**2. Preliminaries**

Let (*M*˜ , *g*˜) be a 2*n*-dimensional manifold, ∇˜ an affine connection on *M*˜ , and *g*˜ a Riemannian metric on *M* ˜ . Consider *T* ˜ ∈ Γ(*TM*˜ (1,2)) the torsion tensor field of ∇˜ .

A pair (∇˜ , *g*˜) is called a *statistical structure* on *M*˜ if the torsion tensor field *T*˜ vanishes and ∇ ˜ *g* ˜ ∈ Γ(*TM*˜ (0,3)) is symmetric.

A Riemannian manifold (*M*˜ , *g*˜) is called a *statistical manifold* if it is endowed with a pair of torsion-free affine connections ∇ ˜ and ∇ ˜ ∗ satisfying

$$Z \circ (X, Y) = \overline{\mathfrak{g}}(\nabla\_Z X, Y) + \overline{\mathfrak{g}}(X, \nabla\_Z^\* Y)\_r$$

for any *X*, *Y*, *Z* ∈ Γ(*TM*˜ ). Denote (*M*˜ , *g*˜, ∇˜ ) as the statistical manifold. The connections ∇˜ and ∇˜ ∗ are named *dual connections* or *conjugate connections*.

**Remark 1.** *If* (*M,* ˜ *g,*˜ ∇˜ ) *is a statistical manifold, then we remark that*


$$
\vec{\nabla} + \vec{\nabla}^\* = 2\vec{\nabla}^0,\tag{1}
$$

*where* ∇ ˜ 0 *is the Levi–Civita connection on M.* ˜

Let *M* be an *m*-dimensional submanifold of a 2*n*-dimensional statistical manifold (*M*˜ , *g*˜) and *g* the induced metric on *M*. The Gauss formulas are given by

$$\begin{aligned} \nabla\_X \mathcal{Y} &= \nabla\_X \mathcal{Y} + h(X, \mathcal{Y})\_\prime \\ \bar{\nabla}\_X^\* \mathcal{Y} &= \nabla\_X^\* \mathcal{Y} + h^\*(X, \mathcal{Y})\_\prime \end{aligned}$$

for any *X*,*Y* ∈ <sup>Γ</sup>(*TM*), where *h* and *h*∗ are symmetric and bilinear (0, 2)-tensors, called the *imbedding curvature tensor* of *M* in *M* ˜ for ∇ ˜ and ∇ ˜ ∗, respectively.

Denote the curvature tensor fields of ∇ and ∇ ˜ by *R* and *R* ˜ , respectively. Then, the *Gauss equation* concerning the connection ∇ ˜ is ([41])

$$\overline{\mathfrak{g}}(\overline{\mathcal{R}}(X,\mathcal{Y})Z,\mathcal{W}) = \overline{\mathfrak{g}}(\mathcal{R}(X,\mathcal{Y})Z,\mathcal{W}) + \overline{\mathfrak{g}}(h(X,Z),h^\*(\mathcal{Y},\mathcal{W})) - \overline{\mathfrak{g}}(h^\*(X,\mathcal{W}),h(\mathcal{Y},Z)),\tag{2}$$

for any *X*,*Y*, *Z*, *W* ∈ <sup>Γ</sup>(*TM*).

In addition, denote the curvature tensor fields of the connections ∇∗ and ∇ ˜ ∗ by *R*∗ and *R* ˜ ∗ , respectively. Then the *Gauss equation* concerning the connection ∇ ˜ ∗ is ([41])

$$\overline{\varrho}(\overline{R}^\*(X,\mathcal{Y})Z,\mathcal{W}) = \overline{\varrho}(R^\*(X,\mathcal{Y})Z,\mathcal{W}) + \overline{\varrho}(h^\*(X,Z),h(\mathcal{Y},\mathcal{W})) - \overline{\varrho}(h(X,\mathcal{W}),h^\*(\mathcal{Y},Z)),\tag{3}$$

for any *X*,*Y*, *Z*, *W* ∈ <sup>Γ</sup>(*TM*).

If *M* is a submanifold of a statistical manifold (*M* ˜ , *g* ˜ , ∇ ˜ ), then (*<sup>M</sup>*, *g*, ∇) is also a statistical manifold with the induced metric *g* and the induced connection ∇.

Let *S* be the *statistical curvature tensor field of a statistical manifold* (*<sup>M</sup>*, *g*, ∇), where *S* ∈ Γ(*TM*(1,3)) is defined by [40]

$$S(X,Y)Z = \frac{1}{2}\{R(X,Y)Z + R^\*(X,Y)Z\},\tag{4}$$

for *X*,*Y*, *Z* ∈ <sup>Γ</sup>(*TM*).

