of *N*˜ . Then, (*<sup>N</sup>*, ∇, *g*) is also a statistical manifold with the statistical structure (∇, *g*) on *N* induced from (∇˜ , *g*˜), and we call (*<sup>N</sup>*, ∇, *g*) a statistical submanifold.

The fundamental equations in the geometry of Riemannian submanifolds are the Gauss and Weingarten formulae and the equations of Gauss, Codazzi, and Ricci (cf. [4,5,27]). In the statistical setting, the Gauss and Weingarten formulae are defined respectively by [28]:

$$\begin{aligned} \nabla\_E F &= \nabla\_E F + h(E, F)\_\prime & \nabla\_E^\ast F &= \nabla\_E^\ast F + h^\ast(E, F)\_\prime \\ \bar{\nabla}\_E \xi &= -A\_\xi(E) + \nabla\_E^\perp \xi\_\prime & \bar{\nabla}\_E^\ast \xi &= -A\_\xi^\ast(E) + \nabla\_E^{\perp \ast} \xi\_\prime \end{aligned} \tag{1}$$

for any *E*, *F* ∈ Γ(*TN*) and *ξ* ∈ <sup>Γ</sup>(*T*⊥*N*), where ∇˜ and ∇˜ ∗ (resp., ∇ and ∇∗) are the dual connections on *N*˜ (resp., on *N*).

The symmetric and bilinear imbedding curvature tensor of *N* in *N* ˜ with respect to ∇ ˜ and ∇ ˜ ∗ is denoted as *h* and *h*<sup>∗</sup>, respectively. The relation between *h* (resp. *h*∗) and *<sup>A</sup>ξ* (resp. *A*∗*ξ* ) is defined by [28]:

$$\begin{aligned} \lg(h(E, F), \mathfrak{F}) &= \lg(A\_{\mathfrak{f}}^{\*} E, F), \\ \lg(h^{\*}(E, F), \mathfrak{F}) &= \lg(A\_{\mathfrak{f}} E, F), \end{aligned} \tag{2}$$

for any *E*, *F* ∈ Γ(*TN*) and *ξ* ∈ <sup>Γ</sup>(*T*⊥*N*).

Let *R* ˜ and *R* be the curvature tensor fields of ∇ ˜ and ∇, respectively. The corresponding Gauss, Codazzi, and Ricci equations are given by [28]:

$$\begin{aligned} \tilde{\mathcal{g}}(\tilde{\mathcal{R}}(E,F)G,H) &= \mathcal{g}(R(E,F)G,H) + \tilde{\mathcal{g}}(h(E,G),h^\*(F,H)) \\ &- \tilde{\mathcal{g}}(h^\*(E,H),h(F,G)), \end{aligned} \tag{3}$$

$$\begin{aligned} \langle \tilde{R}(E,F)G \rangle^{\perp} &= \nabla\_E^{\perp}h(F,G) - h(\nabla\_F F, G) - h(F, \nabla\_F G) \\ &- \{ \nabla\_F^{\perp}h(E,G) - h(\nabla\_F E, G) - h(E, \nabla\_F G) \}, \end{aligned} \tag{4}$$

$$
\tilde{\mathcal{g}}(\tilde{\mathcal{R}}^\perp(E,F)\tilde{\mathcal{g}},\eta) = \tilde{\mathcal{g}}(\mathcal{R}(E,F)\tilde{\mathcal{g}},\eta) + \mathcal{g}([A\_{\tilde{\mathcal{G}}'}^\ast A\_{\tilde{\eta}}]E,F),
\tag{5}
$$

for any *E*, *F*, *G*, *H* ∈ Γ(*TN*) and *ξ*, *η* ∈ <sup>Γ</sup>(*T*⊥*N*), where *R*<sup>⊥</sup> is the Riemannian curvature tensor on *T*⊥*N*.

Similarly, *R* ˜ ∗ and *R*∗ are respectively the curvature tensor fields with respect to ∇ ˜ ∗ and ∇∗. We can obtain the duals of all Equations (3)–(5) with respect to ∇ ˜ ∗ and ∇∗. Furthermore,

$$\bar{S} = \frac{1}{2}(\bar{R} + \bar{R}^\*) \text{ and } S = \frac{1}{2}(R + R^\*) \tag{6}$$

are respectively the curvature tensor fields of *N* ˜ and *N* given by [7]. Thus, the sectional curvature K∇,∇∗ on *N* of *N*˜ is defined by [29,30]:

$$\begin{aligned} \mathbb{K}^{\nabla, \nabla^\*} (E \wedge F) &= g(\mathcal{S}(E, F)F, E) \\ &= \frac{1}{2} (\mathcal{g}(\mathcal{R}(E, F)F, E) + \mathcal{g}(\mathcal{R}^\*(E, F)F, E)), \end{aligned} \tag{7}$$

for any orthonormal vectors *E*, *F* ∈ *TpN*, *p* ∈ *N*.

Suppose that dim(*N*) = *m* and dim(*N*˜ ) = *n*. Let {*<sup>e</sup>*1, ... ,*em*} and {*em*+1, ... ,*en*} be respectively the orthonormal basis of *TpN* and *T*⊥*p N* for *p* ∈ *N*. Then, the scalar curvature *<sup>σ</sup>*∇,∇∗ of *N* is given by:

$$\sigma^{\nabla,\nabla^\*} = \sum\_{1 \le i < j \le m} \mathbb{K}^{\nabla,\nabla^\*} (\mathfrak{e}\_i \wedge \mathfrak{e}\_j). \tag{8}$$

The normalized scalar curvature *ρ* of *N* is defined as:

$$
\rho^{\nabla,\nabla^\*} = \frac{2\sigma^{\nabla,\nabla^\*}}{m(m-1)}.
$$

The mean curvature vectors H and H∗ of *N* in *N* ˜ are:

$$\mathcal{H} = \frac{1}{m} \sum\_{i=1}^{m} h(e\_i, e\_i), \quad \mathcal{H}^\* = \frac{1}{m} \sum\_{i=1}^{m} h^\*(e\_i, e\_i).$$

> Furthermore, we set:

$$h\_{ij}^{a} = \lg(h(e\_i, e\_j), e\_a), \ h\_{ij}^{\*a} = \lg(h^\*(e\_i, e\_j), e\_a),$$

for *i*, *j* ∈ {1, . . . , *<sup>m</sup>*}, *a* ∈ {*m* + 1, . . . , *<sup>n</sup>*}.

A statistical manifold (*N*˜ , ∇˜ , *g*˜) is said to be *of constant curvature c*˜ ∈ R, denoted by *N*˜ (*c*˜), if the following curvature equation holds:

$$\mathcal{S}(E,F)G = \vec{\varepsilon}(\mathcal{g}(F,G)E - \mathcal{g}(E,G)F), \ \forall E, F, G \in \Gamma(T\widehat{N}).\tag{9}$$

## **3. Basics on Statistical Warped Product Manifolds**

**Definition 1.** *[3] Let* (*<sup>N</sup>*1, *g*1) *and* (*<sup>N</sup>*2, *g*2) *be two (pseudo)-Riemannian manifolds and* f > 0 *be a differentiable function on N*1*. Consider the natural projections π* : *N*1 × *N*2 → *N*1 *and π* : *N*1 × *N*2 → *N*2*. Then, the warped product N* = *N*1 <sup>×</sup>f *N*2 *with warping function* f *is the product manifold N*1 × *N*2 *equipped with the Riemannian structure such that:*

$$\lg(E, F) = \lg\_1(\pi\_\* E, \pi\_\* F) + \mathfrak{f}^2(u)\lg\_2(\pi\_\*' E, \pi\_\*' F),\tag{10}$$

*for E*, *F* ∈ <sup>Γ</sup>(*<sup>T</sup>*(*<sup>u</sup>*,*<sup>v</sup>*)*<sup>N</sup>*)*, u* ∈ *N*1*, and v* ∈ *N*2*, where* ∗ *denotes the tangent map.*

Let *<sup>χ</sup>*(*<sup>N</sup>*1) and *<sup>χ</sup>*(*<sup>N</sup>*2) be the set of all vector fields on *N*1 × *N*2, which is the horizontal lift of a vector field on *N*1 and the vertical lift of a vector field on *N*2, respectively. We have *<sup>T</sup>*(*<sup>N</sup>*1 × *<sup>N</sup>*2) = *<sup>χ</sup>*(*<sup>N</sup>*1) ⊕ *<sup>χ</sup>*(*<sup>N</sup>*2). Thus, it can be seen that *<sup>π</sup>*∗(*χ*(*<sup>N</sup>*1)) = <sup>Γ</sup>(*TN*1) and *π* ∗(*χ*(*<sup>N</sup>*2)) = <sup>Γ</sup>(*TN*2). Therefore, *<sup>π</sup>*∗(*X*) = *E*1 ∈ <sup>Γ</sup>(*TN*1), *<sup>π</sup>*∗(*Y*) = *F*1 ∈ <sup>Γ</sup>(*TN*1), *π*∗(*U*) = *E*2 ∈ <sup>Γ</sup>(*TN*2) and *π*∗(*V*) = *F*2 ∈ <sup>Γ</sup>(*TN*2), for any *X*,*Y* ∈ *<sup>χ</sup>*(*<sup>N</sup>*1) and *U*, *V* ∈ *<sup>χ</sup>*(*<sup>N</sup>*2).

