Design and implementation of a smart Internet of Things chest pain center based on deep learning

Feng Li; Zhongao Bi; Hongzeng Xu; Yunqi Shi; Na Duan; Zhaoyu Li; Feng Li; Zhongao Bi; Hongzeng Xu; Yunqi Shi; Na Duan; Zhaoyu Li

doi:10.3934/mbe.2023840

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 10: 18987-19011. doi: 10.3934/mbe.2023840

Previous Article Next Article

Research article Special Issues

Design and implementation of a smart Internet of Things chest pain center based on deep learning

1.
School of Information and Electronic Engineering, Zhejiang Gongshang University, Hangzhou 310018, China
2.
School of Computer Science and Engineering, Nanyang Technological University, 639798, Singapore
3.
Department of Cardiology, The People's Hospital of Liaoning Province, Liaoning, Shenyang 110011, China
4.
Department of Cardiology, The Second Affiliated Hospital Zhejiang University School of Medicine, Hangzhou 310000, China

Academic Editor: Junxin Chen

Received: 05 July 2023 Revised: 13 August 2023 Accepted: 27 August 2023 Published: 10 October 2023

The data input process for most chest pain centers is not intelligent, requiring a lot of staff to manually input patient information. This leads to problems such as long processing times, high potential for errors, an inability to access patient data in a timely manner and an increasing workload. To address the challenge, an Internet of Things (IoT)-driven chest pain center is designed, which crosses the sensing layer, network layer and application layer. The system enables the construction of intelligent chest pain management through a pre-hospital app, Ultra-Wideband (UWB) positioning, and in-hospital treatment. The pre-hospital app is provided to emergency medical services (EMS) centers, which allows them to record patient information in advance and keep it synchronized with the hospital's database, reducing the time needed for treatment. UWB positioning obtains the patient's hospital information through the zero-dimensional base station and the corresponding calculation engine, and in-hospital treatment involves automatic acquisition of patient information through web and mobile applications. The system also introduces the Bidirectional Long Short-Term Memory (BiLSTM)-Conditional Random Field (CRF)-based algorithm to train electronic medical record information for chest pain patients, extracting the patient's chest pain clinical symptoms. The resulting data are saved in the chest pain patient database and uploaded to the national chest pain center. The system has been used in Liaoning Provincial People's Hospital, and its subsequent assistance to doctors and nurses in collaborative treatment, data feedback and analysis is of great significance.

Keywords:

Citation: Feng Li, Zhongao Bi, Hongzeng Xu, Yunqi Shi, Na Duan, Zhaoyu Li. Design and implementation of a smart Internet of Things chest pain center based on deep learning[J]. Mathematical Biosciences and Engineering, 2023, 20(10): 18987-19011. doi: 10.3934/mbe.2023840

Related Papers:

[1]	Oğuzhan Bahadır . On lightlike geometry of indefinite Sasakian statistical manifolds. AIMS Mathematics, 2021, 6(11): 12845-12862. doi: 10.3934/math.2021741
[2]	Fatemah Mofarreh, S. K. Srivastava, Anuj Kumar, Akram Ali . Geometric inequalities of $\mathcal{PR}$ -warped product submanifold in para-Kenmotsu manifold. AIMS Mathematics, 2022, 7(10): 19481-19509. doi: 10.3934/math.20221069
[3]	Aliya Naaz Siddiqui, Mohammad Hasan Shahid, Jae Won Lee . On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature. AIMS Mathematics, 2020, 5(4): 3495-3509. doi: 10.3934/math.2020227
[4]	Mehmet Atçeken, Tuğba Mert . Characterizations for totally geodesic submanifolds of a $K$ -paracontact manifold. AIMS Mathematics, 2021, 6(7): 7320-7332. doi: 10.3934/math.2021430
[5]	Ibrahim Al-Dayel, Meraj Ali Khan . Ricci curvature of contact CR-warped product submanifolds in generalized Sasakian space forms admitting nearly Sasakian structure. AIMS Mathematics, 2021, 6(3): 2132-2151. doi: 10.3934/math.2021130
[6]	Mohd. Aquib, Amira A. Ishan, Meraj Ali Khan, Mohammad Hasan Shahid . A characterization for totally real submanifolds using self-adjoint differential operator. AIMS Mathematics, 2022, 7(1): 104-120. doi: 10.3934/math.2022006
[7]	Tanumoy Pal, Ibrahim Al-Dayel, Meraj Ali Khan, Biswabismita Bag, Shyamal Kumar Hui, Foued Aloui . Generalized warped product submanifolds of Lorentzian concircular structure manifolds. AIMS Mathematics, 2024, 9(7): 17997-18012. doi: 10.3934/math.2024877
[8]	Biswabismita Bag, Meraj Ali Khan, Tanumoy Pal, Shyamal Kumar Hui . First chen inequality for biwarped product submanifold of a Riemannian space forms. AIMS Mathematics, 2025, 10(4): 9917-9932. doi: 10.3934/math.2025454
[9]	Rongsheng Ma, Donghe Pei . $\ast$ -Ricci tensor on $(\kappa, \mu)$ -contact manifolds. AIMS Mathematics, 2022, 7(7): 11519-11528. doi: 10.3934/math.2022642
[10]	Mohammad Aamir Qayyoom, Rawan Bossly, Mobin Ahmad . On CR-lightlike submanifolds in a golden semi-Riemannian manifold. AIMS Mathematics, 2024, 9(5): 13043-13057. doi: 10.3934/math.2024636

Abstract

1. Introduction

The study of statistical manifolds lies at the intersection of differential geometry and information theory (see ^[1,17]), where the geometry of parameter spaces of statistical models is examined through the lens of Riemannian and affine geometry. These manifolds, equipped with structures such as the Fisher information metric and connections, provide a rich framework for understanding various aspects of statistical inference and information processing. The significance of statistical manifolds extends to numerous applications, including machine learning, information theory, and theoretical physics, making them an essential topic of study in modern mathematics. The exponential family is a class of probability distributions that is commonly used in statistics and information theory. This family provides a rich example of a statistical manifold that can help illustrate the key concepts of Riemannian metrics and affine connections.

Sasakian geometry ^[24] has emerged as a fundamental area in differential geometry, characterized by its deep connections to Kähler and contact geometry. Sasakian manifolds are contact metric manifolds that exhibit a rich structure, allowing them to be seen as odd-dimensional counterparts to Kähler manifolds. These manifolds find applications in diverse areas, such as string theory, CR geometry, and even in the study of certain types of foliations. Furuhata et al. introduced the statistical counterpart of Sasakian manifolds in ^[8,10]. They continued their study in ^[11], examining Sasakian statistical manifolds from the perspective of warped products within statistical geometry, and also explored the concept of invariant submanifolds in the same ambient space. Kazan et al. ^[13] investigated Sasakian statistical manifolds with semi-symmetric metric connections, providing examples to support their findings. Lee et al. ^[14] established optimal inequalities for submanifolds in the same ambient space, expressed in terms of Casorati curvatures with a pair of affine connection and its conjugate affine connection. Uddin et al. ^[21] introduced the concept of nearly Sasakian statistical structures, presenting a non-trivial example and discussing different classes of submanifolds, including invariant and anti-invariant, within these manifolds. More recently, a new notion of mixed $3$ -Sasakian statistical manifolds was investigated in ^[16].

In ^[5], Chen introduced the CR $\delta$ -invariant for CR-submanifolds of Kähler manifolds and established a sharp inequality involving this invariant for anti-holomorphic warped product submanifolds in complex space forms. Drawing inspiration from this research, Al-Solamy et al. ^[2,3] gave an optimal inequality for this invariant specifically for anti-holomorphic submanifolds in complex space forms. This inequality was extended by proving an optimal inequality for the contact CR $\delta$ -invariant on contact CR-submanifolds in Sasakian space forms ^[15]. Siddiqui et al. ^[19] developed equivalent inequalities for this invariant in the context of a generic submanifold in trans-Sasakian generalized Sasakian space forms. More recently, Siddiqui et al. ^[20] gave two optimal inequalities involving the CR $\delta$ -invariant for generic statistical submanifolds in holomorphic statistical manifolds of constant holomorphic sectional curvature.

