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Statistical Killing vector field is introduced and Hopf hypersurfaces of complex space forms with the condition of being structural statistical Killing vector field are studied. It is shown that these hypersurfaces have at most three distinct constant principal curvatures. [ABSTRACT FROM AUTHOR]

Let (M , ∇ , g) be a statistical manifold and T M be its tangent bundle endowed with a twisted Sasaki metric G. This paper serves two primary objectives. The first objective is to investigate the curvature properties of the tangent bundle T M . The second objective is to explore conformal vector fields and Ricci, Yamabe, and gradient Ricci–Yamabe solitons on the tangent bundle T M according to the twisted Sasaki metric G. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to study statistical submersions with parallel almost complex structures. First, we define the notion of the generalized Kähler-like statistical submersion and give examples of the Kähler-like statistical submersions. In addition, we investigate total space and fibers under certain conditions. After, we introduce some results on J-invariant, J ∗ -invariant and anti-invariant generalized Kähler-like statistical submersions. [ABSTRACT FROM AUTHOR]

Purpose – In this paper, we give some properties of the α-connections on statistical manifolds and we study the α-conformal equivalence where we develop an expression of curvature R¯ for ∇¯ in relation to those for ∇ and ∇^. Design/methodology/approach – In the first section of this paper, we prove some results about the α-connections of a statistical manifold where we give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds treated in [1, 3], and we construct some examples. Findings – We give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples. Originality/value – We give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.

We determine the general natural metrics G on the total space T M of the tangent bundle of a Riemannian manifold (M , g) such that the Schouten–van Kampen connection ∇ ¯ associated to the Levi-Civita connection of G is (quasi-)statistical. We prove that the base manifold must be a space form and in particular, when G is a natural diagonal metric, (M , g) must be locally flat. We prove that there exist one family of natural diagonal metrics and two families of proper general natural metrics such that (T M , ∇ ¯ , G) is a statistical manifold and one family of proper general natural metrics such that (T M ∖ { 0 } , ∇ ¯ , G) is a quasi-statistical manifold. [ABSTRACT FROM AUTHOR]

One of the most fundamental interests in submanifold theory is to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of submanifolds and find their applications. In this respect, the first author established, in 1993, a basic inequality involving the first δ -invariant, δ (2) , and the squared mean curvature of submanifolds in real space forms, known today as the first Chen inequality or Chen's first inequality. Since then, there have been many papers dealing with this inequality. The purpose of this article is, thus, to present a comprehensive survey on recent developments on this inequality performed by many geometers during the last three decades. [ABSTRACT FROM AUTHOR]

Let (M,∇,g) be a statistical manifold and TM be its tangent bundle endowed with a twisted Sasaki metric G. This paper serves two primary objectives. The first objective is to investigate the curvature properties of the tangent bundle TM. The second objective is to explore conformal vector fields and Ricci, Yamabe, and gradient Ricci–Yamabe solitons on the tangent bundle TM according to the twisted Sasaki metric G.

The gauge equation is a generalization of the conjugacy relation for the Koszul connection to bundle morphisms that are not isomorphisms. The existence of nontrivial solution to this equation, especially when duality is imposed upon related connections, provides important information about the geometry of the manifolds under consideration. In this article, we use the gauge equation to introduce spectral sequences that are further specialized to Hessian structures.

In this paper, we introduce an indefinite Lorentzian para-Sasakian (LPSasakian) statistical manifold and study lightlike submanifold of an indefinite LPSasakian statistical manifold. We also introduce some relations among induced geometrical objects with respect to dual connections in a lightlike submanifold of an indefinite LP-Sasakian statistical manifold. One example related to this concept is also presented. Finally, we show that an invariant lightlike submanifold of an indefinite LP-Sasakian statistical manifold is an indefinite LP-Sasakian statistical manifold. [ABSTRACT FROM AUTHOR]

Our purpose in this article is to study anti-invariant statistical submersions from holomorphic statistical manifolds. Firstly we introduce holomorphic statistical submersions satisfying the certain condition, after we give anti-invariant statistical submersions satisfying the certain condition. And we supported our results with examples. [ABSTRACT FROM AUTHOR]