Recall the following general result from [6] for a dualistic structure on the warped product manifold *N*1 <sup>×</sup>f *N*2.

**Proposition 1.** *Let* (*g*1, ∇*N*1 , <sup>∇</sup>*N*1<sup>∗</sup>) *and* (*g*2, ∇*N*2 , <sup>∇</sup>*N*2<sup>∗</sup>) *be dualistic structures on N*1 *and N*2*, respectively. For X*,*Y* ∈ *<sup>χ</sup>*(*<sup>N</sup>*1) *and U*, *V* ∈ *<sup>χ</sup>*(*<sup>N</sup>*2)*, D, D*∗ *on N*1 × *N*2 *satisfy:*



*where* ∇*N*1 *E*1 *F*1 = *<sup>π</sup>*∗(*DXY*)*,* ∇*N*1<sup>∗</sup> *E*1 *F*1 = *<sup>π</sup>*∗(*D*<sup>∗</sup>*X<sup>Y</sup>*)*,* ∇*N*2 *E*2 *F*2 = *<sup>π</sup>*∗(*DUV*)*, and* ∇*N*2<sup>∗</sup> *E*2 *F*2 = *<sup>π</sup>*∗(*D*<sup>∗</sup>*U<sup>V</sup>*)*. Then,* (*g*˜, *D*, *D*∗) *is a dualistic structure on N*1 × *N*2*.*

Furthermore, Todjihounde [6] derived the curvature of the statistical warped product *N* ˜ = *N*1 <sup>×</sup>f *N*2 in terms of the curvature tensors *R*1 and *R*2 of *N*1 and *N*2, respectively, and its warping function f.

**Lemma 1.** *Let* (*N*˜ = *N*1 <sup>×</sup>f *N*2, *D*, *D*<sup>∗</sup>, *g*˜) *be a statistical warped product manifold. For X*,*Y*, *Z* ∈ *<sup>χ</sup>*(*<sup>N</sup>*1) *and U*, *V*, *W* ∈ *<sup>χ</sup>*(*<sup>N</sup>*2)*, we have:*


*(e) R* ˜(*<sup>X</sup>*, *V*)*W* = − f−<sup>1</sup>*g*˜(*<sup>V</sup>*, *<sup>W</sup>*)*DX*(*grad* f)*,*

*(f) R* ˜(*<sup>U</sup>*, *V*)*W* = *<sup>R</sup>*2(*<sup>E</sup>*2, *<sup>F</sup>*2)*<sup>G</sup>*2 + ||*grad* f||<sup>2</sup>[*g*2(*<sup>U</sup>*, *W*)*V* − *g*2(*<sup>V</sup>*, *<sup>W</sup>*)*U*]*,*

*where R* ˜ *denotes the curvature tensor field of* (*N*˜ = *N*1 <sup>×</sup>f *N*2, *D*, *D*<sup>∗</sup>, *g*˜) *and Hess*f(*<sup>X</sup>*,*<sup>Y</sup>*) = *X*(*Y*f) − (∇*N*1 *X Y*)f *is the Hessian function of* f *with respect to* ∇*N*1 *.*

The next result from [9] provides the Ricci tensor *Ric* ˜ of the statistical warped product manifold.

**Lemma 2.** *Let* (*N*˜ = *N*1 <sup>×</sup>f *N*2, *D*, *D*<sup>∗</sup>, *g*˜) *be a statistical warped product manifold. For X*,*Y* ∈ *<sup>χ</sup>*(*<sup>N</sup>*1) *and U*, *V* ∈ *<sup>χ</sup>*(*<sup>N</sup>*2)*, we have:*


*where Ric*1 *and Ric*2 *are the Ricci tensors of N*1 *and N*2*, respectively, and* Δf = *div*(*grad* f) *is the Laplacian of* f *with respect to D.*

We recall the following result from [31]. This result is useful in some Riemannian problems like the study of the distance between two manifolds, of the extremes of sectional curvature and is applied successfully in the demonstration of the Chen inequality.

Let (*<sup>N</sup>*, *g*) be a Riemannian submanifold of a Riemannian manifold (*N*˜ , *g*˜), and let *f* : *N*˜ → R be a differentiable function. Let:

$$\min\_{\mathbf{x}\_0 \in \mathcal{N}} f(\mathbf{x}\_0) \tag{11}$$

be the constrained extremum problem.

**Theorem 1.** *If x* ∈ *N is the solution of the problem (11), then:*


$$\Theta(E, F) = \operatorname{Hess}\_f(E, F) + \overline{\g}(h'(E, F), (\operatorname{grad} f)(x))^2$$

*is positive semi-definite, where h is the second fundamental form of N in N*˜ *and grad f denotes the gradient of f .*

#### **4. Statistical Solitons on Statistical Warped Product Manifolds**

The Ricci solitons model the formation of singularities in the Ricci flow, and they correspond to self-similar solutions. R. Hamilton [32] introduced the study of Ricci solitons as fixed or stationary points of the Ricci flow in the space of the metrics on Riemannian manifolds modulo diffeomorphisms and scaling. Since then, many researchers studied Ricci solitons for different reasons and in different ambient spaces (for example [33–35]). A complete Riemannian manifold (*N*˜ , *g*˜) is called a *Ricci soliton* (*N*˜ , *g*˜, *ζ*, *λ*) if there exists a smooth vector field *ζ* and a constant *λ* ∈ R such that:

$$2\bar{R}ic = 2\lambda\bar{\varrho} - \mathcal{L}\_{\zeta}\bar{\varrho}\_{\prime}$$

where L*ζ* denotes the Lie derivative along *ζ* and *Ric* ˜ is the Ricci tensor of *g*˜.

A generalization of Ricci solitons in the framework of manifolds endowed with an arbitrary linear connection ∇ ˜ , different from the Levi–Civita connection of *g*˜, is defined in [36] as follows:

Let (*N*˜ , ∇˜ ) be a manifold and *ζ* ∈ *χ*(*N*˜ ). A triple (*g*˜, *ζ*, *λ*) is called a ∇˜ *-Ricci soliton* if ∇ ˜ *ζ* + Q ˜ + *λI* = 0 holds, where Q ˜ is the Ricci operator of *N* ˜ defined by *g*˜(Q˜ *E*, *F*) = *Ric* ˜ (*<sup>E</sup>*, *<sup>F</sup>*), for vector fields *E*, *F* on *N* ˜ .

The statistical manifold (*N*˜ , ∇˜ , *g*˜) is called *Ricci-symmetric* if the Ricci operator Q˜ with respect to ∇ ˜ (equivalently, the dual operator Q ˜ ∗ with respect to ∇ ˜ ∗) is symmetric (cf. [36,37]).

Based on these, we have the following.

**Definition 2.** *A pair* (*ζ*, *λ*) *is called a statistical soliton on a Ricci-symmetric statistical manifold* (*N*˜ , ∇˜ , *g*˜) *if the triple* (*g*˜, *ζ*, *λ*) *is* ∇˜ *-Ricci and* ∇˜ ∗*-Ricci solitons, i.e., we have:*

$$
\vec{\nabla}\mathcal{J} + \vec{\mathcal{Q}} + \lambda I = 0,\tag{12}
$$

*and:*

$$
\nabla^\* \mathcal{J} + \mathcal{Q}^\* + \lambda I = 0,\tag{13}
$$

*where g*˜(Q˜ *E*, *F*) = *Ric* ˜ (*<sup>E</sup>*, *F*) *and g*˜(Q˜ <sup>∗</sup>*E*, *F*) = *Ric* ˜ <sup>∗</sup>(*<sup>E</sup>*, *<sup>F</sup>*)*, for all vector fields on N*˜ *, and Ric* ˜ *and Ric* ˜ ∗ *denote the Ricci tensor fields with respect to* ∇ ˜ *and* ∇ ˜ ∗*, respectively.*

The main purpose of this section is to study the problem: *under what conditions does the base manifold or fiber manifold of the statistical warped product manifold become a statistical soliton?*

Let (*<sup>N</sup>*1, ∇*N*1 , ∇*N*1∗, *g*1) and (*<sup>N</sup>*2, ∇*N*2 , ∇*N*2∗, *g*2) be the Ricci-symmetric statistical manifolds. Denote the Ricci-symmetric statistical warped product manifold by (*N*˜ = *N*1 <sup>×</sup>f *N*2, *D*, *D*<sup>∗</sup>, *g*˜ = *g*1 + f<sup>2</sup>*g*2). Let *ζ* = (*ζ*1, *ζ*2) ∈ *χ*(*N*˜ ) be a vector field on *N*˜ . Then, the pair (*ζ*, *λ*) on (*N*˜ , ∇˜ , *g*˜) is called a *statistical soliton* if the triple (*g*˜, *ζ*, *λ*) is both *D*-Ricci and *D*∗-Ricci solitons, given by (12) and (13).