Building on the aforementioned findings, this paper explores the generic submanifolds in a Sasakian statistical manifold. We then derive an optimal inequality for the contact CR $\delta$ -invariant on contact CR-submanifolds in a Sasakian statistical manifold of constant $\phi$ -sectional curvature.

2. Preliminaries

Definition 2.1. Let $(M, G)$ be a Riemannian manifold, where $G$ is a Riemannian metric. A pair $(\tilde{D}, G)$ is called a statistical structure on $M$ if

(1) $\tilde{D}$ is of torsion-free, and

(2) The Codazzi equation: $(\tilde{D}_{X} G)(Y, Z) = (\tilde{D}_{Y} G)(X, Z)$ holds for any $X, Y, Z \in \Gamma(TM)$ .

Here $\tilde{D}$ is an affine connection on $M$ . A manifold equipped with such a statistical structure is referred to as a statistical manifold.

For an affine connection $\tilde{D}$ on $(M, G)$ , the dual connection $\tilde{D*}$ of $\tilde{D}$ with respect to $G$ is defined by the formula

$\begin{eqnarray*} XG(Y , Z) = G(\tilde{D}_{X}Y , Z) + G(Y , \tilde{D*}_{X} Z). \end{eqnarray*}$

We denote $\tilde{D}^{0}$ as the Levi-Civita connection of $G$ , which satisfies the relation: $2\tilde{D}^{0} = \tilde{D}+\tilde{D*}$ .

Given a statistical structure $(\tilde{D}, G)$ on $M$ , the statistical curvature tensor field $\tilde{\mathcal{R}} \in \Gamma(TM^{(1, 3)})$ is defined as

$\begin{eqnarray*} \tilde{\mathcal{R}} = \frac{1}{2}[\tilde{R}+ \tilde{R*}]. \end{eqnarray*}$

For a point $q \in M$ and a plane $\mathcal{L} = X \wedge Y$ spanned by orthonormal vectors $X, Y \in T_{q}M$ , the statistical sectional curvature $\tilde{S}^{\tilde{D}, \tilde{D*}}$ of $(M, \tilde{D}, G)$ for $X \wedge Y$ is defined as

$\begin{eqnarray*} \tilde{S}^{\tilde{D}, \tilde{D*}}(X \wedge Y ) = G(\tilde{\mathcal{R}}(X, Y )Y, X). \end{eqnarray*}$

For two statistical manifolds $(N, D, g)$ and $(M, \tilde{D}, G)$ , an immersion $h : N \rightarrow M$ , $h$ is called a statistical immersion if the statistical structure induced by $h$ from $(\tilde{D}, G)$ coincides with $(D, g)$ .

For any $X, Y \in \Gamma(TN)$ , the corresponding Gauss formulas ^[22] are

$\begin{eqnarray*} \tilde{D}_{X} Y & = & D_{X}Y + B(X, Y), \\ \tilde{D*}_{X} Y & = & {D*}_{X}Y + {B*}(X, Y), \end{eqnarray*}$

where $B$ and ${B*}$ are symmetric and bilinear, called the imbedding curvature tensors of $N$ in $M$ for $\tilde{D}$ and $D$ , respectively. Next, we have the linear transformations $A$ and ${A*}$ defined by

$\begin{eqnarray*} g(A_{V}X, Y)& = &G({B*}(X, Y), V), \\ g({A*}_{V}X, Y)& = &G(B(X, Y), V), \end{eqnarray*}$

for any $V\in \Gamma(TN^{\perp})$ . Further, in ^[22] the corresponding Weingarten formulas are as follows:

$\begin{eqnarray*} \tilde{D}_{X} V & = & D^{\perp}_{X}V - A_{V}X, \\ \tilde{D*}_{X} V & = & {D*}^{\perp}_{X}V - {A*}_{V}X, \end{eqnarray*}$

where the connections $D^{\perp}$ and ${D*}^{\perp}$ are Riemannian dual connections with respect to the induced metric on $\Gamma(TN^{\perp})$ .

The mean curvature vector field of a $r$ -dimensional statistical submanifold $(N, D, g)$ in any statistical manifold $(M, \tilde{D}, G)$ with respect to both affine connections is as follows:

$\begin{eqnarray*} H = \frac{1}{r} trace_{G}(B), \quad {H*} = \frac{1}{r} trace_{G}({B*}). \end{eqnarray*}$

We respectively symbolize the Riemannian curvature tensors of $\tilde{D}$ (respectively, $\tilde{D*}$ ) and $D$ (respectively, ${D*}$ ) by $\tilde{R}$ (respectively, $\tilde{R*}$ ) and $R$ (respectively, ${R*}$ ). Then, the corresponding Gauss equations for conjugate affine connection are given by ^[22]

$\begin{eqnarray} \tilde{R}_{X, Y, Z, W}& = &R_{X, Y, Z, W} + G(B (X, Z), {B*} (YW))-G ({B*} (X, W), B(Y, Z)), \end{eqnarray}$

(2.1)

$\begin{eqnarray} \tilde{R*}_{X, Y, Z, W}& = &{R*}_{X, Y, Z, W} + G({B*} (X, Z), B (YW)) -G (B (X, W), {B*}(Y, Z)), \end{eqnarray}$

(2.2)

where $\tilde{R}_{X, Y, Z, W} = G(\tilde{R}(X, Y)Z, W)$ and $\tilde{R*}_{X, Y, Z, W} = G(\tilde{R*}(X, Y)Z, W)$ . Thus, we have the Gauss formula for both affine connections:

$\begin{eqnarray} 2\tilde{\mathcal{R}}_{X, Y, Z, W}& = &2 \mathcal{R}_{X, Y, Z, W} + G(B (X, Z), {B*} (Y, W)) -G ({B*} (X, W), B(Y, Z)) \\ &&+ G({B*} (X, Z), B (Y, W))-G (B (X, W), {B*}(Y, Z)), \end{eqnarray}$

(2.3)

where $2 \mathcal{R} = R+{R*}$ .

The Codazzi equation for both affine connections:

$\begin{eqnarray} 2\tilde{\mathcal{R}}_{X, Y, Z, V}& = & G(((\tilde{D}_{X}B) (Y, Z)), V)-G((\tilde{D}_{Y}B)(X, Z), V) \\ &&+G(({\tilde{D}*}_{X}{B*} )(Y, Z), V)-G(({\tilde{D}*}_{Y}{B*})(X, Z), V). \end{eqnarray}$

(2.4)

Definition 2.2. A quadruplet $(\tilde{D}, G, \phi, \xi)$ is called a Sasakian statistical structure on $M$ if the following formula holds:

$\begin{eqnarray*} K_{X}\phi Y = -\phi K_{X}Y, \end{eqnarray*}$

where $K_{X}Y = \tilde{D}_{X}Y-\tilde{D}^{0}_{X}Y$ satisfies $K_{X}Y = K_{Y}X$ and $G(K_{X}Y, Z) = G(Y, K_{X}Z)$ .

Theorem 2.3. Let $(\tilde{D}, G)$ be a statistical structure and $(G, \phi, \xi)$ an almost contact metric structure on $M$ . Then $(\tilde{D}, G, \phi, \xi)$ is a Sasakian statistical structure on $M$ if and only if

$\begin{eqnarray*} \tilde{D}_{X}(\phi Y ) - \phi \tilde{D*}_{X}Y & = & G(\xi, Y )X - G(X, Y )\xi\\ \tilde{D}_{X}\xi & = & \phi X + G(\tilde{D}_{X}\xi, \xi)\xi. \end{eqnarray*}$

Let $(M, \tilde{D}, G, \phi, \xi)$ be a Sasakian statistical manifold, and $c \in \mathbb{R}$ . The Sasakian statistical structure is said to be of constant $\phi$ -sectional curvature $c$ if

$\begin{eqnarray*} \tilde{\mathcal{R}}(X, Y)Z & = & \frac{c + 3}{4}\{G(Y, Z)X -G(X, Z)Y\}+\frac{c - 1}{4}\{G(\phi Y, Z)\phi X\\ &&- G(\phi X, Z)\phi Y - 2G(\phi X, Y)\phi Z -G(Y, \xi )G(Z, \xi )X \\ &&+ G(X, \xi )G(Z, \xi )Y + G(Y, \xi )G(Z, X)\xi -G(X, \xi )G(Z, Y)\xi \}, \end{eqnarray*}$

holds for $X, Y, Z \in \Gamma(TM)$ .