If *π* = *span*R{*<sup>u</sup>*1, *<sup>u</sup>*2} is a 2-dimensional subspace of *TpM*, for *p* ∈ *M*, then the *sectional curvature* of *M* is defined by [40]:

$$\sigma(\pi) = \frac{g(\mathcal{S}(u\_1, u\_2)u\_2, u\_1)}{g(u\_1, u\_1)g(u\_2, u\_2) - g^2(u\_1, u\_2)}.\tag{5}$$

Let {*<sup>e</sup>*1, ...,*em*} be an orthonormal basis of the tangent space *TpM*, for *p* ∈ *M*, and let {*em*+1, ...,*e*2*n*} be an orthonormal basis of the normal space *T*⊥*p M*. The *scalar curvature τ* at *p* is given by

$$\pi(p) = \sum\_{1 \le i < j \le m} \sigma(\varepsilon\_i \wedge \varepsilon\_j) = \sum\_{1 \le i < j \le m} \lg(S(\varepsilon\_i, \varepsilon\_j) e\_{j \prime} e\_i),\tag{6}$$

and the *normalized scalar curvature ρ* of *M* is defined as

$$
\rho = \frac{2\pi}{m(m-1)}.\tag{7}
$$

The *mean curvature* vector fields of *M*, denoted by *H* and *H*<sup>∗</sup>, are given by

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

From Equation (1), we ge<sup>t</sup> 2*h*<sup>0</sup> = *h* + *h*∗ and 2*H*<sup>0</sup> = *H* + *H*<sup>∗</sup>, where *h*0 and *H*<sup>0</sup> are the second fundamental form and the mean curvature field of *M*, respectively, with respect to the Levi–Civita connection ∇<sup>0</sup> on *M*.

The *squared mean curvatures* of the submanifold *M* in *M* ˜ have the expressions

$$\left\|\left\|\boldsymbol{H}\right\|\right\|^{2} = \frac{1}{m^{2}} \sum\_{a=m+1}^{2n} \left(\sum\_{i=1}^{m} h\_{ii}^{a}\right)^{2}, \quad \left\|\left\|\boldsymbol{H}^{\*}\right\|\right\|^{2} = \frac{1}{m^{2}} \sum\_{a=m+1}^{2n} \left(\sum\_{i=1}^{m} h\_{ii}^{\*a}\right)^{2},$$

where *hαij* = *<sup>g</sup>*˜(*h*(*ei*,*ej*),*eα*) and *<sup>h</sup>*<sup>∗</sup>*<sup>α</sup>ij* = *<sup>g</sup>*˜(*h*∗(*ei*,*ej*),*eα*), for *i*, *j* ∈ {1, ..., *<sup>m</sup>*}, *α* ∈ {*m* + 1, ..., <sup>2</sup>*n*}.

Denote by C and C∗ the *Casorati curvatures* of the submanifold *M*, defined by the squared norms of *h* and *h*<sup>∗</sup>, respectively, over the dimension *m*, as follows:

$$\begin{aligned} \mathcal{C} &= \frac{1}{m} ||h||^2 = \frac{1}{m} \sum\_{\alpha=m+1}^{2n} \sum\_{i,j=1}^{m} \left( h\_{ij}^{\alpha} \right)^2, \\\\ \mathcal{C}^\* &= \frac{1}{m} ||h^\*||^2 = \frac{1}{m} \sum\_{\alpha=m+1}^{2n} \sum\_{i,j=1}^{m} \left( h\_{ij}^{\*\alpha} \right)^2. \end{aligned}$$

Let *L* be an *s*-dimensional subspace of *TpM*, *s* ≥ 2 and let {*<sup>e</sup>*1, ... ,*es*} be an orthonormal basis of *L*. Hence, the Casorati curvatures C(*L*) and C∗(*L*) of *L* are given by

$$\mathcal{C}(L) = \frac{1}{s} \sum\_{\substack{\mathfrak{a} = m+1 \ i,j=1}}^{2n} \sum\_{i,j=1}^{s} \left( h\_{ij}^{\mathfrak{a}} \right)^2, \quad \mathcal{C}^\*(L) = \frac{1}{s} \sum\_{\substack{\mathfrak{a} = m+1 \ i,j=1}}^{2n} \sum\_{i,j=1}^{s} \left( h\_{ij}^{\ast \mathfrak{a}} \right)^2.$$

The *normalized δ-Casorati curvatures δ*C (*m* − 1) and ˆ*δ*C (*m* − 1) of the submanifold *Mn* are given by

$$\left. \delta\_{\mathcal{C}}(m-1) \right|\_{p} = \frac{1}{2} \mathcal{C} \left|\_{p} + \frac{m+1}{2m} \inf \{ \mathcal{C}(L) | L \text{ a hyperplane of } T\_{p}M \} \right|$$

and

$$\left. \delta\_{\mathcal{C}}(m-1) \right|\_{p} = 2\mathcal{C} \mid\_{p} - \frac{2m-1}{2m} \text{sup} \{ \mathcal{C}(L) \, | \, L \text{ a hyperplane of } T\_p M \} .$$

Moreover, the *dual normalized δ*∗*-Casorati curvatures δ*∗C (*m* − 1) and *δ* ∗ C (*m* − 1) of the submanifold *M* in *M* ˜ are defined as

$$\left. \delta\_{\mathcal{C}}^{\*} (m - 1) \right|\_{p} = \frac{1}{2} \mathcal{C}^{\*} \left. \vert\_{p} + \frac{m + 1}{2m} \text{inf} \left\{ \mathcal{C}^{\*} (L) \vert L \text{ a hyperplane of } T\_{p}M \right\} \right|\_{p}$$

and

$$|\hat{\delta}^\*\_{\mathcal{C}}(m-1)|\_{\mathcal{P}} = 2\mathcal{C}^\* \, |\_{\mathcal{P}} - \frac{2m-1}{2m} \text{sup} \{ \mathcal{C}^\*(L) | L \text{ a hyperplane of } T\_{\mathcal{P}}M \} .$$