It follows from Lemma 2 that the Ricci tensor of *N* ˜ is given as below:

$$\begin{split} \tilde{Ric} &= Ric\_1 - \mathfrak{f}^{-1} \dim(N\_2) H \text{ess} \mathfrak{f} + Ric\_2 \\ &- [\mathfrak{f}(\Delta \mathfrak{f}) + (\dim(N\_2) - 1) ||\mathfrak{grad} \mathfrak{f}||^2] \mathbb{g}\_2. \end{split} \tag{14}$$

Thus, (12) and (13) can be rewritten as:

$$\begin{aligned} &\nabla^{N\_1}\zeta\_1 + \nabla^{N\_2}\zeta\_2 + \operatorname{Ric}\_1 - \mathfrak{f}^{-1}\dim(N\_2)H\mathrm{ess}\_{\mathfrak{f}} + \operatorname{Ric}\_2\\ &- [\mathfrak{f}(\Delta \mathfrak{f}) + (\dim(N\_2) - 1)||\operatorname{grad}\mathfrak{f}||^2]\mathfrak{g}\mathfrak{z} + \lambda\mathfrak{g}\_1 + \lambda\mathfrak{f}^2\mathfrak{g}\mathfrak{z} = 0,\end{aligned} \tag{15}$$

and:

$$\begin{aligned} \nabla^{N\_1\*} \zeta\_1 + \nabla^{N\_2\*} \zeta\_2 + Ric\_1^\* - \mathfrak{f}^{-1} \dim(N\_2) H \text{ess}\_{\mathfrak{f}}^{D^\*} + \text{Ric}\_2^\* \\ - [f(\Delta^{D^\*} \mathfrak{f}) + (\dim(N\_2) - 1) || \operatorname{grad} f ||^2] \mathfrak{g}\_2 + \lambda \mathfrak{g}\_1 + \lambda \mathfrak{f}^2 \mathfrak{g}\_2 &= 0,\end{aligned} \tag{16}$$

respectively.

> Throughout this section, we use the statistical warped products as Ricci-symmetric. We give the following results by applying Lemma 2:

**Lemma 3.** *Let* (*N*˜ = R <sup>×</sup>f *N*2, *D*, *D*<sup>∗</sup>, *g*˜) *be a statistical warped product manifold, where* (<sup>R</sup>, ∇R, *dz*<sup>2</sup>) *is a trivial statistical manifold of dimension one and* dim(*<sup>N</sup>*2) = *k. Then, for U*, *V* ∈ *<sup>χ</sup>*(*<sup>N</sup>*2)*, we have:*

*(a) Ric* ˜ (*∂<sup>z</sup>*, *∂z*) = − *<sup>k</sup>*f−1¨f*,*


**Proposition 2.** *Let* (*ζ*, *λ*) *be a statistical soliton on statistical warped product manifold* (*N*˜ = R <sup>×</sup>f *N*2, *D*, *D*<sup>∗</sup>, *g*˜ = *dz*<sup>2</sup> + f<sup>2</sup>*g*2) *with* dim(R) = 1 *and* dim(*<sup>N</sup>*2) = *k. Then:*

$$H \text{ess}\_{\mathfrak{f}} = \frac{\mathfrak{f}\lambda}{k}.$$

**Proof.** Since *N* ˜ is a statistical soliton, then from (6), we have:

$$
\mathfrak{F}(\bar{\nabla}\_{\partial z}\zeta, \partial z) + \bar{\mathcal{R}}ic(\partial z, \partial z) + \lambda \,\mathfrak{F}(\partial z, \partial z) \,\, = \, 0 \,\, .
$$

By taking into account Lemma 3 and *Ric*1(*∂<sup>z</sup>*, *∂z*) = 0, we get:

$$-\mathfrak{F}(\zeta, \nabla\_{\partial z}^\* \partial z) - k \mathfrak{f}^{-1} Hess\_{\mathfrak{f}}(\partial z, \partial z) + \lambda \mathfrak{F}(\partial z, \partial z) = 0\_{\mathfrak{f}}$$

which gives *Hess*f(*∂<sup>z</sup>*, *∂z*)=( f*λk*)*g*˜(*∂<sup>z</sup>*, *∂z*).

**Theorem 2.** *Let ζ* = (*∂<sup>z</sup>*, *ζ*2) ∈ *χ*(*N*˜ ) *be a vector field on statistical warped product manifold* (*N*˜ = R <sup>×</sup>f *N*2, *D*, *D*<sup>∗</sup>, *g*˜ = *dz*<sup>2</sup> + f<sup>2</sup>*g*2) *with* dim(R) = 1 *and* dim(*<sup>N</sup>*2) = *k. If* (*ζ*, *λ*) *is a statistical soliton on N, then:* ˜

*(a)* (*<sup>N</sup>*2, *g*2, *ζ*2, *<sup>λ</sup>*2) *is a statistical soliton on* (*<sup>N</sup>*2, ∇*N*2 , ∇*N*2∗, *g*2)*, where λ*2 = (*k* − 1)[f¨f − ˙f<sup>2</sup>]*,*

$$f(b) \quad f(z) = az + b \text{ if } \lambda = 0,$$

*(c)* f(*z*) = cosh(*az* + *b*) *if λ* = 0*,*

*where a*, *b* ∈ R*.*

**Proof.** From Equation (15) and Lemma 3, we have:

$$\begin{aligned} \nabla^{N\_1} \partial z + \nabla^{N\_2} \mathcal{J}\_2 &+ Ric\_1 - k \mathfrak{f}^{-1} \mathfrak{f} + Ric\_2 \\ - (\mathfrak{f} \vec{\mathfrak{f}} + (k - 1) \mathfrak{f}^2) \mathfrak{g}\_2 + \lambda \mathfrak{g}\_1 + \lambda \mathfrak{f}^2 \mathfrak{g}\_2 &= 0. \end{aligned}$$

Note *<sup>g</sup>*1(∇*N*<sup>1</sup> *∂z∂<sup>z</sup>*, *∂z*) = 0 and *Ric*1(*∂<sup>z</sup>*, *∂z*) = 0. Thus, the above equation becomes:

$$\nabla^{N\_2}\zeta\_2 - k\mathfrak{f}^{-1}\ddot{\mathfrak{f}} + Ric\_2 - (\mathfrak{f}\ddot{\mathfrak{f}} + (k-1)\dot{\mathfrak{f}}^2)\mathfrak{g}\_2 + \lambda\mathfrak{g}\_1 + \lambda\mathfrak{f}^2\mathfrak{g}\_2 = 0,$$

from which we get:

$$
\lambda = k \mathfrak{f}^{-1} \mathfrak{f}.\tag{17}
$$

$$\nabla^{\mathcal{N}\_2} \mathbb{Z}\_2 + Ric\_2 + [\lambda \mathfrak{f}^2 - (\mathfrak{f} \mathfrak{f} + (k - 1) \mathfrak{f}^2)] \mathbb{g}\_2 = 0. \tag{18}$$

Putting (17) into the Equation (18), we arrive at:

$$
\nabla^{N\_2} \zeta\_2 + Ric\_2 + (k - 1)[\mathfrak{f}\mathfrak{f} - \mathfrak{f}^2] \wp\_2 = 0.
$$

Similarly, by using (16), we derive:

$$\nabla^{N\_2 \ast} \mathcal{J}\_2 + Ric\_2^\ast + (k - 1)[\mathfrak{f}\sharp - \mathfrak{f}^2] \mathcal{g}\_2 = 0.$$

Thus, (*<sup>N</sup>*2, *g*2, *ζ*2,(*k* − 1)[f¨f − ˙f<sup>2</sup>]) is a statistical soliton provided that (*k* − 1)[f¨f − ˙f2] is constant. On the other hand, by using (17), we have the following cases:


where *a*, *b* are real constants.