Definition 2.4. Let $(N, D, g)$ be a statistical manifold in a Sasakian statistical manifold $(M, \tilde{D}, G, \phi, \xi)$ . For $X, Y \in \Gamma(TN)$ .

(1) $N$ is said to be doubly totally contact umbilical, if

$\begin{eqnarray*} B(X, Y) = \bigg[g(X, Y)-\eta(X)\eta(Y)\bigg]V^{\perp}+ \eta(X)B(Y, \xi)+\eta(Y)B(X, \xi), \end{eqnarray*}$

and

$\begin{eqnarray*} {B*}(X, Y) = \bigg[g(X, Y)-\eta(X)\eta(Y)\bigg]V^{\perp}+ \eta(X){B*}(Y, \xi) +\eta(Y){B*}(X, \xi), \end{eqnarray*}$

where $V^{\perp}$ represents any vector field normal to $N$ .

(2) $N$ is said to be doubly totally contact geodesic if $V^{\perp} = 0$ , that is,

$\begin{eqnarray*} B(X, Y) = \eta(X)B(Y, \xi)+\eta(Y)B(X, \xi), \end{eqnarray*}$

and

$\begin{eqnarray*} {B*}(X, Y) = \eta(X){B*}(Y, \xi)+\eta(Y){B*}(X, \xi). \end{eqnarray*}$

Definition 2.5. A statistical submanifold $(N, D, g)$ in a Sasakian statistical manifold $(M, \tilde{D}, G, \phi, \xi)$ is called a generic statistical submanifold (or simply generic submanifold) in $M$ if

(1) $\phi T_{q}N^{\perp} \subset T_{q}N$ , $q \in N$ , and

(2) $\xi$ is tangent to $N$ .

The tangent space $T_{q}N$ at any point $q \in N$ is decomposed as

$\begin{eqnarray*} T_{q}N = \mathcal{H}_{q}N \oplus \phi T_{q}N^{\perp}, \end{eqnarray*}$

where $\mathcal{H}_{q}N$ denotes the orthogonal complement of $\phi T_{q}N^{\perp}$ . Consequently, we have

$\begin{eqnarray*} \phi \mathcal{H}_{q}N = \mathcal{H}_{q}N \backslash \{\xi\}. \end{eqnarray*}$

3. On generic statistical submanifolds

Lemma 3.1. Let $(N, D, g)$ be a generic submanifold in a Sasakian statistical manifold $(M, \tilde{D}, G, \phi, \xi)$ . Then we have

$\begin{eqnarray*} A_{{\mathbf F}Y}Z = A_{{\mathbf F}Z}Y, \quad {A*}_{{\mathbf F}Y}Z = {A*}_{{\mathbf F}Z}Y, \end{eqnarray*}$

for $Y, Z \in \Gamma(\phi TN^{\perp})$ .

Proof. For $X, Y\in \Gamma(TN)$ , we know

$\begin{eqnarray*} \tilde{D}_{X}\phi Y - \phi \tilde{D*}_{X} Y = \eta(Y)X-g(X, Y)\xi. \end{eqnarray*}$

Since $\phi X = {\mathbf P}X + {\mathbf F}X$

$\begin{array}{c} D_{X} {\mathbf P} Y + B(X, {\mathbf P} Y) + D^{\perp}_{X} {\mathbf F}Y -A_{{\mathbf F}Y}X -\phi {D*}_{X}Y - \phi {B*}(X, Y) = \eta(Y)X-g(X, Y)\xi.\\ D_{X} {\mathbf P} Y + B(X, {\mathbf P} Y) + D^{\perp}_{X} {\mathbf F}Y -A_{{\mathbf F}Y}X -{\mathbf P} {D*}_{X}Y -{\mathbf F} {D*}_{X}Y- \phi {B*}(X, Y) = \eta(Y)X-g(X, Y)\xi. \end{array}$

In comparison, we have

$\begin{eqnarray*} \phi {B*}(X, Y) = D_{X} {\mathbf P} Y - {\mathbf P} {D*}_{X}Y - A_{{\mathbf F}Y}X -\eta(Y)X+g(X, Y)\xi, \end{eqnarray*}$

and

$\begin{eqnarray*} B(X, {\mathbf P} Y) & = & - D^{\perp}_{X} {\mathbf F}Y + {\mathbf F} {D*}_{X}Y. \end{eqnarray*}$

In particular, for $Y, Z\in \Gamma(\phi TN^{\perp})$

$\begin{eqnarray*} G( \phi {B*}(X, Y), Z) = g(D_{X} {\mathbf P} Y, Z) - g({\mathbf P} {D*}_{X}Y, Z) - g(A_{{\mathbf F}Y}X, Z) -\eta(Y)g(X, Z)+\eta(Z)g(X, Y). \end{eqnarray*}$

We notice that ${\mathbf P}\phi T_{q}N^{\perp} = 0$ and $\phi {\mathbf P} T_{q}N \subset \mathcal{H}_{q}N$ . So, we have

$\begin{eqnarray*} G( \phi {B*}(X, Y), Z) = - g(A_{{\mathbf F}Y}X, Z), \end{eqnarray*}$

which can be reduced further as

$\begin{eqnarray*} g(A_{{\mathbf F}Z}Y, X) = g(X, A_{{\mathbf F}Y}Z), \end{eqnarray*}$

that is, $A_{{\mathbf F}Y}Z = A_{{\mathbf F}Z}Y$ . Similarly, one can show that ${A*}_{{\mathbf F}Y}Z = {A*}_{{\mathbf F}Z}Y$ . □

Lemma 3.1 implies the following result:

Proposition 3.2. Let $(N, D, g)$ be a generic submanifold in a Sasakian statistical manifold $(M, \tilde{D}, G, \phi, \xi)$ of codimension greater than 1. If $N$ is doubly totally contact umbilical, then $N$ is doubly totally contact geodesic.

Proposition 3.3. Let $(M(c), \tilde{D}, G, \phi, \xi)$ be a $(2s+1)$ -dimensional Sasakian statistical manifold of constant $\phi$ -sectional curvature $c$ and $(N, D, g)$ be an $(r+1)$ -dimensional generic submanifold in $M(c)$ , with $r > s$ and $r\geq 3$ . If $N$ is of codimension greater than 1 and totally contact umbilical, then $c = -3$ .

Proof. When $M$ is not a statistical hypersurface, that is, $N$ is of codimension greater than or equal to $2$ , then Proposition 3.2 implies that $B$ and ${B*}$ of $N$ are of the following form:

$\begin{eqnarray*} B(Y, Z) = g(Y, \xi){\mathbf F}Z+g(Z, \xi){\mathbf F}Y, \quad {B*}(Y, Z) = g(Y, \xi){\mathbf F}Z+g(Z, \xi){\mathbf F}Y. \end{eqnarray*}$

The covariant derivatives of $B$ and $B^{*}$ are defined as

$\begin{eqnarray*} (\tilde{D}_{X} B)(Y, Z)& = &D^{\perp}_{X}(B(Y, Z))-B(D_{X}Y, Z)-B(Y, D_{X}Z)\\ & = & g(Y, {D*}_{X}\xi){\mathbf F}Z+g(Z, {D*}_{X}\xi){\mathbf F}Y+g(Y, \xi)D^{\perp}_{X}{\mathbf F}Z\\ &&+g(Z, \xi)D^{\perp}_{X}{\mathbf F}Y-g(Z, \xi){\mathbf F}D_{X}Y-g(Y, Z){\mathbf F}D_{X}Z, \end{eqnarray*}$

and

$\begin{eqnarray*} ({\tilde{D}*}_{X} {B*})(Y, Z)& = &{D*}_{X}({B*}(Y, Z))-{B*}({D*}_{X}Y, Z)-{B*}(Y, {D*}_{X}Z)\\ & = & g(Y, D_{X}\xi){\mathbf F}Z+g(Z, D_{X}\xi){\mathbf F}Y+g(Y, \xi){D*}^{\perp}_{X}{\mathbf F}Z\\ &&+g(Z, \xi){D*}^{\perp}_{X}{\mathbf F}Y-g(Z, \xi){\mathbf F}{D*}_{X}Y-g(Y, Z){\mathbf F}{D*}_{X}Z. \end{eqnarray*}$