A denoising algorithm was proposed for point cloud with high-density noise. The algorithm utilized geometric metrics on the statistical manifold and applied the idea of clustering K-means based on local statistical characteristics between noise and valid data. First, by calculating the expectation and covariance matrix of the data points, the point cloud with high-density noise was projected onto the Gaussian distribution family manifold, aiming to form the parameter point cloud. Afterwards, the geometry metrics were assigned to the manifold, and the K-means algorithm was applied to cluster the parameter point cloud, so as to classify the valid data and noise. Furthermore, in order to analyze the robustness of the means with different metrics, the approximate values of their influence functions were calculated, respectively. Finally, simulation analysis was conducted using the algorithm based on geometric metrics to verify the effectiveness in point cloud denoising. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to find some conditions under which the tangent bundle T M has a dualistic structure. Then, we introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution and the tangent bundle of a statistical manifold too. Moreover, we also study the mutual curvatures of a statistical manifold M and its tangent bundle T M and we investigate their relations. More precisely, we obtain the mutual curvatures of well-known connections on the tangent bundle T M (the complete, horizontal, and Sasaki connections) and we study the vanishing of them. [ABSTRACT FROM AUTHOR]

Let (M, g) be a Riemannian manifold and TM be its tangent bundle. The purpose of this paper is to study statistical structures on TM with respect to the metrics Gf 1 = cg + v(fg) and Gf 2 = sgf + hg, where f is a smooth function on M, cg is the complete lift of g, v(fg) is the vertical lift of fg, sgf is a metric obtained by rescaling the Sasaki metric by a smooth function f and hg is the horizontal lift of g. Moreover, we give some results about Killing vector fields on TM with respect to these metrics. [ABSTRACT FROM AUTHOR]

In information geometry, there has been extensive research on the deep connections between differential geometric structures, such as the Fisher metric and the α-connection, and the statistical theory for statistical models satisfying regularity conditions. However, the study of information geometry for non-regular statistical models is insufficient, and a one-sided truncated exponential family (oTEF) is one example of these models. In this paper, based on the asymptotic properties of maximum likelihood estimators, we provide a Riemannian metric for the oTEF. Furthermore, we demonstrate that the oTEF has an α = 1 parallel prior distribution and that the scalar curvature of a certain submodel, including the Pareto family, is a negative constant. [ABSTRACT FROM AUTHOR]

Introduction Let p(x,ζ) be the set of parametric probability distribution with parameter ζ=ζ1,…,ζn∊Rn. This set is called a statistical model or manifold. The distance between two points is measured by the Fisher metric. In general, statistical manifolds are Riemannian manifolds of distributions endowed with the Fisher information metric. On the other hand, one of the most important structures on odd dimensional Riemannian manifolds is the almost contact structure. Recently, statistical manifolds equipped with almost contact structures are studied by many authors. In this paper, we introduce statistical almost contact-like and statistical cosymplectic manifolds on a Riemannian manifold. We recall the basic definitions and define statistical cosymplectic manifolds and their invariant submanifolds. We prove that an invariant submanifold of a statistical cosymplectic manifold with tangent structure vector field is a cosymplectic and minimal-like submanifold. Also, we prove if the structure vector field be normal to the submanifold then the submanifold is a statistical Keahler-like manifold. Finally, we construct two examples to illustrate some results of the paper. Statistical almost contact-like manifolds Let M,g be a Riemannian manifold with the Levi-Civita connection ∇. M,g is called a statistical manifold if there exists an affine and torsion free connection ∇ such that for all U,V,W∊τM ∇UgV,W=∇VgU,W. Moreover, an affine and torsion free connection ∇* is called a dual connection with respect to g, if UgV,W=g∇UV,W+gV,∇*U W. An almost contact manifold (M,φ,ξ,η) with Riemannian metric g is an almost contact-like manifold if it has another (1,1)-tensor field φ* satisfying gφU,V=-gU,φ*V, gU, ξ= ηU. Let (M,φ,ξ,η) be an almost contact-like manifold, then for all U,V∊τ(M) the following relations hold gφU, φ*V=gU,V- ηUηV, φ*2U=-U+ηUξ. Definition. An almost contact-like manifold (M,∇,φ,ξ,η,g) with statistical structure (∇,g) is a statistical almost contact-like manifold. Moreover, M,∇,φ,ξ,η,g is called a statistical cosymplectic manifold if ∇UφV=0. M is an invariant submanifold of a statistical cosymplectic manifold M,∇,φ,ξ,η,g, if for all U∊τ(M) we have φU∊τM, φ*U∊τM. Submanifolds of statistical cosymplectic manifolds We show that the manifold M,∇,φ,ξ,η,g is a statistical cosymplectic manifold if and only if (M,∇*,φ*,ξ,η,g) is a statistical cosymplectic manifold. Moreover we prove the following theorems. Theorem. Any invariant submanifold of a statistical cosymplectic manifold with tangent structure vector field ξ, is a statistical cosymplectic and minimal-like submanifold. Theorem. Let M be a submanifold of statistical cosymplectic manifold (M,∇,φ,ξ,η,g) such that the structure vector field ξ is normal to M. Then for any vector field U∊τ(M) we have A*ξU=0, ∇⊥Uξ=η∇Uξξ. Theorem. Let M,∇,φ,ξ,η,g be a statistical cosymplectic manifold. If M is a submanifold of M and the structure vector field ξ is normal to M then R⊥U,Vξ=0, ∀U,V∊τM. Theorem. Let M be an invariant submanifold of statistical cosymplectic manifold M,∇,φ,ξ,η,g and ξ is normal to M. Then M is a statistical Keahler-like manifold. Conclusion We introduce statistical cosymplectic manifolds and investigate some properties of their tensors. We define invariant and anti-invariant submanifolds and study invariant submanifolds with normal and tangent structure vector fields. We prove that an invariant submanifold of a statistical cosymplectic manifold with tangent structure vector field is a cosymplectic and minimal-like submanifold. Also we show if the structure vector field is normal to the submanifold then that is a statistical Keahler-like manifold