> Before proving the next result, we state the following:

**Lemma 4.** *Let* (*N*˜ = *N*1 <sup>×</sup>f R, *D*, *D*<sup>∗</sup>, *g*˜) *be a statistical warped product manifold, where* (<sup>R</sup>, ∇R, *dz*<sup>2</sup>) *is a trivial statistical manifold of dimension one and* dim(*<sup>N</sup>*1) = *k. For X*,*Y* ∈ *<sup>χ</sup>*(*<sup>N</sup>*1)*, we have:*


**Theorem 3.** *Let ζ* = (*ζ*1, *∂z*) ∈ *χ*(*N*˜ ) *be a vector field on statistical warped product manifold* (*N*˜ = *N*1 <sup>×</sup>f R, *D*, *D*<sup>∗</sup>, *g*˜ = *g*1 + f<sup>2</sup>*dz*<sup>2</sup>) *with* dim(R) = 1 *and* dim(*<sup>N</sup>*1) = *k. Suppose that Hess*f = 0*. Then,* (*ζ*, *λ*) *is a statistical soliton on N if and only if* ˜ (*ζ*1, *λ* = f−<sup>1</sup>(<sup>Δ</sup>f)) *is a statistical soliton on N*1*.*

**Proof.** Since *<sup>g</sup>*2(∇*N*<sup>1</sup> *∂z ∂<sup>z</sup>*, *∂z*) = 0 and *Ric*2(*∂<sup>z</sup>*, *∂z*) = 0, then by using Equation (15) and Lemma 4, we get:

$$\nabla^{\aleph\_1} \mathcal{J}\_1 + \mathrm{Ric}\_1 - \mathfrak{f}(\Delta \mathfrak{f}) \mathcal{g}\_2 + \lambda \mathcal{g}\_1 + \lambda \mathfrak{f}^2 \mathcal{g}\_2 = 0.$$

Therefore, we have:

$$
\nabla^{N\_1} \mathbb{Z}\_1 + \mathrm{Ric}\_1 + \lambda \mathbb{g}\_1 = 0. \tag{19}
$$

Furthermore, f−<sup>1</sup>(<sup>Δ</sup>f) = *λ* = *constant*. Putting this into (19), we get:

$$\nabla^{\mathcal{N}\_1} \mathcal{J}\_1 + Ric\_1 + \mathfrak{f}^{-1}(\Delta \mathfrak{f}) \mathcal{J}\_1 = 0.$$

Similarly, by using (16), we obtain:

$$\nabla^{\mathcal{N}\_1 \ast} \mathcal{J}\_1 + \mathring{R} \mathring{c}\_1^\ast + \mathfrak{f}^{-1} (\Delta^\ast \mathfrak{f}) \mathcal{g}\_1 = 0.$$

Since f−<sup>1</sup>(<sup>Δ</sup>f) is constant, (*<sup>N</sup>*1, *g*1, *ζ*1, *λ* = f−<sup>1</sup>(<sup>Δ</sup>f)) is a statistical soliton. Conversely, if (*ζ*1, *λ* = f−<sup>1</sup>(<sup>Δ</sup>f)) is a statistical soliton on *N*1, then:

$$\begin{aligned} \nabla^{N\_1} \mathcal{J}\_1 + \nabla^{N\_2} \partial z + Ric\_1 - \mathfrak{f}^{-1} k\_2 \operatorname{Hess}\_{\mathfrak{f}} &+ Ric\_2 \\ - [\mathfrak{f}(\Delta \mathfrak{f}) + (k\_2 - 1) |\operatorname{grad} \mathfrak{f}|]^2 \mathfrak{g}\_2 \\ &= \nabla^{N\_1} \mathcal{J}\_1 + Ric\_1 + \mathfrak{f}^{-1} (\Delta \mathfrak{f}) \mathfrak{g}\_1 - \mathfrak{f}^{-1} (\Delta \mathfrak{f}) \mathfrak{g}\_1 - \mathfrak{f} (\Delta \mathfrak{f}) \mathfrak{g}\_2 \\ &= - \mathfrak{f}^{-1} (\Delta \mathfrak{f}) \mathfrak{g}\_1 - \mathfrak{f} (\Delta \mathfrak{f}) \mathfrak{g}\_2 = - \mathfrak{f}^{-1} (\Delta \mathfrak{f}) (\mathfrak{g}\_1 + \mathfrak{g}\_2) \\ &= -\lambda \mathfrak{g} .\end{aligned}$$

Thus, *Dζ* + Q ˜ + *λI* = 0. Similarly, *D*∗*ζ* + Q ˜ ∗ + *λI* = 0. Hence, (*ζ*, *λ*) is a statistical soliton on *N* ˜ .

An immediate consequence of Theorem 3 is as follows:

**Corollary 1.** *Let* (*N*˜ , *g*˜, *ζ*, *λ*) *be a Statistical soliton on statistical manifold* (*N*˜ = *N*1 <sup>×</sup>f R, *D*, *D*<sup>∗</sup>, *g*˜ = *g*1 + f<sup>2</sup>*dz*<sup>2</sup>) *with dim*(R) = 1 *and dim*(*<sup>N</sup>*1) = *k. If Hess*f = *g*<sup>1</sup>*,* ∈ *<sup>C</sup>*<sup>∞</sup>(*<sup>N</sup>*1)*, then* (*<sup>N</sup>*1, *g*1, *ζ*1, f−<sup>1</sup>(<sup>Δ</sup>f) − f−<sup>1</sup>) *is a statistical soliton.*

## **5. B.Y. Chen Inequality**

A universal sharp inequality for submanifolds in a Riemannian manifold of constant sectional curvature was established in [38], known as the first Chen inequality. The main purpose of this section is to establish the corresponding inequality for statistical warped product manifolds statistically immersed in a statistical manifold of constant curvature.

Let *ϕ* : *N* = *N*1 <sup>×</sup>f *N*2 → *N* ˜ (*c*˜) be an isometric statistical immersion of a warped product *N*1 <sup>×</sup>f *N*2 into a statistical manifold of constant sectional curvature *c*˜. We denote by *r*, *k*, and *m* = *r* + *k* the dimensions of *N*1, *N*2, and *N*1 × *N*2, respectively. Since *N*1 <sup>×</sup>f *N*2 is a statistical warped product, we have:

$$
\nabla\_{E\_1} E\_2 = \nabla\_{E\_2} E\_1 = (E\_1 \ln \mathfrak{f}) E\_{2\mathfrak{e}}
$$

for unit vector fields *E*1 and *E*2 tangent to *N*1 and *N*2, respectively. Hence, we derive:

$$\mathbb{K}(E\_1 \wedge E\_2) = \frac{1}{f} \{ (\nabla\_{E\_1} E\_1) \mathfrak{f} - E\_1^2 \mathfrak{f} \}. \tag{20}$$

If we choose a local orthonormal frame {*<sup>e</sup>*1, ... ,*em*} such that {*<sup>e</sup>*1, ... ,*er*} are tangent to *N*1 and {*er*+1,...,*er*+*<sup>k</sup>* = *em*} are tangent to *N*2, then we have:

$$\frac{\Delta \mathbf{f}}{\mathbf{f}} = \sum\_{i=-1}^{r} \mathbb{K}(\mathbf{c}\_{i} \wedge \mathbf{c}\_{j}),\tag{21}$$

for each *j* = *r* + 1, . . . , *m*.

On the other hand, let *E*1 and *E*2 be two unit local vector fields tangent to *N*1 and *N*2, respectively, such that *e*1 = *E*1 and *er*+1 = *E*2. By taking into account Equations (3), (6), and (9), we derive (7) as follows:

$$\begin{split} \mathbb{K}^{\nabla,\nabla^{\*}}(\boldsymbol{e}\_{1}\wedge\boldsymbol{e}\_{r+1}) &= \frac{\mathbb{E}}{2} \{ 2\{ \mathbf{g}(\boldsymbol{e}\_{r+1},\boldsymbol{e}\_{r+1})\mathbf{g}(\boldsymbol{e}\_{1},\boldsymbol{e}\_{1}) - 2\mathbf{g}(\boldsymbol{e}\_{1},\boldsymbol{e}\_{r+1})\mathbf{g}(\boldsymbol{e}\_{r+1},\boldsymbol{e}\_{1}) \} \\ &+ \frac{1}{2} \{ \mathbf{g}(\boldsymbol{h}^{\*}(\boldsymbol{e}\_{1},\boldsymbol{e}\_{1}),\boldsymbol{h}(\boldsymbol{e}\_{r+1},\boldsymbol{e}\_{r+1})) \\ &+ \mathbf{g}(\boldsymbol{h}(\boldsymbol{e}\_{1},\boldsymbol{e}\_{1}),\boldsymbol{h}^{\*}(\boldsymbol{e}\_{r+1},\boldsymbol{e}\_{r+1})) - 2\mathbf{g}(\boldsymbol{h}(\boldsymbol{e}\_{1},\boldsymbol{e}\_{r+1}),\boldsymbol{h}^{\*}(\boldsymbol{e}\_{1},\boldsymbol{e}\_{r+1})) \} \\ &= \overline{\boldsymbol{\varepsilon}} + \frac{1}{2} \sum\_{\boldsymbol{a}=\boldsymbol{m}+1}^{\boldsymbol{n}} \left\{ h\_{11}^{\*4}h\_{r+1,r+1}^{\boldsymbol{a}} + h\_{11}^{\boldsymbol{a}}h\_{r+1,r+1}^{\boldsymbol{a}} - 2h\_{1,r+1}^{\boldsymbol{a}}h\_{1,r+1}^{\boldsymbol{a}} \right\}. \end{split}$$