So, we derive

$\begin{eqnarray*} (\tilde{D}_{X} B)(Y, Z) = g(Y, {\mathbf P}X){\mathbf F}Z+g(Z, {\mathbf P}X){\mathbf F}Y +g({D*}_{X}\xi, \xi)g(Y, \xi){\mathbf F}Z+g({D*}_{X}\xi, \xi)g(Z, \xi){\mathbf F}Y, \end{eqnarray*}$

$\begin{eqnarray*} (\tilde{D}_{Y} B)(X, Z) = g(X, {\mathbf P}Y){\mathbf F}Z+g(Z, {\mathbf P}Y){\mathbf F}X +g({D*}_{Y}\xi, \xi)g(X, \xi){\mathbf F}Z+g({D*}_{Y}\xi, \xi)g(Z, \xi){\mathbf F}X, \end{eqnarray*}$

$\begin{eqnarray*} ({\tilde{D}*}_{X} {B*})(Y, Z) = g(Y, {\mathbf P}X){\mathbf F}Z+g(Z, {\mathbf P}X){\mathbf F}Y +g(D_{X}\xi, \xi)g(Y, \xi){\mathbf F}Z+g(D_{X}\xi, \xi)g(Z, \xi){\mathbf F}Y, \end{eqnarray*}$

and

$\begin{eqnarray*} ({\tilde{D}*}_{Y} {B*})(X, Z) = g(X, {\mathbf P}Y){\mathbf F}Z+g(Z, {\mathbf P}Y){\mathbf F}X +g(D_{Y}\xi, \xi)g(X, \xi){\mathbf F}Z+g(D_{Y}\xi, \xi)g(Z, \xi){\mathbf F}X. \end{eqnarray*}$

Further, we use these expressions in the Codazzi equation (2.4) as

$\begin{eqnarray*} 2\tilde{\mathcal{R}}_{X, Y, Z, V}& = & G(((\tilde{D}_{X}B) (Y, Z)), V)-G((\tilde{D}_{Y}B)(X, Z), V) \nonumber\\ &&+G(({\tilde{D}*}_{X}{B*} )(Y, Z), V)-G(({\tilde{D}*}_{Y}{B*})(X, Z), V)\nonumber\\ &&\frac{c-1}{4}\bigg[g({\mathbf P}Y, Z){\mathbf F}X-g({\mathbf P}X, Z){\mathbf F}Y-2g({\mathbf P}X, Y){\mathbf F}Z\bigg]\nonumber \\ & = &2g(Y, {\mathbf P}X){\mathbf F}Z+g(Z, {\mathbf P}X){\mathbf F}Y-g(Z, {\mathbf P}Y){\mathbf F}X\nonumber\\ &&+g(Y, \xi)\bigg[g(D_{X}\xi, \xi)+g({D*}_{X}\xi, \xi)\bigg]+g(X, \xi)\bigg[g(D_{Y}\xi, \xi)+g({D*}_{Y}\xi, \xi)\bigg]. \end{eqnarray*}$

From which we arrive at

$\begin{eqnarray*} &&\frac{c+3}{4}\bigg[-g({\mathbf P}Y, Z){\mathbf F}X+g({\mathbf P}X, Z){\mathbf F}Y+2g({\mathbf P}X, Y){\mathbf F}Z\bigg]\nonumber \\ &&+g(Y, \xi)\bigg[g(D_{X}\xi, \xi)+g({D*}_{X}\xi, \xi)\bigg]+g(X, \xi)\bigg[g(D_{Y}\xi, \xi)+g({D*}_{Y}\xi, \xi)\bigg] = 0. \end{eqnarray*}$

For $Y \in \mathcal{H}_{q}N$ , we put $Z = {\mathbf P}Y$ . Then ${\mathbf F}Y = 0$ and ${\mathbf FP}Y = 0$ . Thus, we conclude that $c = -3$ . □

4. A geometric inequality

Following the analogy of Chen's CR $\delta$ -invariant, Mihai et al. ^[15] defined the contact CR $\delta$ -invariant for an odd-dimensional contact CR-submanifold in a Sasakian space form. Here, we define the statistical Chen's CR $\delta$ -invariant on a $(r + 1)$ -dimensional contact CR-submanifold $(N, D, g)$ in the $(2s+1)$ -dimensional Sasakian statistical manifold $(M(c), \tilde{D}, G, \phi, \xi)$ of constant $\phi$ -sectional curvature $c$ as follows:

$\begin{eqnarray*} \delta^{D, D*}(\mathcal{D})(q) = scal^{D, D*} (q) - scal^{D, D*}(\mathcal{D}_{q}), \end{eqnarray*}$

where $scal^{D, D*}$ and $scal^{D, D*}(\mathcal{D})$ denote the scalar curvature of $N$ and the scalar curvature of the invariant distribution $\mathcal{D} \subset TN$ , respectively.

Orthonormal frames on differentiable manifolds provide a powerful tool for simplifying complex geometric and physical problems. They offer a structured way to understand local properties of the manifold (metric tensors), aid in defining connections and curvature, and play a crucial role in both theoretical and applied contexts like Riemannian and Lorentzian geometry. If $\dim(\mathcal{D}) = 2\alpha + 1$ and $\dim(\mathcal{D}^{\perp}) = \beta$ and let $\{v_{0} = \xi, v_{1}, v_{2}, \cdots, v_{r}\}$ be an orthonormal frame on $N$ such that $\{v_{0}, v_{1}, \cdots, v_{2\alpha}\}$ are tangent to $\mathcal{D}$ and $\{v_{2\alpha + 1}, \cdots, v_{\beta}\}$ are tangent to $\mathcal{D}^{\perp}$ . Then the partial mean curvature vectors $H1$ and $H2$ (respectively, $H1*$ and $H2*$ for both affine connections) of $N$ are given by

$\begin{eqnarray*} &&H1 = \frac{1}{2\alpha + 1} \sum\limits_{I = 0}^{2\alpha}B(v_{I}, v_{I}), \quad H2 = \frac{1}{\beta} \sum\limits_{a = 2\alpha + 1}^{2 \alpha + \beta} B(v_{a}, v_{a}), \\ &&H1* = \frac{1}{2\alpha + 1} \sum\limits_{I = 0}^{2\alpha}B(v_{I}, v_{I}), \quad H2* = \frac{1}{\beta} \sum\limits_{a = 2\alpha + 1}^{2 \alpha + \beta} B(v_{a}, v_{a}). \end{eqnarray*}$

A contact CR-submanifold $(N, D, g)$ of a Sasakian manifold $(M, \tilde{D}, G, \phi, \xi)$ is said to be doubly minimal if $H = H* = 0$ . Likewise, it is referred to as doubly $\mathcal{D}$ -minimal or doubly $\mathcal{D}^{\perp}$ -minimal if $H1 = H1* = 0$ or $H2 = H2* = 0$ , respectively.