Denote by *δ*C (*r*; *m* − 1) and ˆ*δ*C (*r*; *m* − <sup>1</sup>), the *generalized normalized δ-Casorati curvatures* of *M*, defined in [10] as

$$\delta\_{\mathcal{C}}(r; m - 1)|\_{\mathcal{P}} = r \mathcal{C} \mid\_{\mathcal{P}} + a(r) \inf \{ \mathcal{C}(L) \mid L \text{ a hyperplane of } T\_{\mathcal{P}}M \}\_r$$

if 0 < *r* < *m*(*m* − <sup>1</sup>), and

$$\delta\_{\mathcal{C}}(r; m - 1)|\_p = r \mathcal{C} \mid\_p + a(r) \sup \{ \mathcal{C}(L) \mid L \text{ a hyperplane of } T\_p M \} \,,$$

if *r* > *m*(*m* − <sup>1</sup>), for *a*(*r*) set as

$$a(r) = \frac{(m-1)(r+m)(m^2-m-r)}{mr}.$$

where *r* ∈ R+ and *r* = *m*(*m* − <sup>1</sup>).

Furthermore, denote by *δ*∗C (*r*; *m* − 1) and ˆ*δ*∗C (*r*; *m* − 1) the *dual generalized normalized δ*∗*-Casorati curvatures* of the submanifold *M*, defined as follows:

$$\left|\delta^\*\_{\mathcal{C}}(r; m - 1)\right|\_{\mathcal{P}} = r \mathcal{C}^\* \left| \, \_{\mathcal{P}} + a(r) \inf \{ \mathcal{C}^\*(L) \mid L \text{ a hyperplane of } T\_{\mathcal{P}}M \} \right|\_{\mathcal{P}}$$

if 0 < *r* < *m*(*m* − <sup>1</sup>), and

$$|\delta^\*\_{\mathcal{C}}(r; m - 1)|\_p = r \cdot \mathcal{C}^\* \mid\_p + a(r) \sup \{ \mathcal{C}^\*(L) \mid L \text{ a hyperplane of } T\_p M \},$$

 if *r* > *m*(*m* − <sup>1</sup>), for *a*(*r*) set above.

A statistical submanifold (*<sup>M</sup>*, *g*, ∇) of (*M*˜ , *g*˜, ∇˜ ) is called *totally geodesic* with respect to the connection ∇ ˜ if the second fundamental form *h* of *M* for ∇ ˜ vanishes identically [40].

Let *M* ˜ be an almost complex manifold with almost complex structure *J* ∈ Γ(*TM*˜ (1,1)). A quadruplet (*M*˜ , ∇˜ , *g*˜, *J*) is called a *holomorphic statistical manifold* if


where *ω* is defined by *<sup>ω</sup>*(*<sup>X</sup>*,*<sup>Y</sup>*) = *g*˜(*<sup>X</sup>*, *JY*), for any *X*,*Y* ∈ Γ(*TM*˜ ).

For a holomorphic statistical manifold, the following formula holds:

$$\overline{\mathfrak{g}}(\overline{\mathcal{S}}(Z,\mathcal{W})|Y,JX) = \overline{\mathfrak{g}}(\overline{\mathcal{S}}(fZ,J\mathcal{W})Y,X) = \overline{\mathfrak{g}}(\overline{\mathcal{S}}(Z,\mathcal{W})Y,X),\tag{8}$$

for any *X*,*Y*, *Z*, *W* ∈ Γ(*TM*˜ ).

A holomorphic statistical manifold (*M* ˜ , ∇ ˜ , *g* ˜ , *J*) is said to be of *constant holomorphic sectional curvature c* ∈ R if the following formula holds [42]:

$$\bar{S}(\mathbf{X},\mathbf{Y})Z = \frac{c}{4}\{\tilde{\varrho}(\mathbf{Y},\mathbf{Z})\mathbf{X} - \tilde{\varrho}(\mathbf{X},\mathbf{Z})\mathbf{Y} + \tilde{\varrho}(\mathbf{J}\mathbf{Y},\mathbf{Z})\mathbf{J}\mathbf{X} - \tilde{\varrho}(\mathbf{J}\mathbf{X},\mathbf{Z})\mathbf{J}\mathbf{Y} + 2\tilde{\varrho}(\mathbf{X},\mathbf{J}\mathbf{Y})\mathbf{J}\mathbf{Z}\},\tag{9}$$

for any *X*,*Y*, *Z* ∈ Γ(*TM*˜ ), where *S*˜ is the statistical curvature tensor field of *M*˜ .