We rewrite the terms of the RHS of the previous equation as:

$$\begin{split} \mathbb{K}^{\nabla,\nabla^\*} (\boldsymbol{c}\_1 \wedge \boldsymbol{c}\_{r+1}) &= \mathbb{\tilde{c}} + \frac{1}{2} \sum\_{a=m+1}^n \left\{ (h\_{11}^a + h\_{11}^{\*a}) (h\_{r+1,r+1}^a + h\_{r+1,r+1}^{\*a}) \right\} \\ &- (h\_{1,r+1}^a + h\_{1,r+1}^{\*a})^2 + (h\_{1,r+1}^a)^2 + (h\_{1,r+1}^{\*a})^2 \\ &- h\_{11}^a h\_{r+1,r+1}^a - h\_{11}^{\*a} h\_{r+1,r+1}^{\*a} \boldsymbol{\}. \end{split}$$

Since, 2*h*<sup>0</sup> = *h* + *h*<sup>∗</sup>, we get:

$$\begin{aligned} \mathbb{K}^{\nabla\_{\boldsymbol{\tau}}\nabla^{\boldsymbol{\tau}}}(\boldsymbol{e}\_{1}\wedge\boldsymbol{e}\_{r+1}) &= \tilde{\boldsymbol{c}} + \frac{1}{2} \sum\_{a=m+1}^{n} \left\{ 4h\_{11}^{0a}h\_{r+1,r+1}^{0a} \\ &- (h\_{11}^{a}h\_{r+1,r+1}^{a} - (h\_{1,r+1}^{a})^{2}) \\ &- (h\_{11}^{\*a}h\_{r+1,r+1}^{\*a} - (h\_{1,r+1}^{\*a})^{2}) - 4(h\_{1,r+1}^{0a})^{2} \right\}. \end{aligned}$$

Thus, we have:

$$\mathbb{K}^{\nabla,\nabla^\*} (c\_1 \wedge c\_{r+1}) = \tilde{c} + \sum\_{a=m+1}^n \left\{ 2(h\_{11}^{0a} h\_{r+1,r+1}^{0a} - (h\_{1,r+1}^{0a})^2) \right.$$

$$-\frac{1}{2} (h\_{11}^a h\_{r+1,r+1}^a - (h\_{1,r+1}^a)^2) - \frac{1}{2} (h\_{11}^{\*a} h\_{r+1,r+1}^{\*a} - (h\_{1,r+1}^{\*a})^2) \}. \tag{22}$$

Using the Gauss equation for the Levi–Civita connection, we arrive at:

$$\mathbb{K}^0(\mathfrak{e}\_1 \wedge \mathfrak{e}\_{r+1}) = \overline{\varepsilon} - \sum\_{a=m+1}^n \left\{ (h\_{1,r+1}^{0a})^2 - h\_{11}^{0a} h\_{r+1,r+1}^{0a} \right\}\_{r \ge 0}$$

which can be rewritten as:

$$\sum\_{a=-m+1}^{n} \left\{ (h\_{1,r+1}^{0a})^2 - h\_{11}^{0a} h\_{r+1,r+1}^{0a} \right\} \\ = \mathbb{K}^0(\varepsilon\_1 \wedge \varepsilon\_{r+1}) - \overline{\varepsilon}. \tag{23}$$

 Substituting (23) into (22), we get:

$$\begin{split} \mathbb{K}^{\nabla,\nabla^\*} (\boldsymbol{c}\_1 \wedge \boldsymbol{c}\_{r+1}) &= 2\mathbb{K}^0 (\boldsymbol{c}\_1 \wedge \boldsymbol{c}\_{r+1}) - \overline{\boldsymbol{c}} - \frac{1}{2} \sum\_{a=m+1}^n \left\{ h\_{11}^a h\_{r+1,r+1}^a \\ &- (h\_{1,r+1}^a)^2 + h\_{11}^{\*a} h\_{r+1,r+1}^{\*a} - (h\_{1,r+1}^{\*a})^2 \right\}. \end{split} \tag{24}$$

Furthermore, we derive (8) as:

$$\begin{split} \sigma^{\nabla,\nabla^{\*}} &= \frac{m(m-1)\bar{\varepsilon}}{2} + \frac{1}{2} \sum\_{a=m+1}^{n} \sum\_{i$$

By a similar argumen<sup>t</sup> as above, we deduce that:

$$\sigma^{\nabla,\nabla^\*} = \frac{m(m-1)\overline{c}}{2} + \frac{1}{2} \sum\_{a=-m+1}^n \sum\_{i
$$-\frac{1}{2}(h\_{ii}^a h\_{jj}^a - (h\_{ij}^a)^2) - \frac{1}{2}(h\_{ii}^{\*a}h\_{jj}^{\*a} - (h\_{ij}^{\*a})^2) \}. \tag{25}$$
$$

Again by the Gauss equation for the Levi–Civita connection, we find that:

$$
\sigma^0 = \frac{m(m-1)\bar{c}}{2} + \sum\_{\substack{a=m+1 \ i$$

or:

$$\sum\_{a=-m+1}^{n} \sum\_{i$$

Inserting (26) into (25), we have:

$$\begin{aligned} \sigma^{\nabla,\nabla^{\ast}} &= \quad 2\sigma^{0} - \frac{m(m-1)\bar{c}}{2} - \frac{1}{2} \sum\_{a=-m+1}^{n} \sum\_{i$$

By subtracting (24) from (27), we can state the following result:

**Lemma 5.** *Let N* = *N*1 <sup>×</sup>f *N*2 *be an m-dimensional statistical warped product submanifold immersed into an n-dimensional statistical manifold of constant sectional curvature c. Then:* ˜

$$\begin{split} \sigma^{\nabla,\nabla^{\ast}} - \mathbb{K}^{\nabla,\nabla^{\ast}} \left( \varepsilon\_{1} \wedge \varepsilon\_{r+1} \right) &= 2(\sigma^{0} - \mathbb{K}^{0} (\varepsilon\_{1} \wedge \varepsilon\_{r+1})) - \frac{(m-2)(m+1)\varepsilon}{2} \\ &- \frac{1}{2} \sum\_{a=m+1}^{n} \sum\_{i$$

Further, we have:

$$\begin{split} \sigma^{\nabla,\nabla^{\*}} - \mathbb{K}^{\nabla,\nabla^{\*}} \left( \boldsymbol{e}\_{1} \wedge \boldsymbol{e}\_{r+1} \right) &\geq 2 \left( \sigma^{0} - \mathbb{K}^{0} (\boldsymbol{e}\_{1} \wedge \boldsymbol{e}\_{r+1}) \right) - \frac{(m-2)(m+1)\varepsilon}{2} \\ &- \frac{1}{2} \sum\_{a=-m+1}^{n} \sum\_{i$$

or we write it as:

$$\begin{split} 2\left(\sigma^{0} - \mathbb{K}^{0}(\boldsymbol{e}\_{1}\wedge\boldsymbol{e}\_{r+1})\right) &\leq \sigma^{\nabla,\nabla^{\*}} - \mathbb{K}^{\nabla,\nabla^{\*}}\left(\boldsymbol{e}\_{1}\wedge\boldsymbol{e}\_{r+1}\right) + \frac{(m-2)(m+1)\tilde{c}}{2} \\ &\quad + \frac{1}{2}\sum\_{\substack{a=m+1\\a\text{ even}}}^{n} \left\{ \sum\_{i$$

We use an optimization technique: For *a* ∈ [*m* + 1, *<sup>n</sup>*], we consider the quadratic forms:

$$
\phi\_{\mathfrak{a}} : \mathbb{R}^{\mathfrak{m}} \to \mathbb{R}, \quad \phi\_{\mathfrak{a}}^{\*} : \mathbb{R}^{\mathfrak{m}} \to \mathbb{R}.
$$

given by:

$$\Phi\_a(h\_{11'}^a, \dots, h\_{mm}^a) \ = \sum\_{i$$

and:

$$\phi\_a^\*(h\_{11'}^{\*a}, \dots, h\_{mm}^{\*a}) \ = \sum\_{i$$

The constrained extremum problem is max *φa* subject to:

$$Q: h\_{11}^a + \dots + h\_{nm}^a = f^a, \quad (t^a \text{ is any constant}).$$

The partial derivatives of *φa* are:

$$\begin{array}{rcl} \frac{\partial \phi\_a}{\partial h\_{11}^a} &=& \sum\_{i=2}^m h\_{ii}^a - h\_{r+1,r+1}^a \\ \frac{\partial \phi\_a}{\partial h\_{r+1,r+1}^a} &=& \sum\_{\substack{i \in \overline{1,m} \ r+1}} h\_{ii}^a - h\_{11}^a \\ \frac{\partial \phi\_a}{\partial h\_{II}^a} &=& \sum\_{\substack{i \in \overline{1,m} \ \{l\}}} h\_{ii}^a & l \in [r+2, m] .\end{array}$$