According to the definition of the contact CR $\delta$ -invariant, we have

$\begin{eqnarray*} \delta^{D, D*}(\mathcal{D})& = &\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta} S^{D, D*}(\xi, v_{a}) + \sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta} S^{D, D*}(v_{I}, v_{a}) \\ &&+ \frac{1}{2} \sum\limits_{2\alpha+1\leq a \neq b \leq 2\alpha+\beta} S^{D, D*}(v_{a}, v_{b})\\ & = & \bigg(\frac{c+3}{4}\bigg)\bigg(\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta} g(v_{a}, v_{a})\bigg)-\bigg(\frac{c-1}{4}\bigg)\bigg(\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta} g(v_{a}, v_{a})\bigg)\\ &&+ \frac{\beta(4\alpha +\beta-1)}{2}\bigg(\frac{c+3}{4}\bigg)\\ &&+\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}[G(B(v_{I}, v_{I}), {B*}(v_{a}, v_{a}))-G(B(v_{I}, v_{a}), {B*}(v_{I}, v_{a}))]\\ &&+\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}[G({B*}(v_{I}, v_{I}), B(v_{a}, v_{a}))-G({B*}(v_{I}, v_{a}), B(v_{I}, v_{a}))]\\ &&+\frac{1}{2}\sum\limits_{a, b = 2\alpha+1}^{2\alpha +\beta}[G(B(v_{a}, v_{a}), {B*}(v_{b}, v_{b}))-G(B(v_{a}, v_{b}), {B*}(v_{a}, v_{b}))]\\ &&+\frac{1}{2}\sum\limits_{a, b = 2\alpha+1}^{2\alpha +\beta}[G({B*}(v_{a}, v_{a}), B(v_{b}, v_{b}))-G({B*}(v_{a}, v_{b}), B(v_{a}, v_{b}))]. \end{eqnarray*}$

We use ${B*}(X, \xi) = B(X, \xi) = 0$ and obtain

$\begin{eqnarray*} \delta^{D, D*}(\mathcal{D})& = & \beta\bigg(1+ \frac{(4\alpha +\beta-1)(c+3)}{8}\bigg) \\ &&+\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}[G(B(v_{I}, v_{I}), {B*}(v_{a}, v_{a}))-G(B(v_{I}, v_{a}), {B*}(v_{I}, v_{a}))]\\ &&+\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}[G({B*}(v_{I}, v_{I}), B(v_{a}, v_{a}))-G({B*}(v_{I}, v_{a}), B(v_{I}, v_{a}))]\\ &&+\frac{1}{2}\sum\limits_{a, b = 2\alpha+1}^{2\alpha +\beta}[G(B(v_{a}, v_{a}), {B*}(v_{b}, v_{b}))-G(B(v_{a}, v_{b}), {B*}(v_{a}, v_{b}))]\\ &&+\frac{1}{2}\sum\limits_{a, b = 2\alpha+1}^{2\alpha +\beta}[G({B*}(v_{a}, v_{a}), B(v_{b}, v_{b}))-G({B*}(v_{a}, v_{b}), B(v_{a}, v_{b}))]. \end{eqnarray*}$

Given that $2B^{0} = B+{B*}$ , we deduce

$4H^{02} = H^{2} + {H*}^{2} + 2G(H, {H*})$

and

$4B_{\mathcal{D}^{\perp}}^{02} = B_{\mathcal{D}^{\perp}}^{2} + {B*}_{\mathcal{D}^{\perp}}^{2} + 2G(B_{\mathcal{D}^{\perp}}, {B*}_{\mathcal{D}^{\perp}}).$

So, we have

$\begin{eqnarray*} \delta^{D, D*}(\mathcal{D})& = & \beta\bigg(1+ \frac{(4\alpha +\beta-1)(c+3)}{8}\bigg) \\ &&+4\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}[G(B^{0}(v_{I}, v_{I}), B^{0}(v_{a}, v_{a}))-G(B^{0}(v_{I}, v_{a}), B^{0}(v_{I}, v_{a}))]\\ &&-\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}[G(B(v_{I}, v_{I}), B(v_{a}, v_{a}))-G(B(v_{I}, v_{a}), B(v_{I}, v_{a}))]\\ &&-\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}[G({B*}(v_{I}, v_{I}), {B*}(v_{a}, v_{a}))-G({B*}(v_{I}, v_{a}), {B*}(v_{I}, v_{a}))]\\ &&+2\sum\limits_{a, b = 2\alpha+1}^{2\alpha +\beta}[G(B^{0}(v_{a}, v_{a}), B^{0}(v_{b}, v_{b}))-G(B^{0}(v_{a}, v_{b}), B^{0}(v_{a}, v_{b}))]\\ &&-\frac{1}{2}\sum\limits_{a, b = 2\alpha+1}^{2\alpha +\beta}[G(B(v_{a}, v_{a}), B(v_{b}, v_{b}))-G(B(v_{a}, v_{b}), B(v_{a}, v_{b}))]\\ &&-\frac{1}{2}\sum\limits_{a, b = 2\alpha+1}^{2\alpha +\beta}[G({B*}(v_{a}, v_{a}), {B*}(v_{b}, v_{b}))-G({B*}(v_{a}, v_{b}), {B*}(v_{a}, v_{b}))]. \end{eqnarray*}$

From ^[15], we have

$\begin{eqnarray*} &&4\bigg[\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}G(B^{0}(v_{I}, v_{I}), B^{0}(v_{a}, v_{a}))+\frac{1}{2}\sum\limits_{a, b = 2\alpha+1}^{2\alpha +\beta}G(B^{0}(v_{a}, v_{a}), B^{0}(v_{b}, v_{b}))\\ &&-\frac{1}{2}G(B^{0}(v_{a}, v_{b}), B^{0}(v_{a}, v_{b}))\bigg]\\ & = & 2(2\alpha + \beta +1)^{2}H^{02} -2(2\alpha +1)^{2}{H1}^{02}-2B_{\mathcal{D}^{\perp}}^{02}. \end{eqnarray*}$

Similarly, we have

$\begin{eqnarray*} &&-\bigg[\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}G(B(v_{I}, v_{I}), B(v_{a}, v_{a}))+\frac{1}{2}\sum\limits_{a, b = 2\alpha+1}^{2\alpha +\beta}G(B(v_{a}, v_{a}), B(v_{b}, v_{b}))\\ &&-\frac{1}{2}G(B(v_{a}, v_{b}), B(v_{a}, v_{b}))\bigg]\\ & = & -\frac{(2\alpha + \beta +1)^{2}}{2}H^{2} +\frac{(2\alpha +1)^{2}}{2}{H1}^{2}+\frac{1}{2}B_{\mathcal{D}^{\perp}}^{2}, \end{eqnarray*}$

and

$\begin{eqnarray*} &&-\bigg[\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}G({B*}(v_{I}, v_{I}), {B*}(v_{a}, v_{a}))+\frac{1}{2}\sum\limits_{a, b = 2\alpha+1}^{2\alpha +\beta}G({B*}(v_{a}, v_{a}), {B*}(v_{b}, v_{b}))\\ &&-\frac{1}{2}G({B*}(v_{a}, v_{b}), {B*}(v_{a}, v_{b}))\bigg]\\ & = &- \frac{(2\alpha + \beta +1)^{2}}{2}{H*}^{2} +\frac{(2\alpha +1)^{2}}{2}{H1*}^{2}+\frac{1}{2}{B*}_{\mathcal{D}^{\perp}}^{2}. \end{eqnarray*}$

So, we derive

$\begin{eqnarray*} \delta^{D, D*}(\mathcal{D})& = & \beta\bigg(1+ \frac{(4\alpha +\beta-1)(c+3)}{8}\bigg) \\ &&+2(2\alpha + \beta +1)^{2}H^{02} -\frac{(2\alpha + \beta +1)^{2}}{2}(H^{2} + {H*}^{2})\\ &&-2(2\alpha +1)^{2}{H1}^{02}+\frac{(2\alpha +1)^{2}}{2}({H1}^{2}+{H1*}^{2})\\ &&-2B_{\mathcal{D}^{\perp}}^{02}+\frac{1}{2}(B_{\mathcal{D}^{\perp}}^{2}+{B*}_{\mathcal{D}^{\perp}}^{2})\\ &&-\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}[4B_{Ia}^{02}-B_{Ia}^{2}-{B*}_{Ia}^{2}]. \end{eqnarray*}$

In the final equation, we use the following inequalities for both affine connections (analogous to those obtained for the Levi-Civita connection in ^[15]):

$\begin{eqnarray*} B_{\mathcal{D}^{\perp}}^{2} \geq \frac{3\beta^{2}}{\beta + 2}{H2}^{2}, \quad {B*}_{\mathcal{D}^{\perp}}^{2} \geq \frac{3\beta^{2}}{\beta + 2}{H2*}^{2} \end{eqnarray*}$

with equality holds if and only if the following conditions are met:

(1) $B^{a}_{aa} = 3B^{a}_{bb}$ and ${B*}^{a}_{aa} = 3{B*}^{a}_{bb}$ , for $2\alpha + 1 \leq a \neq b \leq 2\alpha + \beta$ ;

(2) $B^{a}_{bc} = 0$ and ${B*}^{a}_{bc} = 0$ for $a, b, c \in \{2\alpha + 1, \cdots, 2\alpha + \beta\}$ , $a\neq b \neq c$ .