In this paper, we introduce and study Golden-like statistical manifolds. We obtain some basic inequalities for curvature invariants of statistical submanifolds in Golden-like statistical manifolds. Also, in support of our definition, we provide a couple of examples.

We determine the general natural metrics G on the total space TM of the tangent bundle of a Riemannian manifold (M,g) such that the Schouten–van Kampen connection ∇¯ associated to the Levi-Civita connection of G is (quasi-)statistical. We prove that the base manifold must be a space form and in particular, when G is a natural diagonal metric, (M,g) must be locally flat. We prove that there exist one family of natural diagonal metrics and two families of proper general natural metrics such that (TM,∇¯,G) is a statistical manifold and one family of proper general natural metrics such that (TM∖{0},∇¯,G) is a quasi-statistical manifold.

One of the most fundamental interests in submanifold theory is to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of submanifolds and find their applications. In this respect, the first author established, in 1993, a basic inequality involving the first δ-invariant, δ(2), and the squared mean curvature of submanifolds in real space forms, known today as the first Chen inequality or Chen’s first inequality. Since then, there have been many papers dealing with this inequality. The purpose of this article is, thus, to present a comprehensive survey on recent developments on this inequality performed by many geometers during the last three decades.

We study lift metrics and lift connections on the tangent bundle TM of a Riemannian manifold (M, g). We also investigate the statistical and Codazzi couples of TM and their consequences on the geometry of M. Finally, we prove a result on 1-Stein and Osserman structures on TM, whenever TM is equipped with the complete lift connection. [ABSTRACT FROM AUTHOR]

In this paper, we establish some inequalities between the normalized δ -Casorati curvatures and the scalar curvature (i.e., between extrinsic and intrinsic invariants) of spacelike statistical submanifolds in Sasaki-like statistical manifolds, endowed with a semi-symmetric metric connection. Moreover, we study the submanifolds satisfying the equality cases of these inequalities. We also present an appropriate example. [ABSTRACT FROM AUTHOR]

A denoising algorithm was proposed for point cloud with high-density noise. The algorithm utilized geometric metrics on the statistical manifold and applied the idea of clustering K-means based on local statistical characteristics between noise and valid data. First, by calculating the expectation and covariance matrix of the data points, the point cloud with high-density noise was projected onto the Gaussian distribution family manifold, aiming to form the parameter point cloud. Afterwards, the geometry metrics were assigned to the manifold, and the K-means algorithm was applied to cluster the parameter point cloud, so as to classify the valid data and noise. Furthermore, in order to analyze the robustness of the means with different metrics, the approximate values of their influence functions were calculated, respectively. Finally, simulation analysis was conducted using the algorithm based on geometric metrics to verify the effectiveness in point cloud denoising.