**Remark 2** ([43])**.** *Let (M* ˜ , *g* ˜ , *J) be a Kähler manifold. If we define a connection* ∇ ˜ *as* ∇ ˜ = ∇*g* ˜ + *K, where K* ∈ Γ(*TM*˜ (1,2)) *satisfying the conditions*

$$K(X,Y) = K(Y,X),\tag{10}$$

$$\overline{\mathcal{g}}(\mathcal{K}(X,Y),Z) = \overline{\mathcal{g}}(Y,\mathcal{K}(X,Z)),\tag{11}$$

$$K(X, fY) = -fK(X, Y)\_\prime \tag{12}$$

*for any X*,*Y*, *Z* ∈ Γ(*TM*˜ )*, then (M*˜ , ∇˜ , *g*˜, *J) is a holomorphic statistical manifold.*

Let *M* be an *m*-dimensional statistical submanifold of a holomorphic statistical manifold (*M*˜ , ∇˜ , *g*˜, *J*). For any vector field *X* tangent to *M* we can decompose

$$fX = PX + FX\_{\prime} \tag{13}$$

where *PX* and *FX* are the tangent component and the normal component, respectively, of *JX*. Given a local orthonormal frame {*<sup>e</sup>*1,*e*2, ··· ,*em*} of *M*, then the squared norm of *P* is expressed by

$$\|\|P\|\|^2 = \sum\_{i,j=1}^m \mathbf{g}^2(Pe\_{i\prime}e\_j).$$

Next, we consider the constrained extremum problem

$$\min\_{x \in M} f(x),\tag{14}$$

where *M* is a Riemannian submanifold of a Riemannian manifold (*M*˜ , *g*˜), and *f* : *M*˜ → R is a function of differentiability class *C*2.

**Theorem 1** ([44])**.** *If M is complete and connected,* (*grad f*)(*p*) ∈ *T*⊥*p M for a point p* ∈ *M, and the bilinear form* A : *TpM* × *TpM* → R *defined by*

$$\mathcal{A}(X,Y) = \text{Hess}(f)(X,Y) + \tilde{\mathfrak{g}}(h^0(X,Y), \text{grad}f),\tag{15}$$

*is positive definite in p, then p is the optimal solution of the Problem* (14)*.*

**Remark 3** ([44])**.** *If the bilinear form* A *defined by Equation* (15) *is positive semi-definite on the submanifold M, then the critical points of f* |*M are global optimal solutions of the Problem* (14)*.*

## **3. Main Inequalities**

**Theorem 2.** *Let M be an m-dimensional statistical submanifold of a* 2*n-dimensional holomorphic statistical manifold (M* ˜ , ∇ ˜ , *g* ˜ , *J) of constant holomorphic sectional curvature c. Then we have (i)*

$$\begin{aligned} 2\tau &\le \quad \delta^0\_{\tilde{C}}(r; m - 1) + m\mathcal{C}^0 - 2m^2||H^0||^2 \\ &+ \quad m^2\tilde{\varrho}(H, H^\*) + \frac{3c}{4}||P||^2 + \frac{c}{4}m(m - 1), \end{aligned} \tag{16}$$

*for any real number r such that* 0 < *r* < *m*(*m* − <sup>1</sup>)*, where δ*0C (*r*; *m* − 1) = *δ*C (*<sup>r</sup>*;*m*−<sup>1</sup>)+*δ*∗C (*<sup>r</sup>*;*m*−<sup>1</sup>) 2 *and* C<sup>0</sup> = C+C∗ 2 *; and*

*(ii)*

$$\begin{aligned} 2\pi &\le \quad \tilde{\mathcal{C}}\_{\mathcal{C}}^{0}(r; m - 1) + m\mathcal{C}^{0} - 2m^{2}||H^{0}||^{2} \\ &+ \quad m^{2}\tilde{\mathcal{g}}(H, H^{\*}) + \frac{3c}{4}||P||^{2} + \frac{c}{4}m(m - 1), \end{aligned} \tag{17}$$

*for any real number r such that r* > *m*(*m* − <sup>1</sup>)*, where* ˆ*δ*0C (*r*; *m* − 1) = ˆ *δ*C (*<sup>r</sup>*;*m*−<sup>1</sup>)+ ˆ*δ*∗C (*<sup>r</sup>*;*m*−<sup>1</sup>) 2 .

*Moreover, the equality cases of Inequalities* (16) *and* (17) *hold identically at all points p* ∈ *M if and only if the following condition is satisfied:*

$$h + h^\* = 0,\tag{18}$$

*where h and h*∗ *are the imbedding curvature tensors of the submanifold associated to the dual connections* ∇ ˜ *and* ∇ ˜ ∗*, respectively.*

**Proof.** The relations (Equations (2)–(4)) imply

$$\begin{aligned} 2\overline{\mathfrak{g}}(\overline{\mathcal{S}}(X,Y)Z,\mathcal{W}) &= 2\overline{\mathfrak{g}}(\mathcal{S}(X,Y)Z,\mathcal{W}) - \overline{\mathfrak{g}}(h(Y,Z),h^\*(X,\mathcal{W})) + \overline{\mathfrak{g}}(h(X,Z),h^\*(Y,\mathcal{W})) \\ &- \overline{\mathfrak{g}}(h^\*(Y,Z),h(X,\mathcal{W})) + \overline{\mathfrak{g}}(h^\*(X,Z),h(Y,\mathcal{W})), \end{aligned} \tag{19}$$

where *X*,*Y*, *Z*, *W* ∈ <sup>Γ</sup>(*TM*).