Thus, we find that

$\begin{eqnarray} \delta^{D, D*}(\mathcal{D})&\geq& \beta\bigg(1+ \frac{(4\alpha +\beta-1)(c+3)}{8}\bigg) \\ &&+2(2\alpha + \beta +1)^{2}H^{02} -\frac{(2\alpha + \beta +1)^{2}}{2}(H^{2} + {H*}^{2}\\ &&-2(2\alpha +1)^{2}{H1}^{02}+\frac{(2\alpha +1)^{2}}{2}({H1}^{2}+{H1*}^{2})\\ &&-2B_{\mathcal{D}^{\perp}}^{02}+\frac{3\beta^{2}}{2(\beta + 2)}({H2}^{2}+{H2*}^{2})\\ &&-\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}[4B_{Ia}^{02}-B_{Ia}^{2}-{B*}_{Ia}^{2}]. \end{eqnarray}$

(4.1)

By drawing on the analogy with ^[3] and Lemma 3.1, we obtain the following inequalities:

$\begin{eqnarray} \sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}(B_{Ia}^{2}+{B*}_{Ia}^{2})+\frac{(2\alpha+ 1)^{2}}{2}(H1^{2} + {H1*}^{2}) +\frac{3\beta^{2}}{4(\beta + 2)}({H2}^{2}+{H2*}^{2})\\ \geq \frac{3\beta^{2}}{4(\beta+2)}({H2}^{2} + {H2*}^{2}). \end{eqnarray}$

(4.2)

By substituting (4.2) into (4.1), we obtain

$\begin{eqnarray} &&\delta^{D, D*}(\mathcal{D})- \beta\bigg(1+ \frac{(4\alpha +\beta-1)(c+3)}{8}\bigg) \\ &&-2(2\alpha + \beta +1)^{2}H^{02} +2(2\alpha +1)^{2}{H1}^{02}+ 2B_{\mathcal{D}^{\perp}}^{02}\\ &&+4\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}B_{Ia}^{02}\\ &\geq& \frac{(2\alpha +1)^{2}}{2}({H1}^{2}+{H1*}^{2})-\frac{(2\alpha + \beta +1)^{2}}{2}(H^{2} + {H*}^{2})\\ &&+\frac{3\beta^{2}}{2(\beta + 2)}({H2}^{2}+{H2*}^{2})+\sum\limits_{I = 1}^{2\alpha}\sum\limits_{a = 2\alpha+1}^{2\alpha +\beta}(B_{Ia}^{2}-{B*}_{Ia}^{2})\\ &\geq& \frac{3\beta^{2}}{2(\beta+2)}({H2}^{2} + {H2*}^{2})-\frac{(2\alpha + \beta +1)^{2}}{2}(H^{2} + {H*}^{2}). \end{eqnarray}$

(4.3)

On the other hand, we have the following relation for the contact CR $\delta$ -invariant $\delta^{0}(\mathcal{D})$ of $N$ with respect to Levi-Civita connection, given by ^[15]

$\begin{eqnarray} \delta^{0}(\mathcal{D}) & = &\frac{(2\alpha + \beta + 1)^{2}}{2} H^{02} + \beta \bigg(1+ (4\alpha + \beta- 1)\bigg)\frac{ c + 3}{8} \\ &&-\frac{(2\alpha + 1)^{2}}{2} {H1}^{02} -\sum\limits_{I = 1}^{2\alpha} \sum\limits_{a = 2\alpha + 1}^{2\alpha+\beta}B^{02}(v_{I}, v_{a})- \frac{1}{2}B_{\mathcal{D}^{\perp}}^{02}. \end{eqnarray}$

(4.4)

Putting (4.4) into (4.3), we obtain

$\begin{eqnarray*} &&\delta^{D, D*}(\mathcal{D})+3\beta\bigg(1+ \frac{(4\alpha +\beta-1)(c+3)}{8}\bigg)-4\delta^{0}(\mathcal{D})\nonumber \\ &\geq& \frac{3\beta^{2}}{2(\beta+2)}({H2}^{2} + {H2*}^{2})-\frac{(2\alpha + \beta + 1)^{2}}{2}(H^{2} + {H*}^{2}). \end{eqnarray*}$

Hence, we have:

Theorem 4.1. Let $(M(c), \tilde{D}, G, \phi, \xi)$ be a $(2s+1)$ -dimensional Sasakian statistical manifold of constant $\phi$ -sectional curvature $c$ and $(N, D, g)$ be a $(r+1)$ -dimensional generic submanifold in $M(c)$ , with $\dim(\mathcal{D}) = 2\alpha + 1$ and $\dim(\mathcal{D}^{\perp}) = \beta$ . Then

$\begin{eqnarray} \delta^{D, D*}(\mathcal{D})&\geq& 4\delta^{0}(\mathcal{D})-3\beta\bigg(1+ \frac{(4\alpha +\beta-1)(c+3)}{8}\bigg)\\ &&+ \frac{3\beta^{2}}{2(\beta+2)}({H2}^{2} + {H2*}^{2})-\frac{(r+1)^{2}}{2}(H^{2} + {H*}^{2}). \end{eqnarray}$

(4.5)

Furthermore, the equality in (4.5) holds identically if and only if the following conditions are met:

(1) $N$ is doubly $\mathcal{D}$ -minimal,

(2) $N$ is mixed totally geodesic with respect to both affine connections, and

(3) there exists an orthonormal frame $\{v_{2\alpha+1}, v_{2\alpha+2}, \cdots, v_{2\alpha+\beta}\}$ of $\mathcal{D}^{\perp}$ such that

(a) $B^{a}_{aa} = 3B^{a}_{bb}$ and ${B*}^{a}_{aa} = 3{B*}^{a}_{bb}$ , for $2\alpha + 1 \leq a \neq b \leq 2\alpha + \beta$ ,

(b) $B^{a}_{bc} = 0$ and ${B*}^{a}_{bc} = 0$ for $a, b, c \in \{2\alpha + 1, \cdots, 2\alpha + \beta\}$ , $a\neq b \neq c$ .

It is evident that the equality case of (4.3) holds identically when $N$ is doubly $\mathcal{D}$ -minimal, and mixed totally geodesic with respect to both affine connections. It is worth noting that the equality in (4.5) holds identically if and only if the three conditions from Theorem 4.1 are met.

5. Conclusion and Remarks

(1) In the early 1990s, the renowned author B.-Y. Chen introduced the concept of $\delta$ -invariants (see ^[4,6,7]) to address an open question concerning minimal immersions proposed by S.S. Chern in the 1960s, as well as to explore applications of the well-known Nash embedding theorem. Chen specifically defined the CR $\delta$ -invariant for anti-holomorphic submanifolds in complex space forms. Building on this work, we extended this study to the statistical version of contact CR $\delta$ -invariant.

(2) In fact, Furuhata et al. introduced a novel notion of $U$ sectional curvature for statistical manifolds $(M, \tilde{D}, G)$ in ^[12] as follows:

$\begin{eqnarray*} \tilde{S}^{U}(X\wedge Y)& = &G(U(X, Y)Y, X)\\ & = &2G(\tilde{R}^{0}(X, Y)Y, X)-G(\tilde{\mathcal{R}}(X, Y)Y, X)\\ & = & (2\tilde{S}^{0}-\tilde{S}^{\tilde{D}, \tilde{D*}})(X\wedge Y), \end{eqnarray*}$

where $\tilde{R}^{0}$ is the Riemannian curvature tensor for $\tilde{D}^{0}$ on $M$ . They also defined a corresponding $\delta$ -invariant $\delta^{U}$ based on this new concept of $U$ sectional curvature for statistical manifolds. It would be of significant interest to reformulate such an optimal inequality by defining a new notion for the (contact) CR $\delta^{U}$ -invariant for (contact) CR-submanifolds in a (respectively, Sasakian statistical manifold) holomorphic statistical manifold.