The purpose of this paper is to find some conditions under which the tangent bundle TM has a dualistic structure. Then, we introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution and the tangent bundle of a statistical manifold too. Moreover, we also study the mutual curvatures of a statistical manifold M and its tangent bundle TM and we investigate their relations. More precisely, we obtain the mutual curvatures of well-known connections on the tangent bundle TM (the complete, horizontal, and Sasaki connections) and we study the vanishing of them.

The physical meaning and mathematical meaning of nonmetricity on Weyl manifold and the application of dual affine connection for it are discussed. In viewpoint from affine connection, Weyl 1-form in Weyl manifold is recognized as nonmetricity and causes scale change with conformal invariance under parallel transport. On manifolds expressing spacetime and material space, Weyl 1-form is equivalent to the extra coupling field to field expressed as Riemannian manifold, protruding from Riemannian manifold. The scale change with conformal invariance corresponds to the similar change in the magnitude of the field with the volumetric distortion of each manifold. Weyl 1-form is the change rate of volume element (scale) in Riemannian manifold. Therefore, nonmetricity in Weyl manifold is defined as expansion or contraction rate as similar volumetric distortion due to the extra coupled field. The volumetric distortion as nonmetricity on Riemannian manifold can be canceled out by setting dual affine connection on Riemann–Cartan–Weyl manifold, which is used in statistical manifold. Moreover, the role of nonmetricity in modified gravity theory in the framework of symmetric teleparallel manifold and the meaning of nonmetricity on statistical manifold are discussed based on this concept. [ABSTRACT FROM AUTHOR]

In this study, some identities involving the Riemannian curvature invariants are presented on lightlike hypersurfaces of a statistical manifold in the Lorentzian settings. Several inequalities characterizing lightlike hypersurfaces are obtained. These inequalities are also investigated on lightlike hypersurfaces of Lorentzian statistical space forms. [ABSTRACT FROM AUTHOR]

With the globalization of higher education, academic evaluation is increasingly valued by the scientific and educational circles. Although the number of published papers of academic evaluation methods is increasing, previous research mainly focused on the method of assigning different weights for various indicators, which can be subjective and limited. This paper investigates the evaluation of academic performance by using the statistical K-means (SKM) algorithm to produce clusters. The core idea is mapping the evaluation data from Euclidean space to Riemannian space in which the geometric structure can be used to obtain accurate clustering results. The method can adapt to different indicators and make full use of big data. By using the K-means algorithm based on statistical manifolds, the academic evaluation results of universities can be obtained. Furthermore, through simulation experiments on the top 20 universities of China with the traditional K-means, GMM and SKM algorithms, respectively, we analyze the advantages and disadvantages of different methods. We also test the three algorithms on a UCI ML dataset. The simulation results show the advantages of the SKM algorithm. [ABSTRACT FROM AUTHOR]

The main focus of this paper is concern to the study on the point-wise Osserman structure on 4-dimensional Lorentzian Lie group. In this paper we study on the spectrum of the Jacobi operator and spectrum of the skew-symmetric curvature operator on the non-abelian 4-dimensional Lie group G, whenever G equipped with an orthonormal left invariant pseudo-Riemannian metric g of signature (¡;+;+; +), i.e, Lorentzian metric, where e1 is a unit time-like vector. The Lie algebra structure in dimension four has key role in our investigation, also in this case we study on the classification of 1-Stein and mixed IP spaces. At the end we show that G does not admit any space form and Einstein structures. [ABSTRACT FROM AUTHOR]

In information geometry, there has been extensive research on the deep connections between differential geometric structures, such as the Fisher metric and the α-connection, and the statistical theory for statistical models satisfying regularity conditions. However, the study of information geometry for non-regular statistical models is insufficient, and a one-sided truncated exponential family (oTEF) is one example of these models. In this paper, based on the asymptotic properties of maximum likelihood estimators, we provide a Riemannian metric for the oTEF. Furthermore, we demonstrate that the oTEF has an α = 1 parallel prior distribution and that the scalar curvature of a certain submodel, including the Pareto family, is a negative constant.