For *p* ∈ *M*, we choose {*<sup>e</sup>*1, ...,*em*} and {*em*+1, ...,*e*2*n*} orthonormal bases of *TpM* and *T*⊥*p M*, respectively. For *X* = *Z* = *ei* and *Y* = *W* = *ej* with *i*, *j* ∈ {1, ..., *<sup>m</sup>*}, from the Equation (19), it follows that

$$2\tau(p) \quad = \quad m^2 \overline{\g}(H, H^\*) - \sum\_{1 \le i, j \le m} \overline{\g}(h^\*(e\_i, e\_j), h(e\_i, e\_j)) \tag{20}$$

$$+ \frac{c}{4} (m^2 - m + 3\|P\|^2).$$

Denoting 2*H*<sup>0</sup> = *H* + *H*∗ and 2C<sup>0</sup> = C + C∗, Equation (20) becomes

$$\begin{aligned} 2\pi(p) &= 2m^2 \|H^0\|^2 - \frac{m^2}{2} \|H\|^2 - \frac{m^2}{2} \|H^\*\|^2 \\ &- 2m\mathcal{C}^0 + \frac{m}{2} (\mathcal{C} + \mathcal{C}^\*) + \frac{c}{4} (m^2 - m + 3\|P\|^2). \end{aligned} \tag{21}$$

Let P be the quadratic polynomial defined by

$$\begin{aligned} \mathcal{P} &= -r\mathcal{C}^0 + a(r)\mathcal{C}^0(L) + \frac{m}{2}(\mathcal{C} + \mathcal{C}^\*) - \frac{m^2}{2}(\|H\|^2 + \|H^\*\|^2) \\ &- 2\tau(p) + \frac{c}{4}(m^2 - m + 3\|P\|^2), \end{aligned} \tag{22}$$

where *L* is a hyperplane of *TpM*.

We consider that the hyperplane *L* is spanned by the tangent vectors *e*1, ...,*em*−1, without loss of generality. Therefore, we ge<sup>t</sup>

$$\mathcal{P} = \sum\_{a=m+1}^{2n} \left[ \frac{2m+r}{m} \sum\_{i,j=1}^{m} (h\_{ij}^{0a})^2 + a(r) \frac{1}{m-1} \sum\_{i,j=1}^{m-1} (h\_{ij}^{0a})^2 - 2 \left( \sum\_{i=1}^{m} h\_{ii}^{0a} \right)^2 \right]. \tag{23}$$

Then, Equation (23) yields

$$\begin{split} \mathcal{P} &= \sum\_{a=m+1}^{2n} \left\{ \left[ \frac{2(2m+r)}{m} + \frac{2a(r)}{m-1} \right] \sum\_{1 \le i < j \le m-1} (h\_{ij}^{0\alpha})^2 + \left[ \frac{2(2m+r)}{m} + \frac{2a(r)}{m-1} \right] \sum\_{i=1}^{m-1} (h\_{iim}^{0\alpha})^2 \right\} \\ &+ \left( \frac{2m+r}{m} + \frac{a(r)}{m-1} - 2 \right) \sum\_{i=1}^{m-1} (h\_{ii}^{0\alpha})^2 \\ &- 4 \sum\_{1 \le i < j \le m} h\_{ii}^{0\alpha} h\_{jj}^{0\alpha} + \left( \frac{2m+r}{m} - 2 \right) (h\_{mm}^{0\alpha})^2 \Big] \\ &\geq \sum\_{a=m+1}^{2n} \left[ \frac{r(m-1) + a(r)m}{m(m-1)} \sum\_{i=1}^{m-1} (h\_{ii}^{0\alpha})^2 + \left( \frac{r}{m} \right) (h\_{mm}^{0\alpha})^2 - 4 \sum\_{1 \le i < j \le m} h\_{ii}^{0\alpha} h\_{jj}^{0\alpha} \right]. \end{split}$$

Let *fα* be a quadratic form defined by *fα* : R*m* → R for any *α* ∈ {*m* + 1, ..., <sup>2</sup>*n*},

$$\begin{aligned} f\_a(h\_{11}^{\text{On}}, h\_{22}^{\text{On}}, \dots, h\_{mm}^{\text{On}}) \quad &= \sum\_{i=1}^{m-1} \frac{r(m-1) + a(r)m}{m(m-1)} (h\_{ii}^{\text{On}})^2 \\ &+ \frac{r}{m} (h\_{mm}^{\text{On}})^2 - 4 \sum\_{1 \le i < j \le m} h\_{ii}^{\text{On}} h\_{jj}^{\text{Out}} \dots \end{aligned}$$

We investigate the constrained extremum problem

> min *fα*

with the constraint

$$Q: h\_{11}^{\Omega\alpha} + h\_{22}^{\Omega\alpha} + \dots + h\_{mm}^{\Omega\alpha} = k^{\alpha},$$

where *kα* is a real constant.

> We obtain the system of first-order partial derivatives:

$$\begin{cases} \frac{\partial f\_{\alpha}}{\partial h\_{ii}^{0\alpha}} = 2 \frac{r(m-1) + a(r)m}{m(m-1)} h\_{ii}^{0\alpha} - 4 \left( \sum\_{k=1}^{m} h\_{kk}^{0\alpha} - h\_{ii}^{0\alpha} \right) = 0\\ \frac{\partial f\_{\alpha}}{\partial h\_{mmu}^{0\alpha}} = \frac{2r}{m} h\_{mmu}^{0\alpha} - 4 \sum\_{k=1}^{m-1} h\_{kk}^{0\alpha} = 0, \end{cases}$$

for every *i* ∈ {1, ..., *m* − <sup>1</sup>}, *α* ∈ {*m* + 1, ..., <sup>2</sup>*n*}.