(3) It would be of significant interest to check whether Proposition 3.3 is valid for any codimension. We have already established its validity for codimension greater than or equal to $2$ . Thus, the remaining case to consider is when $N$ is a hypersurface, that is, of codimension 1.

(4) From ^[9,18], we note that a contact CR-submanifold $(N, D, g)$ in a Sasakian statistical manifold $(M, \tilde{D}, G, \phi, \xi)$ is said to be mixed foliate with respect to $\tilde{D}$ (respectively, ${\tilde{D}*}$ ) if $N$ is mixed totally geodesic with respect to $\tilde{D}$ (respectively, ${\tilde{D}*}$ ) and $\mathcal{D}$ is completely integrable. Now, let us consider a $(r+1)$ -dimensional generic submanifold $(N, D, g)$ in a $(2s+1)$ -dimensional Sasakian statistical manifold $(M(c), \tilde{D}, G, \phi, \xi)$ of constant $\phi$ -sectional curvature $c$ , where $\dim(\mathcal{D}) = 2\alpha + 1$ and $\dim(\mathcal{D}^{\perp}) = \beta$ . If $N$ satisfies the equality case of (4.5) and $\mathcal{D}$ is integrable, then it follows from Theorem 4.1 that $N$ is mixed foliate with respect to $\tilde{D}$ (respectively, ${\tilde{D}*}$ ).

(5) In ^[10], Furuhata et al. constructed Sasakian statistical structures on the $(2s+1)$ -dimensional unit hypersphere $\mathbb{S}$ in $(2s+2)$ -dimensional Euclidean space $\mathbb{R}$ . They showed that $\mathbb{S}$ is a Sasakian statistical manifold of constant statistical sectional curvature 1 and of constant $\phi$ -sectional curvature $c = 1$ as well by setting $K_{X}Y = G(X, \xi)G(Y, \xi)\xi$ . Building in this, we consider $N = \mathbb{S}^{s_{1}}(r_{1})\times \cdots \times \mathbb{S}^{s_{k}}(r_{k})$ and an immersion $N\rightarrow \mathbb{S}^{m+k+1}$ from ^[23], where $m = s_{1}+s_{2}+\cdots+s_{k}$ , $\sum_{i = 1}^{k}r_{i}^{2} = 1$ . It is straightforward to observe that $N$ is a generic statistical submanifold of a Sasakian statistical manifold $\mathbb{S}^{m+k+1}$ , provided that all $s_{i}$ are odd.

(6) Let $\mathbb{E}^{2s+1}(-3)$ be a Sasakian space form of constant $\phi$ -sectional curvature $-3$ with Sasakian structure $(G, \phi, \xi)$ on $\mathbb{E}^{2s+1}(-3)$ as:

$\begin{eqnarray*} &&G = \left ( \large \begin{array}{ccc} \frac{1}{4}(\delta_{ij}+y^{i}y{^j})& 0 & \frac{-1}{4}y^{i}\\ 0 & \frac{1}{4} \delta_{ij} & 0 \\ \frac{1-}{4}y^{j} & 0 & \frac{1}{4} \\ \end{array} \right ), \quad \phi = \left ( \large \begin{array}{ccc} 0& \delta^{i}_{j} & 0\\ -\delta^{i}_{j} & 0 & 0 \\ 0 & y^{j} & 0 \\ \end{array} \right ), \\ \end{eqnarray*}$

$\begin{eqnarray*} &&\xi = (0, 0, \cdots, 0, 0, 2), \quad G(\cdot, \xi) = (-y^{1}, \cdots, -y^{s}, 0, \cdots, 0, 1), \end{eqnarray*}$

where $(x^{1}, \cdots, x^{n}, y^{1}, \cdots, y^{n}, z)$ denotes the cartesian coordinates. Now, we can construct a Sasakian statistical structure on $(\mathbb{E}^{2s+1}(-3), G, \phi, \xi)$ by setting $K_{X}Y = G(X, \xi)G(Y, \xi)\xi$ satisfies Definition 2.2.

Further, we consider $N^{2s} = \mathbb{S}^{2s-1}\times \mathbb{E}^{1}$ . Then $N^{2s}$ is doubly totally contact umbilical submanifold of $\mathbb{E}^{2s+1}(-3)$ .

Author contributions

Aliya Naaz Siddiqui: Conceptualization, Methodology, Formal analysis, Writing-Original draft preparation, Writing-Reviewing and Editing. Meraj Ali Khan: Visualization, Investigation, Supervision. Amira Ishan: Project administration, Funding acquisition. All authors have read and approved the final version of the manuscript for publication.

Acknowledgment

The authors would like to acknowledge the Deanship of Graduate Studies and Scientific Research, Taif University, for funding this work.

Conflict of interest

The authors declare there are no conflicts of interest.