For an optimal solution (*ha*11, ... , *hamm*) of the above problem and grad (*φa*) normal at *Q*, we obtain:

$$(h\_{11}^a, h\_{22}^a, \dots, h\_{mm}^a) \;=\;(0, a^a, \dots, a^a). \tag{31}$$

As *ta* = ∑*mi*=<sup>1</sup> *haii* = (*m* − <sup>1</sup>)*αa*, then we have:

$$
\alpha^a = \frac{t^a}{m-1}.\tag{32}
$$

As *φa* is obtained from the similar function studied in [39] by subtracting some square terms, *φa*|*Q* will have the Hessian semi-negative definite. Consequently, the point in (31), together with (32) is a global maximum point, and hence, we calculate:

$$\begin{aligned} \phi\_{\mathfrak{a}} &\le \frac{(m-1)(m-2)(\mathfrak{a}^a)^2}{2} \\ &= \frac{(m-2)(t^a)^2}{2(m-1)} = \frac{m^2(m-2)}{2(m-1)}(\mathcal{H}^a)^2. \end{aligned}$$

Similarly, one gets:

$$
\phi\_a^\* \le \frac{m^2(m-2)}{2(m-1)} (\mathcal{H}^{\*a})^2 \lambda
$$

by considering (30) and the constrained extremum problem max *φ*∗*a* subject to:

$$\mathcal{Q}^\* : h\_{11}^{\*a} + \dots + h\_{mm}^{\*a} = |t^{\*a}| \quad (t^{\*a} \text{ is any constant}).$$

Thus, (28) becomes:

$$\begin{split} 2\left(\sigma^{0} - \mathbb{K}^{0}(\mathfrak{e}\_{1} \wedge \mathfrak{e}\_{r+1})\right) &\leq \sigma^{\nabla,\nabla^{\*}} - \mathbb{K}^{\nabla,\nabla^{\*}}(\mathfrak{e}\_{1} \wedge \mathfrak{e}\_{r+1}) + \frac{(m-2)(m+1)\tilde{\varepsilon}}{2} \\ &+ \frac{m^{2}(m-2)}{4(m-1)}(||\mathcal{H}||^{2} + ||\mathcal{H}^{\*}||^{2}). \end{split}$$

By summarizing, we state the following: **Theorem 4.** *Let N* = *N*1 <sup>×</sup>f *N*2 *be an m-dimensional statistical warped product submanifold immersed into an n-dimensional statistical manifold of constant sectional curvature c. Then:* ˜

$$\begin{split} \sigma^{\nabla,\nabla^{\ast}} - \mathbb{K}^{\nabla,\nabla^{\ast}} (\boldsymbol{e}\_{1} \wedge \boldsymbol{e}\_{r+1}) &\geq 2(\sigma^{0} - \mathbb{K}^{0} (\boldsymbol{e}\_{1} \wedge \boldsymbol{e}\_{r+1})) - \frac{(m-2)(m+1)\overline{\sigma}}{2}, \\ &- \frac{m^{2}(m-2)}{4(m-1)} (||\mathcal{H}||^{2} + ||\mathcal{H}^{\ast}||^{2}). \end{split}$$

By using (20), we obtain:

$$\begin{aligned} \mathbb{K}^{\nabla,\nabla^\*} (\mathfrak{e}\_1 \wedge \mathfrak{e}\_{r+1}) &= \frac{1}{2} (\mathbb{K} (\mathfrak{e}\_1 \wedge \mathfrak{e}\_{r+1}) + \mathbb{K}^\* (\mathfrak{e}\_1 \wedge \mathfrak{e}\_{r+1})) \\ &= \frac{1}{2f} \{ (\nabla\_{\mathfrak{e}\_1} \mathfrak{e}\_1) \mathfrak{f} - \mathfrak{e}\_1^2 \mathfrak{f} + (\nabla\_{\mathfrak{e}\_1}^\* \mathfrak{e}\_1) \mathfrak{f} - \mathfrak{e}\_1^2 \mathfrak{f} \}. \end{aligned}$$

For *b* = 1, 2, . . . ,*r*, we also have:

$$\mathbb{K}^{\nabla,\nabla^\*} (\boldsymbol{\varepsilon}\_b \wedge \boldsymbol{\varepsilon}\_{r+1}) \;= \frac{1}{2\mathfrak{f}} \{ (\nabla\_{\boldsymbol{\varepsilon}\_b} \boldsymbol{\varepsilon}\_b) \mathfrak{f} - \boldsymbol{\varepsilon}\_b^2 \mathfrak{f} + (\nabla\_{\boldsymbol{\varepsilon}\_b}^\* \boldsymbol{\varepsilon}\_b) \mathfrak{f} - \boldsymbol{\varepsilon}\_b^2 \mathfrak{f} \}.$$

By summing up *b* from one to *r*, we find that:

$$\sum\_{b=1}^{r} \frac{1}{2\mathfrak{f}} \{ (\nabla\_{\mathfrak{e}\_b} \varepsilon\_b) \mathfrak{f} - \varepsilon\_b^2 \mathfrak{f} + (\nabla\_{\mathfrak{e}\_b}^\* \varepsilon\_b) \mathfrak{f} - \varepsilon\_b^2 \mathfrak{f} \} \ = \frac{1}{2} (\frac{\Delta^{N\_1} \mathfrak{f}}{\mathfrak{f}} + \frac{\Delta^{N\_1 \*} \mathfrak{f}}{\mathfrak{f}}) = \frac{\Delta^{N\_1 0} \mathfrak{f}}{\mathfrak{f}}.$$

where Δ*N*1 and Δ*N*1<sup>∗</sup> are dual Laplacians of *N*1 and Δ*N*1<sup>0</sup> denotes the Laplacian operator of *N*1 for the Levi–Civita connection [37]. Thus, we have:

**Theorem 5.** *Let N* = *N*1 <sup>×</sup>f *N*2 *be an m-dimensional statistical warped product submanifold immersed into an n-dimensional statistical manifold of constant sectional curvature c*˜*. Then, the scalar curvature <sup>σ</sup>*∇,∇∗ *of N satisfies:*

$$\begin{aligned} \sigma^{\nabla,\nabla^\*} &\geq 2\sigma^0 - \frac{\Delta^{N\_10}\mathfrak{f}}{r\mathfrak{f}} - \frac{(m-2)(m+1)\mathfrak{f}}{2}, \\ &- \frac{m^2(m-2)}{4(m-1)}(||\mathcal{H}||^2 + ||\mathcal{H}^\*||^2). \end{aligned}$$

## **6. Optimal Casorati Inequality**

Let {*<sup>e</sup>*1, ... ,*em*} and {*em*+1, ... ,*en*} be respectively the orthonormal basis of *TpN* and *T*⊥*p N*, *p* ∈ *N*. Then, the squared norm of second fundamental forms *h* and *h*∗ is denoted by C and C∗, respectively, called the Casorati curvatures of *N* in *N* ˜ . Therefore, we have:

$$\mathcal{L} = \frac{1}{m}||h||^2, \quad \mathcal{C}^\* = \frac{1}{m}||h^\*||^2. \tag{33}$$

where:

$$||h||^2 = \sum\_{\substack{a=m+1 \ i,j=1}}^n \sum\_{\substack{i,j=1 \ i\neq j}}^m (h\_{ij}^a)^2 \quad ||h^\*||^2 = \sum\_{\substack{a=m+1 \ i,j=1}}^n \sum\_{\substack{i,j=1 \ i\neq j}}^m (h\_{ij}^{\*a})^2 .$$

If *W* is a *q*-dimensional subspace of *TN*, *q* ≥ 2, and {*<sup>e</sup>*1, ... ,*eq*} an orthonormal basis of *W*. Then, the scalar curvature of the *q*-plane section *W* is:

$$\sigma^{\nabla,\nabla^\*}(W) = \sum\_{1 \le i < j \le q} \mathcal{S}(e\_i, e\_j, e\_j, e\_i)\_{\mathcal{H}}$$

and the Casorati curvatures of the subspace *W* are as follows:

$$\mathcal{C}(\mathcal{W}) \;= \frac{1}{q} \sum\_{a=-m+1}^{n} \sum\_{i,j=-1}^{q} (h\_{ij}^a)^2, \; \mathcal{C}^\*(\mathcal{W}) \;= \frac{1}{q} \sum\_{a=-m+1}^{n} \sum\_{i,j=-1}^{q} (h\_{ij}^{\*a})^2.$$