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely the normalized δ -Casorati curvatures and the scalar curvature of statistical submanifolds in Kenmotsu statistical manifolds of constant ϕ -sectional curvature that are endowed with semi-symmetric metric connection. Furthermore, we investigate the equality cases of these inequalities. We also describe an illustrative example. [ABSTRACT FROM AUTHOR]

We consider a statistical connection ∇ on an almost complex manifold with (pseudo-) Riemannian metric, in particular the Norden metric. We investigate almost Norden (statistical) manifolds under the condition that the almost complex structure J is ∇-recurrent. We provide one example of a complex statistical connection. [ABSTRACT FROM AUTHOR]

The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. The study of inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature was truly initiated in 2018 by Mihai, A. and Mihai, I. who dealt with Chen-Ricci and Euler inequalities. Later on, Siddiqui, A.N., Ahmad K. and Ozel C. came with the study of Casorati inequality for statistical submanifolds in the same ambient space by using algebraic technique. Also, Chen, B.-Y., Mihai, A. and Mihai, I. obtained a Chen first inequality for such submanifolds. In 2020, Mihai, A. and Mihai, I. studied the Chen inequality for δ (2 , 2) -invariant. In the development of this topic, we establish the generalized Wintgen inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Some examples are also discussed at the end. [ABSTRACT FROM AUTHOR]

Our purpose in this article is to study anti-invariant ξ⊥-cosymplectic-like statistical submersions from cosymplectic-like statistical manifolds and an example. Also, we investigate the integrability and the totally geodesicness of the distributions and the geometry of foliations. [ABSTRACT FROM AUTHOR]

Introduction Let p(x,ζ) be the set of parametric probability distribution with parameter ζ=ζ1,…,ζn∊Rn. This set is called a statistical model or manifold. The distance between two points is measured by the Fisher metric. In general, statistical manifolds are Riemannian manifolds of distributions endowed with the Fisher information metric.  On the other hand, one of the most important structures on odd dimensional Riemannian manifolds is the almost contact structure. Recently, statistical manifolds equipped with almost contact structures are studied by many authors. In this paper, we introduce statistical almost contact-like and statistical cosymplectic manifolds on a Riemannian manifold. We recall the basic definitions and define statistical cosymplectic manifolds and their invariant submanifolds. We prove that an invariant submanifold of a statistical cosymplectic manifold with tangent structure vector field is a cosymplectic and minimal-like submanifold. Also, we prove if the structure vector field be normal to the submanifold then the submanifold is a statistical Keahler-like manifold. Finally, we construct two examples to illustrate some results of the paper. Statistical almost contact-like manifolds Let M,g be a Riemannian manifold with the Levi-Civita connection ∇. M,g is called a statistical manifold if there exists an affine and torsion free connection ∇ such that for all U,V,W∊τM ∇UgV,W=∇VgU,W. Moreover, an affine and torsion free connection ∇* is called a dual connection with respect to g, if UgV,W=g∇UV,W+gV,∇*U W. An almost contact manifold (M,φ,ξ,η) with Riemannian metric g is an almost contact-like manifold if it has another (1,1)-tensor field φ* satisfying gφU,V=-gU,φ*V, gU, ξ= ηU. Let (M,φ,ξ,η) be an almost contact-like manifold, then for all U,V∊τ(M) the following relations hold gφU, φ*V=gU,V- ηUηV, φ*2U=-U+ηUξ. Definition. An almost contact-like manifold (M,∇,φ,ξ,η,g) with statistical structure (∇,g) is a statistical almost contact-like manifold. Moreover, M,∇,φ,ξ,η,g is called a statistical cosymplectic manifold if ∇UφV=0. M is an invariant submanifold of a statistical cosymplectic manifold M,∇,φ,ξ,η,g, if for all U∊τ(M) we have φU∊τM, φ*U∊τM. Submanifolds of statistical cosymplectic manifolds We show that the manifold M,∇,φ,ξ,η,g is a statistical cosymplectic manifold if and only if (M,∇*,φ*,ξ,η,g) is a statistical cosymplectic manifold. Moreover we prove the following theorems. Theorem. Any invariant submanifold of a statistical cosymplectic manifold with tangent structure vector field ξ, is a statistical cosymplectic and minimal-like submanifold. Theorem. Let M be a submanifold of statistical cosymplectic manifold (M,∇,φ,ξ,η,g) such that the structure vector field ξ is normal to M. Then for any vector field U∊τ(M) we have A*ξU=0, ∇⊥Uξ=η∇Uξξ. Theorem. Let M,∇,φ,ξ,η,g be a statistical cosymplectic manifold. If M is a submanifold of M and the structure vector field ξ is normal to M then R⊥U,Vξ=0, ∀U,V∊τM. Theorem. Let M be an invariant submanifold of statistical cosymplectic manifold M,∇,φ,ξ,η,g and ξ is normal to M. Then M is a statistical Keahler-like manifold. Conclusion We introduce statistical cosymplectic manifolds and investigate some properties of their tensors. We define invariant and anti-invariant submanifolds and study invariant submanifolds with normal and tangent structure vector fields. We prove that an invariant submanifold of a statistical cosymplectic manifold with tangent structure vector field is a cosymplectic and minimal-like submanifold. Also we show if the structure vector field is normal to the submanifold then that is a statistical Keahler-like manifold [ABSTRACT FROM AUTHOR]