References

[1]	L. Ma, Z. Wang, J. Fan, S. Hu, An essential introduction to the annual report on cardiovascular health and diseases in china (2021) (in Chinese), Chin. Gen. Pract., 25 (2022), 3331–3346. https://doi.org/10.12114/j.issn.1007-9572.2022.0506 doi: 10.12114/j.issn.1007-9572.2022.0506
[2]	W. Wang, X. Yu, B. Fang, D. Y. Zhao, Y. Chen, W. Wei, et al., Cross-modality lge-cmr segmentation using image-to-image translation based data augmentatio, IEEE/ACM Trans. Comput. Biol. Bioinf., 20 (2023), 2367–2375. https://doi.org/10.1109/TCBB.2022.3140306 doi: 10.1109/TCBB.2022.3140306
[3]	Y. Duan, S. Tang, Observation on the effect of bian quefei rescue remote first aid system in pre-hospital first aid for patients with chest pain (in Chinese), Chin. Med. Equip., 14 (2017), 83–86. https://doi.org/10.3969/J.ISSN.1672-8270.2017.10.024 doi: 10.3969/J.ISSN.1672-8270.2017.10.024
[4]	S. Ding, J. Xiong, Y. Zhang, Y. Zhao, H. Zhang, Y. Xu, Clinical application of time management system in chest pain center (in Chinese), Intern. Med. Theory Pract., 16 (2021), 202–204. https://doi.org/10.16138/j.1673-6087.2021.03.013 doi: 10.16138/j.1673-6087.2021.03.013
[5]	H. V. Denysyuk, R. J. Pinto, P. M. Silva, R. P. Duarte, F. A. Marinho, L. Pimenta, et al., Algorithms for automated diagnosis of cardiovascular diseases based on ECG data: A comprehensive systematic review, Heliyon, 9 (2023), e13601. https://doi.org/10.1016/j.heliyon.2023.e13601 doi: 10.1016/j.heliyon.2023.e13601
[6]	H. Costa, C. A. Costa, R. S. Antunes, R. D. R. Righi, P. A. Crocker, V. R. Q. Leithardt, et al., ID-Care: A model for sharing wide healthcare data, IEEE Access, 11 (2023), 33455–33469. https://doi.org/10.1109/ACCESS.2023.3249109 doi: 10.1109/ACCESS.2023.3249109
[7]	H. Chen, D. Xiang, W. Qin, M. Zhou, Y. Tian, J. Liu, et al., A study of regional cooperative emergency care system for st-elevation myocardial infarction patients based on the internet of things, in 2012 IEEE 14th International Conference on e-Health Networking, Applications and Services (Healthcom), (2012), 73–77. https://doi.org/10.1109/HealthCom.2012.6380069
[8]	H. Chen, D. Xiang, W. Qin, M. Zhou, J. Yang, J. Liu, et al., Optimal st-elevation myocardial infarction system by regional cooperative emergency care based on the internet of things, in Smart Health, Springer, (2014), 225–232. https://doi.org/10.1007/978-3-319-08416-9_24
[9]	Y. Cao, S. Miraba, S. Rafiei, A. Ghabussi, F. Bokaei, S. Baharom, et al., Economic application of structural health monitoring and internet of things in efficiency of building information modeling, Smart Struct. Syst., 26 (2020), 559–573. https://doi.org/10.12989/sss.2020.26.5.559 doi: 10.12989/sss.2020.26.5.559
[10]	R. Ma, M. Karimzadeh, A. Ghabussi, Y. Zandi, S. Baharom, A. Selmi, et al., Assessment of composite beam performance using GWO–ELM metaheuristic algorithm, Eng. Comput., 38 (2021), 2083–2099. https://doi.org/10.1007/s00366-021-01363-1 doi: 10.1007/s00366-021-01363-1
[11]	A. Morasaei, A. Ghabussi, S. Aghlmand, M. Yazdani, S. Baharom, H. Assilzade, Simulation of steel–concrete composite floor system behavior at elevated temperatures via multi-hybrid metaheuristic framework, Eng. Comput., 38 (2022), 2567–2582. https://doi.org/10.1007/s00366-020-01228-z doi: 10.1007/s00366-020-01228-z
[12]	K. Tajziehchi, A. Ghabussi, H. Alizadeh, Control and optimization against earthquake by using genetic algorithm, J. Appl. Eng. Sci., 8 (2018), 73–78. https://doi.org/10.2478/jaes-2018-0010 doi: 10.2478/jaes-2018-0010
[13]	D. Xiang, S. Yi, Chest pain centers in china: current status and prospects, Cardiol. Plus, 2 (2017), 18–21. https://doi.org/10.4103/2470-7511.248469 doi: 10.4103/2470-7511.248469
[14]	Q. Xiao, M. Zhou, Y. Tian, Informatization construction and practice of chest pain center (in Chinese), China Digital Med., 2015 (2015), 17–18. https://doi.org/10.3969/j.issn.1673-7571.2015.09.006 doi: 10.3969/j.issn.1673-7571.2015.09.006
[15]	B. G. Choi, J. Y. Park, S. W. Rha, Y. K. Noh, Pre-test probability for coronary artery disease in patients with chest pain based on machine learning techniques, Int. J. Cardiol., 385 (2023), 85–93. https://doi.org/10.1016/j.ijcard.2023.05.041 doi: 10.1016/j.ijcard.2023.05.041
[16]	H. Zhou, W. Feng, H. Li, L. Jin, L. Qian, X. Zhu, et al., Effect evaluation and quality control strategy of grassroots chest pain center construction (in Chinese), Chin. J. Gen. Pract., 19 (2020), 434–437. https://doi.org/10.3760/cma.j.cn114798-20200312-00280 doi: 10.3760/cma.j.cn114798-20200312-00280
[17]	Y. Han, S. Sun, B. Qiao, H. Liu, C. Zhang, B. Wang, et al., Timing of angiography and outcomes in patients with non-st-segment elevation myocardial infarction: insights from the evaluation and management of patients with acute chest pain in china registry, Front. Cardiovasc. Med., 9 (2022), 1000554. https://doi.org/10.3389/fcvm.2022.1000554 doi: 10.3389/fcvm.2022.1000554
[18]	F. Fan, Y. Li, Y. Zhang, J. Li, J. Liu, H. Hao, et al., Deep learning-based classification of mesothelioma improves prediction of patient outcome, J. Am. Heart Assoc., 8 (2019), e013384. https://doi.org/10.1161/JAHA.119.013384 doi: 10.1161/JAHA.119.013384
[19]	J. Li, X. Li, Q. Wang, S. Hu, Y. Wang, F. A. Masoudi, et al., ST-segment elevation myocardial infarction in china from 2001 to 2011 (the china peace-retrospective acute myocardial infarction study): a retrospective analysis of hospital data, The Lancet, 385 (2015), 441–451. https://doi.org/10.1016/S0140-6736(14)60921-1 doi: 10.1016/S0140-6736(14)60921-1
[20]	W. B. Gibler, J. P. Runyon, R. C. Levy, M. R. Sayre, R. Kacich, C. R. Hattemer, et al., A rapid diagnostic and treatment center for patients with chest pain in the emergency department, Annals of emergency medicine, 25 (1995), 1–8. https://doi.org/10.1016/S0196-0644(95)70347-0 doi: 10.1016/S0196-0644(95)70347-0
[21]	K. Tang, Z. Shuai, Z. Li, L. Zhou, J. Gou, Y. Wang, et al., Research status of biomarkers of acute coronary syndrome, Chin. J. Arterioscler., 29 (2021), 451–455. https://doi.org/10.3969/j.issn.1007-3949.2021.05.016 doi: 10.3969/j.issn.1007-3949.2021.05.016
[22]	J. Liu, D. Zhao, Q. Zhang, Recent hospitalization trends for acute myocardial infarction in Beijing, Eur. Heart J., 37 (2016), 3188–3189. https://doi.org/10.1093/eurheartj/ehw472 doi: 10.1093/eurheartj/ehw472
[23]	L. Li, J. Zhao, L. Hou, Y. Zhai, J. Shi, F. Cui, An attention-based deep learning model for clinical named entity recognition of Chinese electronic medical records, BMC Med. Inf. Decis. Making, 19 (2019), 1–11. https://doi.org/10.1186/s12911-019-0933-6 doi: 10.1186/s12911-019-0933-6
[24]	J. Yu, X. Kang, C. Bai, H. Liu, A new text retrieval model for chinese electronic medical records (in Chinese), Comput. Sci., 49 (2022), 32–38. https://doi.org/10.11896/jsjkx.210400198 doi: 10.11896/jsjkx.210400198
[25]	K. Xu, Z. Yang, P. Kang, Q. Wang, W. Liu, Document-level attention-based bilstm-crf incorporating disease dictionary for disease named entity recognition, Comput. Biol. Med., 108 (2019), 122–132. https://doi.org/10.1016/j.compbiomed.2019.04.002 doi: 10.1016/j.compbiomed.2019.04.002
[26]	Q. Wan, J. Liu, L. Wei, B. Ji, A self-attention based neural architecture for Chinese medical named entity recognition, Math. Biosci. Eng., 17 (2020), 3498–3511. https://doi.org/10.3934/mbe.2020197 doi: 10.3934/mbe.2020197
[27]	Z. Long, W. Liu, Z. Zhao, S. Tong, L. Wang, M. Zhou, et al., Case fatality rate of patients with acute myocardial infarction in 253 chest pain centers—china, 2019–2020, China CDC Weekly, 4 (2022), 518–521. https://doi.org/10.46234/ccdcw2022.026 doi: 10.46234/ccdcw2022.026
[28]	D. C. Xiang, Y. Z. Jin, W. Y. Fang, X. Su, B. Yu, Y. Wang, et al., The national chest pain centers program: Monitoring and improving quality of care for patients with acute chest pain in China, Cardiol. Plus, 6 (2021), 187–197. https://doi.org/10.4103/2470-7511.327239 doi: 10.4103/2470-7511.327239
[29]	F. Tong, C. Wang, S. Han, Y. Li, Z. Li, Z. Sun, Differential risk assessment of acute aortic syndrome with chest pain onset less than 3 hours and non-st-segment elevation myocardial infarction (in Chinese), Chin. J. Arterioscler., 8 (2021), 681–687. https://doi.org/10.3969/j.issn.1007-3949.2021.08.006 doi: 10.3969/j.issn.1007-3949.2021.08.006
[30]	J. Chen, Z. Guo, X. Xu, L. Zhang, Y. Teng, Y. Chen, et al., A robust deep learning framework based on spectrograms for Heart sound classification, IEEE/ACM Trans. Comput. Biol. Bioinf., 2023 (2023), 1–12. https://doi.org/10.1109/TCBB.2023.3247433 doi: 10.1109/TCBB.2023.3247433
[31]	J. Chen, S. Sun, L. Zhang, B. Yang, W. Wang, Compressed sensing framework for heart sound acquisition in internet of medical things, IEEE Trans. Ind. Inf., 18 (2021), 2000–2009. https://doi.org/10.1109/TII.2021.3088465 doi: 10.1109/TII.2021.3088465
[32]	J. Wang, C. J. Rao, M. Goh, X. P. Xiao, Risk assessment of coronary heart disease based on cloud-random forest, Artif. Intell. Rev., 56 (2023), 203–232. https://doi.org/10.1007/s10462-022-10170-z doi: 10.1007/s10462-022-10170-z

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)