(1) The normalized Casorati curvatures *δ*C (*m* − 1) and *δ*∗C (*m* − 1) are defined as:

$$\begin{aligned} [\delta\_{\mathcal{C}}(m-1)]\_{\mathcal{P}} &= \frac{1}{2} \mathcal{C}\_{\mathcal{P}} + (\frac{m+1}{2m}) \inf \{ \mathcal{C}(\mathcal{W}) | \mathcal{W} \text{ : a hyperplane of } T\_{\mathcal{P}} N \}, \\ \text{and} \quad [\delta\_{\mathcal{C}}^{\*}(m-1)]\_{\mathcal{P}} &= \frac{1}{2} \mathcal{C}\_{\mathcal{P}}^{\*} + (\frac{m+1}{2m}) \inf \{ \mathcal{C}^{\*}(\mathcal{W}) | \mathcal{W} \text{ : a hyperplane of } T\_{\mathcal{P}} N \}. \end{aligned}$$

(2) The normalized Casorati curvatures *δ* C (*m* − 1) and *δ* ∗ C (*m* − 1) are defined as:

$$[\hat{\delta}\_{\mathcal{C}}(m-1)]\_{p} = 2\mathcal{C}\_{p} - (\frac{2m-1}{2m}) \text{sup} \{ \mathcal{C}(\mathcal{W}) | \mathcal{W} \text{ : a hyperplane of } T\_{\mathcal{P}}N \},$$
 
$$\text{and} \quad [\hat{\delta}\_{\mathcal{C}}^{\*}(m-1)]\_{p} = 2\mathcal{C}\_{p}^{\*} - (\frac{2m-1}{2m}) \text{sup} \{ \mathcal{C}^{\*}(\mathcal{W}) | \mathcal{W} \text{ : a hyperplane of } T\_{\mathcal{P}}N \}.$$

Let *ϕ* : *N* = *N*1 <sup>×</sup>f *N*2 → *N* ˜ (*c*˜) be an isometric statistical immersion of a warped product *N*1 <sup>×</sup>f *N*2 into a statistical manifold of constant sectional curvature *c*˜. If we chose a local orthonormal frame {*<sup>e</sup>*1, ... ,*em*} such that {*<sup>e</sup>*1, ... ,*er*} are tangent to *N*1 and {*er*+1, ... ,*er*+*k* = *em*} are tangent to *N*2, then the two partial mean curvature vectors H1 (resp. H∗1) and H2 (resp. H∗2) of *N* are given by:

$$\mathcal{H}\_1 = \frac{1}{r} \sum\_{i=1}^r h(e\_i, e\_i)\_r \\ \quad \mathcal{H}\_1^\* = \frac{1}{r} \sum\_{i=1}^r h^\*(e\_i, e\_i)\_r$$

and:

$$\mathcal{H}\_2 = \frac{1}{k} \sum\_{j=1}^k h(e\_{r+j}, e\_{r+j}), \quad \mathcal{H}\_2^\* = \frac{1}{k} \sum\_{j=1}^k h^\*(e\_{r+j}, e\_{r+j}).$$

Furthermore, the Casorati curvatures are:

$$\mathcal{C}\_1 \;= \frac{1}{r} \sum\_{a=m+1}^n \sum\_{i,j=1}^r (h\_{ij}^a)^2 \; , \quad \mathcal{C}\_1^\* \;= \frac{1}{r} \sum\_{a=m+1}^n \sum\_{i,j=1}^r (h\_{ij}^{\*a})^2 \; \tag{34}$$

and:

$$\mathcal{C}\_2 = \frac{1}{k} \sum\_{a=-m+1}^n \sum\_{i,j=1}^k (h\_{r+ir+j}^a)^2, \quad \mathcal{C}\_2^\* = \frac{1}{k} \sum\_{a=-m+1}^n \sum\_{i,j=1}^k (h\_{r+ir+j}^{\*a})^2. \tag{35}$$

Equation (21) implies:

$$\frac{k\Delta^{N\_10}\mathfrak{f}}{\mathfrak{f}} = \left. \sigma^{\nabla,\nabla^\*} - \sum\_{1 \le i \le j \le r} \mathbb{K}^{\nabla,\nabla^\*} \left( \boldsymbol{e}\_i \wedge \boldsymbol{e}\_j \right) - \sum\_{r+1 \le l \le s \le m} \mathbb{K}^{\nabla,\nabla^\*} \left( \boldsymbol{e}\_l \wedge \boldsymbol{e}\_s \right).$$

By using (8), the previous equation becomes:

$$\begin{split} 2r^{\nabla\_{\Gamma}\nabla^{\*}} &= \frac{k\mathcal{M}^{1}\mathbb{1}\_{\mathsf{f}}}{\mathsf{f}} + r(r-1)\bar{\varepsilon} + k(k-1)\bar{\varepsilon} + 2r^{2}||\mathcal{H}\_{1}^{0}||^{2} \\ &- \frac{r^{2}}{2}(||\mathcal{H}\_{1}||^{2} + ||\mathcal{H}\_{1}^{\*}||^{2}) - \frac{k^{2}}{2}(||\mathcal{H}\_{2}||^{2} + ||\mathcal{H}\_{2}^{\*}||^{2}) \\ &+ 2k^{2}||\mathcal{H}\_{2}^{0}||^{2} - 2r\mathcal{C}\_{1}^{0} + \frac{r}{2}(\mathcal{C}\_{1} + \mathcal{C}\_{1}^{\*}) \\ &- 2k\mathcal{C}\_{2}^{0} + \frac{k}{2}(\mathcal{C}\_{2} + \mathcal{C}\_{2}^{\*}). \end{split} \tag{36}$$

We define a polynomial *P* in terms of the components of the second fundamental form *h*0 (with respect to the Levi–Civita connection) of *N*.

$$\begin{split} P &= 2r(r-1)\mathcal{C}\_{1}^{0} + (r^{2}-1)\mathcal{C}\_{1}^{0}(\mathcal{W}\_{1}) + \frac{r}{2}(\mathcal{C}\_{1} + \mathcal{C}\_{1}^{\*}) \\ &+ 2k(k-1)\mathcal{C}\_{2}^{0} + (k^{2}-1)\mathcal{C}\_{2}^{0}(\mathcal{W}\_{2}) + \frac{k}{2}(\mathcal{C}\_{2} + \mathcal{C}\_{2}^{\*}) \\ &+ \frac{k\Delta^{N\_{1}0}\mathfrak{f}}{\mathfrak{f}} + r(r-1)\tilde{c} + k(k-1)\tilde{c} - \frac{r^{2}}{2}(||\mathcal{H}\_{1}||^{2} + ||\mathcal{H}\_{1}^{\*}||^{2}) \\ &- \frac{k^{2}}{2}(||\mathcal{H}\_{2}||^{2} + ||\mathcal{H}\_{2}^{\*}||^{2}) - 2\sigma^{\nabla,\nabla^{\*}}. \end{split} \tag{37}$$

Without loss of generality, we assume that *W*1 and *W*2 are respectively spanned by {*<sup>e</sup>*1, ... ,*er*−<sup>1</sup>} and {*er*+1,...,*er*+*k*−<sup>1</sup>}. Then, by (36) and (37), we derive:

*P* = *n* ∑ *a* = *m*+1 *r*∑*i*,*j* = 1 *r* + 3 2 (*h*<sup>0</sup>*aij* )2 + *r* + 1 2 *<sup>r</sup>*−1 ∑*<sup>i</sup>*,*j*=<sup>1</sup>(*h*<sup>0</sup>*aij* )2 − 2( ∑*i* = 1 *h*0*aii* )2 + *n* ∑ *a* = *m*+1 *k*∑*l*,*<sup>s</sup>* = 1 *k* + 3 2 (*h*<sup>0</sup>*als* )2 + *k* + 1 2 *k*−1 ∑*<sup>l</sup>*,*s*=<sup>1</sup>(*h*<sup>0</sup>*als* )2 − 2( ∑*l* = 1 *h*0*all* )2 = *n* ∑ *a* = *m*+1 <sup>2</sup>(*r* + 2) ∑ <sup>1</sup>≤*i*<*j*≤*<sup>r</sup>*−<sup>1</sup>(*h*<sup>0</sup>*aij* )2 + (*r* + 3) *<sup>r</sup>*−1 ∑*<sup>i</sup>*=<sup>1</sup>(*h*<sup>0</sup>*air* )2 + *r <sup>r</sup>*−1 ∑ *i*=1 (*h*<sup>0</sup>*aii* )2 − 4 ∑ 1≤*i*<*j*≤*r* (*h*<sup>0</sup>*aii h*0*ajj* ) + *r* − 1 2 (*h*<sup>0</sup>*arr* )<sup>2</sup>} + *n* ∑ *a* = *m*+1 {2(*k* + 2) ∑ 1≤*l*<*s*≤*k*−1 (*h*<sup>0</sup>*als* )2 + (*k* + 3) *k*−1 ∑ *l*=1 (*h*<sup>0</sup>*alk* )2 + *k k*−1 ∑ *l*=1 (*h*<sup>0</sup>*all* )2 − 4 ∑ 1≤*l*<*s*≤*k* (*h*<sup>0</sup>*all h*<sup>0</sup>*ass* ) + *k* − 1 2 (*h*<sup>0</sup>*akk* )2 ≥ *n* ∑ *a* = *m*+1 *<sup>r</sup>*−1 ∑*i* = 1 *r*(*h*<sup>0</sup>*aii* )2 + *r* − 1 2 (*h*<sup>0</sup>*arr* )2 − 4 ∑1≤*i*<*j*≤*r h*0*aii h*0*ajj* + *n* ∑ *a* = *m*+1 *k*−1 ∑*l* = 1 *k*(*h*<sup>0</sup>*all* )2 + *k* − 1 2 (*h*<sup>0</sup>*akk* )2 − 4 ∑1≤*l*<*s*≤*k h*0*all h*<sup>0</sup>*ass* .