It follows that the constrained critical point is

$$h\_{ii}^{0\alpha} = \frac{2m(m-1)}{(m-1)(2m+r) + ma(r)}k^{\alpha}$$

$$h\_{mm}^{0\alpha} = \frac{2m}{2m+r}k^{\alpha},$$

for any *i* ∈ {1, ..., *m* − <sup>1</sup>}, *α* ∈ {*m* + 1, ..., <sup>2</sup>*n*}.

> For *p* ∈ Q, let A be a 2-form, A : *TpQ* × *TpQ* → R defined by

$$\mathcal{A}(X,Y) = \text{Hess}(f\_a)(X,Y) + \langle h'(X,Y), (\text{grad} f\_a)(p) \rangle\_{\text{tr}}$$

where *h* is the second fundamental form of *Q* in R*m*+<sup>1</sup> and ·,· is the standard inner product on R*<sup>m</sup>*.

The Hessian matrix of *fα* is given by

$$\text{Hess}(f\_{\alpha}) = \begin{pmatrix} \lambda & -4 & \dots & -4 & -4 \\ -4 & \lambda & \dots & -4 & -4 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ -4 & -4 & \dots & \lambda & -4 \\ -4 & -4 & \dots & -4 & \frac{2r}{m} \end{pmatrix} \times$$

where *λ* = 2 (*<sup>m</sup>*−<sup>1</sup>)(*r*+2*m*)+*ma*(*r*) *<sup>m</sup>*(*<sup>m</sup>*−<sup>1</sup>) is a real constant.

The condition ∑*mi*=<sup>1</sup> *Xi* = 0 is satisfied, for a vector field *X* ∈ *TpQ*, as the hyperplane *Q* is totally geodesic in R*<sup>m</sup>*. Then, we achieve

$$\begin{split} \mathcal{A}(X,X) &= \lambda \sum\_{i=1}^{m-1} X\_i^2 + \frac{2r}{m} X\_m^2 - 8 \sum\_{\substack{i,j=1 \ (i \neq j)}}^m X\_i X\_j \\ &= \lambda \sum\_{i=1}^{m-1} X\_i^2 + \frac{2r}{m} X\_m^2 + 4 \left( \sum\_{i=1}^m X\_i \right)^2 - 8 \sum\_{\substack{i,j=1 \ (i \neq j)}}^m X\_i X\_j \\ &= \lambda \sum\_{i=1}^{m-1} X\_i^2 + \frac{2r}{m} X\_m^2 + 4 \sum\_{i=1}^m X\_i^2 \\ &\ge 0. \end{split}$$

Applying Remark 3, the critical point (*h*0*<sup>α</sup>*11, ..., *<sup>h</sup>*0*<sup>α</sup>mm*) of *fα* is the global minimum point of the problem. Since *fα*(*h*0*<sup>α</sup>*11, ..., *<sup>h</sup>*0*<sup>α</sup>mm*) = 0, we ge<sup>t</sup> P ≥ 0.

We have then proved Inequalities (16) and (17), considering infimum and supremum, respectively, over all tangent hyperplanes *L* of *TpM*.

In addition, we study the equality cases of Inequalities (16) and (17). First, we find out the critical points of P

$$h^c = (h\_{11}^{0\,m+1}, h\_{12}^{0\,m+1}, \dots, h\_{m\,m}^{0\,m+1}, \dots, h\_{11}^{0\,2n}, \dots, h\_{m\,m}^{0\,2n})^c$$

as the solutions of following system of linear homogeneous equations:

$$\begin{cases} \frac{\partial \mathcal{P}}{\partial l\_{ii}^{\alpha \alpha}} = 2 \left[ \frac{2m+r}{m} + \frac{a(r)}{m-1} - 2 \right] h\_{ii}^{0\alpha} - 4 \sum\_{k \neq i,k=1}^{m} h\_{kk}^{0\alpha} = 0, \\\\ \frac{\partial \mathcal{P}}{\partial l\_{mm}^{0\alpha}} = 2 \frac{r}{m} h\_{mm}^{0\alpha} - 4 \sum\_{k=1}^{m-1} h\_{kk}^{0\alpha} = 0, \\ \frac{\partial \mathcal{P}}{\partial l\_{ij}^{0\alpha}} = 4 \left[ \frac{2m+r}{m} + \frac{a(r)}{m-1} \right] h\_{ij}^{0\alpha} = 0, \ i \neq j, \\ \frac{\partial \mathcal{P}}{\partial l\_{im}^{0\alpha}} = 4 \left[ \frac{2m+r}{m} + \frac{a(r)}{m-1} \right] h\_{im}^{0\alpha} = 0. \end{cases}$$

The critical points satisfy *h*0*αij* = 0, with *i*, *j* ∈ {1, ..., *m*} and *α* ∈ {*m* + 1, ..., <sup>2</sup>*n*}. On the other hand, we know that P ≥ 0 and P(*hc*) = 0, then the critical point *hc* is a minimum point of P. Consequently, the cases of equality hold in both Inequalities (16) and (17) if and only if *hαij* = <sup>−</sup>*h*<sup>∗</sup>*<sup>α</sup>ij* , for *i*, *j* ∈ {1, ..., *<sup>m</sup>*}, *α* ∈ {*m* + 1, ..., <sup>2</sup>*n*}.