For any *a* ∈ { *m* + 1, . . . , *<sup>n</sup>*}, we define two quadratic forms *φa* : R*r* → R and *ϕa* : R*<sup>k</sup>* → R by:

$$\begin{split} \phi\_{a}(h\_{11}^{0a}, h\_{22}^{0a}, \dots, h\_{r-1,r-1}^{0a}, h\_{rr}^{0a}) \\ = \sum\_{i=1}^{r-1} r(h\_{ii}^{0a})^2 + \frac{r-1}{2}(h\_{rr}^{0a})^2 - 4 \sum\_{1 \le i < j \le r} h\_{ii}^{0a} h\_{jj}^{0a} \, , \end{split} \tag{38}$$

and:

$$\begin{split} &\rho\_{a}(h\_{11}^{0a},h\_{22}^{0a},\dots,h\_{k-1,k-1}^{0a},h\_{kk}^{0a}) \\ & \qquad = \sum\_{l=1}^{k-1} k(h\_{ll}^{0a})^2 + \frac{k-1}{2}(h\_{kk}^{0a})^2 - 4\sum\_{1\le l$$

First, we consider the constrained extremum problem min *φa* subject to:

$$Q: h\_{11}^{0a} + \cdots + h\_{rr}^{0a} = \; t^a, \quad (t^a \text{ is any constant}).$$

 From (38), we find that the critical points

$$h^{0\mathbf{c}} = (h^{0a}\_{11\prime}h^{0a}\_{22\prime}\dots, h^{0a}\_{r-1,r-1\prime}h^{0a}\_{rr})^\dagger$$

of *Q* are the solutions of the following system of linear homogeneous equations.

$$\begin{cases} \frac{\partial \phi\_d}{\partial h\_{il}^{0a}} = 2(r+2)(h\_{il}^{0a}) - 4\sum\_{j=-1}^{r} h\_{jj}^{0a} = 0, \\\\ \frac{\partial \phi\_d}{\partial h\_{ll}^{0a}} = (r-1)h\_{rr}^{0a} - 4\sum\_{j=-1}^{r-1} h\_{jj}^{0a} = 0, \end{cases} \tag{40}$$

.

for *i* ∈ {1, 2, . . . ,*r* − 1} and *a* ∈ { *m* + 1, . . . , *<sup>n</sup>*}. Hence, every solution *h*0*c* has:

$$h\_{ii}^{0a} = \frac{1}{r+1}t^a \prime \quad h\_{rr}^{0a} = \frac{4}{r+3}t^a \prime$$

for *i* ∈ {1, 2, . . . ,*r* − 1} and *a* ∈ { *m* + 1, . . . , *<sup>n</sup>*}.

Now, we fix *x* ∈ *Q*. The bilinear form Θ : *TxQ* × *TxQ* → R has the following expression (cf. Theorem 1):

$$\Theta(E, F) = \operatorname{Hess}\_{\Phi\_{\mathsf{A}}}(E, F) + \langle h'(E, F), \gcd(\phi\_{\mathsf{a}})(x) \rangle\_{\mathsf{A}}$$

where *h* denotes the second fundamental form of *Q* in R*r* and < ·, · > denotes the standard inner product on R*<sup>r</sup>*. The Hessian matrix of *φa* is given by:

$$Hess\_{\Phi\_{\sf s}} = \begin{pmatrix} 2(r+2) & -4 & \dots & -4 & -4 \\ -4 & 2(r+2) & \dots & -4 & -4 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ -4 & -4 & \dots & 2(r+2) & -4 \\ -4 & -4 & \dots & -4 & (r-1) \end{pmatrix}$$

Take a vector *E* ∈ *TxQ*, which satisfies a relation ∑*ri*=<sup>1</sup> *Ei* = 0. As the hyperplane is totally geodesic, i.e., *h* = 0 in R*<sup>r</sup>*, we get:

$$\begin{aligned} \Theta(E,E) &= \operatorname{Hess}\_{\phi\_{\mathbb{P}}}(E,E) \\ &= 2(r+2)\sum\_{i=1}^{r-1} E\_i^2 + (r-1)E\_r^2 - 8\sum\_{i:i\neq j=1}^r E\_i E\_j \\ &= 2(r+2)\sum\_{i=1}^{r-1} E\_i^2 + (r-1)E\_r^2 - 4\left\{ (\sum\_{i=1}^r E\_i)^2 - \sum\_{i=1}^r E\_i^2 \right\} \\ &= 2(r+4)\sum\_{i=1}^{r-1} E\_i^2 + (r+3)E\_r^2 \\ &\ge 0. \end{aligned}$$

However, the point *h*0*c* is the only optimal solution, i.e., the global minimum point of problem, and reaches a minimum *Q*(*h*0*c*) = 0 by considering (39) and the constrained extremum problem min *ϕa* subject to:

$$Q': h\_{11}^{\Omega a} + \dots + h\_{kk}^{\Omega a} = \alpha^a, \quad (\alpha^a \text{ is any constant}).$$

Thus, we have:

$$\begin{split} 2\sigma^{\nabla,\nabla^\*} \le & r(r-1)\mathcal{C}\_1^0 + (r^2-1)\mathcal{C}\_1^0(\mathcal{W}\_1) + \frac{r}{2}(\mathcal{C}\_1 + \mathcal{C}\_1^\*) \\ &+ k(k-1)\mathcal{C}\_2^0 + (k^2-1)\mathcal{C}\_2^0(\mathcal{W}\_2) + \frac{k}{2}(\mathcal{C}\_2 + \mathcal{C}\_2^\*) \\ &+ \frac{k\Delta^{N\_10}\mathfrak{f}}{\mathfrak{f}} + r(r-1)\bar{\varepsilon} + k(k-1)\bar{\varepsilon} \\ &- \frac{r^2}{2}(||\mathcal{H}\_1||^2 + ||\mathcal{H}\_1^\*||^2) - \frac{k^2}{2}(||\mathcal{H}\_2||^2 + ||\mathcal{H}\_2^\*||^2). \end{split}$$

Consequently, we ge<sup>t</sup> immediately the following theorem from the above relation:

**Theorem 6.** *Let N* = *N*1 <sup>×</sup>f *N*2 *be an m-dimensional statistical warped product submanifold immersed into an n-dimensional statistical manifold of constant sectional curvature c. Then, the Casorati curvatures satisfy:* ˜

$$\begin{split} 2\sigma^{\nabla,\nabla^{4}} \leq & r(r-1)C\_{1}^{0} + (r^{2}-1)C\_{1}^{0}(\mathcal{W}\_{1}) + rC\_{1}^{0} \\ &+ k(k-1)C\_{2}^{0} + (k^{2}-1)C\_{2}^{0}(\mathcal{W}\_{2}) + kC\_{2}^{0} \\ &+ \frac{k\Delta^{N\_{1}0}\mathfrak{f}}{\mathfrak{f}} + r(r-1)\bar{\varepsilon} + k(k-1)\bar{\varepsilon} \\ &- \frac{r^{2}}{2}(||\mathcal{H}\_{1}||^{2} + ||\mathcal{H}\_{1}^{\*}||^{2}) - \frac{k^{2}}{2}(||\mathcal{H}\_{2}||^{2} + ||\mathcal{H}\_{2}^{\*}||^{2}), \end{split}$$

*where W*1 *and W*2 *are respectively the hyperplanes of TpN*1 *and TpN*<sup>2</sup>*,* C01 = 12 (C1 + C∗1 )*, and* C02 = 12 (C2 + C∗2 )*.*