**Remark 4.** *Under Equation* (18)*, the submanifold M is totally geodesic with respect to the Levi–Civita connection* ∇ ˜ 0*. Then, the equality cases of Inequalities (16) and (17) hold for all unit tangent vectors at p if and only if p is a totally geodesic point with respect to the Levi–Civita connection.*

By virtue of Theorem 2, the generalized normalized *δ*-Casorati curvatures satisfy Inequalities (16) and (17). If the normalized *δ*-Casorati curvatures *δ*C (*m* − 1) and *δ*∗C (*m* − <sup>1</sup>), respectively, ˆ*δ*C (*m* − 1) and ˆ *δ*∗C(*m* − 1) are involved, then we can state the following result.

**Corollary 1.** *Let M be an m-dimensional statistical submanifold of a* 2*n-dimensional holomorphic statistical manifold (M* ˜ , ∇ ˜ , *g* ˜ , *J) of constant holomorphic sectional curvature c. Then, we have*

*(i)*

$$\begin{split} \rho &\quad \leq \quad \delta\_{\mathcal{C}}^{0}(m-1) + \frac{1}{m-1} \mathcal{C}^{0} - \frac{2m}{m-1} ||H^{0}||^{2} \\ &\quad + \quad \frac{m}{m-1} \mathfrak{F}(H, H^{\*}) + \frac{3c}{4m(m-1)} ||P||^{2} + \frac{c}{4} .\end{split} \tag{24}$$

*where* <sup>2</sup>*δ*0C (*m* − 1) = *δ*C (*m* − 1) + *δ*∗C (*m* − 1) *and* 2C<sup>0</sup> = C + C∗*, and (ii)*

$$\begin{split} \rho &\quad \leq \quad \delta\_C^0(m-1) + \frac{1}{m-1} \mathcal{C}^0 - \frac{2m}{m-1} \|H^0\|^2 \\ &\quad + \quad \frac{m}{m-1} \overline{\mathfrak{z}}(H, H^\*) + \frac{3c}{4m(m-1)} \|P\|^2 + \frac{c}{4} \end{split} \tag{25}$$

*where* 2 ˆ *δ*0C (*m* − 1) = ˆ*δ*C (*m* − 1) + ˆ*δ*∗C (*m* − 1)*.*

*Moreover, the equality cases of Inequalities* (24) *and* (25) *hold identically at all points if and only if h and h*∗ *satisfy the condition in Equation* (18)*, which implies that M is a totally geodesic submanifold with respect to the Levi–Civita connection.*

## **4. An Example**

**Example 1.** *Let* (*<sup>x</sup>*1, *x*2, *y*1, *y*2) *be a standard system on* <sup>R</sup>4*, g the Euclidean metric. Define t* = (*y*21 + *y*22)/2 *(t* ≥ 0*) and the functions u, v on* R<sup>4</sup> *as*

$$u(\mathbf{x}\_1, \mathbf{x}\_2, y\_1, y\_2) = a(t), \ v(\mathbf{x}\_1, \mathbf{x}\_2, y\_1, y\_2) = b(t),$$

*where a is a function a* : [0, ∞) → (0, <sup>∞</sup>)*, and b*(*t*) = −*<sup>a</sup>*(*t*)*a*(*t*)(<sup>2</sup>*ta*(*t*) − *<sup>a</sup>*(*t*))−1*, assuming that a*(*t*) + 2*tb*(*t*) > 0 *for t* ≥ 0*.*

*Let G be a g-natural metric on* R<sup>4</sup> *and J a complex structure defined by Oproiu ([45]) such that* R<sup>4</sup> *is Kählerian, as follows:*

$$\begin{split} G &= -(u + vy\_1^2)dx\_1dx\_1 + 2vy\_1y\_2dx\_1dx\_2 + (u + vy\_2^2)dx\_2dx\_2 + \frac{u + vy\_2^2}{u(u + 2tv)}dy\_1dy\_1 \\ &- 2\frac{vy\_1y\_2}{u(u + 2tv)}dy\_1dy\_2 + \frac{u + vy\_1^2}{u(u + 2tv)}dy\_2dy\_2, \end{split} \tag{26}$$

$$\begin{cases} \begin{aligned} \int \frac{\partial}{\partial x\_1} &= (u + vy\_1^2) \frac{\partial}{\partial y\_1} + vy\_1y\_2 \frac{\partial}{\partial y\_2}, \\ \int \frac{\partial}{\partial x\_2} &= vy\_1y\_2 \frac{\partial}{\partial y\_1} + (u + vy\_2^2) \frac{\partial}{\partial y\_2}, \\ \int \frac{\partial}{\partial y\_1} &= -\frac{u + vy\_2^2}{u(u + 2tv)} \frac{\partial}{\partial x\_1} + \frac{vy\_1y\_2}{u(u + 2tv)} \frac{\partial}{\partial x\_2}, \\ \int \frac{\partial}{\partial y\_2} &= \frac{vy\_1y\_2}{u(u + 2tv)} \frac{\partial}{\partial x\_1} - \frac{u + vy\_1^2}{u(u + 2tv)} \frac{\partial}{\partial x\_2}. \end{aligned} \end{cases} \tag{27}$$

*Let the function u be defined as <sup>u</sup>*(*<sup>x</sup>*1, *x*2, *y*1, *y*2) = <sup>1</sup>+√<sup>1</sup>+4*t* 2 *. Therefore, the function v becomes <sup>v</sup>*(*<sup>x</sup>*1, *x*2, *y*1, *y*2) = 1*. Then, for the metric G and the complex structure J, there exists a tensor field K such that* (R4, ∇˜ := ∇*<sup>G</sup>* + *K*, *g*˜ := *G*, *J*) *is a special Kähler manifold* [46]*. Notice that a holomorphic statistical structure of holomorphic curvature* 0 *is nothing but a special Kähler manifold [43].*

*In this respect, define a* (1, 2)*-tensor field K on* R4*:*

$$\mathcal{K} = \sum\_{i,j,l=1}^{4} k\_{ij}^{l} \frac{\partial}{\partial \mathbf{x}^{l}} \otimes d\mathbf{x}^{i} \otimes d\mathbf{x}^{j}. \tag{28}$$

*Let α*1, ..., *α*7 *be functions on* R<sup>4</sup> *and denote p* := *u* + *vy*21*, q* := *u* + *vy*22*, r* := *u* + 2*tv, s* := *vy*1*y*<sup>2</sup>*. Suppose that α*2 *has the expression*

$$
\alpha\_2 = \frac{1}{2}s(u\_{\mathcal{Y}\_1} + 2y\_1) + \frac{1}{2}q u\_{\mathcal{Y}\_2}.\tag{29}
$$

*Moreover, α*1 *and α*3 *satisfy the equation*

$$\left(a\_2\frac{q}{sur} - a\_3\frac{1}{ur} - a\_1\frac{q}{s}\right)\frac{pur}{q} + a\_1\frac{sur}{q} + a\_2\frac{s}{q} = \frac{1}{2}p(u\_{y1} + 2y\_1) + \frac{1}{2}su\_{y2} \tag{30}$$

*where uy*1 := *∂u ∂y*1 *and uy*2 := *∂u ∂y*2 *.*

*If K performs the conditions in Equations (10)–(12) and also the conditions in Equations (29), (30), then we get* (R4, ∇˜ := ∇*<sup>G</sup>* + *K*, *g*˜ := *G*, *J*) *a special Kähler manifold [46] with K constructed as follows:*

*k*114 = *k*141 = *k*213 = *k*231 = −*<sup>k</sup>*334 = −*<sup>k</sup>*343 = *α*1, *k*411 = *k*312 = *k*321 = *α*2, *k*412 = *k*421 = *k*322 = *α*3, *k*124 = *k*142 = *k*223 = *k*232 = −*<sup>k</sup>*434 = −*<sup>k</sup>*443 = *α*4, *k*222 = −*<sup>k</sup>*424 = −*<sup>k</sup>*442 = *α*5,

*k*111 = *k*112 = *k*121 = −*<sup>k</sup>*323 = −*<sup>k</sup>*332 = 0, *k*212 = *k*221 = −*<sup>k</sup>*324 = −*<sup>k</sup>*342 = *α*7 *qs*, *k*133 = *α*6, *k*211 = *k*314 = *k*341 = 0, *k*214 = *k*241 = −*α*2 *surq* + *α*3 *purq* − *α*1 *sq* , *k*123 = *k*132 = *α*2 *qurp* − *α*4 *sp* − *α*3 *s urp* , *k*311 = *α*1 2*s*<sup>4</sup> − *u*2*r*<sup>2</sup> *sq* + *α*2 *ur* + 2*s*<sup>2</sup> *sq* − *α*3 *pq* , *k*422 = −*α*2 *qp* − *α*4 *u*2*r*<sup>2</sup> *sp* + *α*3 *ur* + 2*s*<sup>2</sup> *sp* , *k*113 = *k*131 = *α*2 *qsur* − *α*3 1*ur* − *α*1 *qs*, *k*133 = −*α*2 *qurp* + *α*3 *s urp* + *α*4 *sp* , *k*144 = −*<sup>k</sup>*224 = −*<sup>k</sup>*242 = *α*2 1*ur* − *α*3 *psur* + *α*4 *ps* , *k*344 = *α*2 *s qur* − *α*3 *pqur* + *α*1 *sq* , *k*333 = −*α*2 *qsur* + *α*3 1*ur* + *α*1 *qs*, *k*122 = −*<sup>k</sup>*423 = −*<sup>k</sup>*432 = −*α*5 *sp* , *k*134 = *k*143 = −*α*6 *sq* − *α*5 *s*2 *urpq* , *k*144 = *α*6 *s*2*q*2 + *α*5 *s*(<sup>2</sup>*s*<sup>2</sup> + *ur*) *urpq*<sup>2</sup> , *k*313 = *k*331 = *k*414 = *k*441 = 0, *k*413 = *k*431 = 0, *k*233 = −*α*6 *ps* , *k*234 = *k*243 = *α*6 *pq* + *α*5 *surq* , *k*244 = −*α*5 *pq* + *s*2 *urq*<sup>2</sup> − *α*6 *spq*2 .

*Then, M* ˜ = (R4, ∇˜ := ∇*<sup>G</sup>* + *K*, *G*, *J*) *is a holomorphic statistical manifold of holomorphic curvature* 0*.*

*Next, let M be any m-dimensional submanifold (m* < 4*) of M* ˜ *. Then, Inequalities* (16) *and* (17) *are satisfied. Moreover, the statistical submanifold M of M* ˜ *attains equality in both these inequalities, provided that M is totally geodesic.*