Let $(M,g,\varphi )$ be an $n$-dimensional locally decomposable Riemann manifold, that is, $g(\varphi X,Y)=g(X,\varphi Y)$ and $\nabla \varphi =0$, where $\nabla $ is Riemann (Levi-Civita) connection of metric $g$. In this paper, we construct a new connection on locally decomposable Riemann manifold, whose name is statistical ($\alpha ,\varphi )$-connection. A statistical $\alpha $-connection is a torsion-free connection such that $% \overline{\nabla }g=\alpha C$, where $C$ is a completely symmetric $(0,3)$% -type cubic form. The aim of this article is to use connection $\overline{% \nabla }$ and product structure $\varphi $ in the same equation, which is possible by writing the cubic form $C$ in terms of the product structure $% \varphi $. We examine some curvature properties of the new connection and give examples of it.

In this paper, a novel unsupervised C ross- M odal retrieval method via S ufficient S cene R epresentation (CMSSR) is proposed. Distinguished from the existing methods which mainly focus on simultaneously preserving the mutually-constrained intra- and inter-modal similarity relation, CMSSR considers data of different modalities as the descriptions of a scene from different views and accordingly integrates information of different modalities to learn a complete common representation containing sufficient information of the corresponding scene. To obtain such common representation, Gaussian Mixture Model (GMM) is firstly utilized to generate statistic representation of each uni-modal data, while the uni-modal spaces are accordingly abstracted as uni-modal statistical manifolds. In addition, the common space is assumed to be a high-dimensional statistical manifold with different uni-modal statistical manifolds as its sub-manifolds. In order to generate sufficient scene representation from uni-modal data, a representation completion strategy based on logistic regression is proposed to effectively complete the missing representation of another modality. Then, the similarity between different multi-modal data can be more accurately reflected by the distance metric in common statistical manifold. Based on the distance metric in common statistical manifold, Iterative Quantization is utilized to further generate binary code for fast cross-modal retrieval. Extensive experiments on three standard benchmark datasets fully demonstrate the superiority of CMSSR compared with several state-of-the-art methods. [ABSTRACT FROM AUTHOR]

We continue our study of the mixed Einstein–Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler–Lagrange equations (which in the case of space-time are analogous to those in Einstein–Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler–Lagrange equations of the mixed Einstein–Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric connections. [ABSTRACT FROM AUTHOR]

In this paper, we establish some inequalities between the normalized δ-Casorati curvatures and the scalar curvature (i.e., between extrinsic and intrinsic invariants) of spacelike statistical submanifolds in Sasaki-like statistical manifolds, endowed with a semi-symmetric metric connection. Moreover, we study the submanifolds satisfying the equality cases of these inequalities. We also present an appropriate example.

With the globalization of higher education, academic evaluation is increasingly valued by the scientific and educational circles. Although the number of published papers of academic evaluation methods is increasing, previous research mainly focused on the method of assigning different weights for various indicators, which can be subjective and limited. This paper investigates the evaluation of academic performance by using the statistical K-means (SKM) algorithm to produce clusters. The core idea is mapping the evaluation data from Euclidean space to Riemannian space in which the geometric structure can be used to obtain accurate clustering results. The method can adapt to different indicators and make full use of big data. By using the K-means algorithm based on statistical manifolds, the academic evaluation results of universities can be obtained. Furthermore, through simulation experiments on the top 20 universities of China with the traditional K-means, GMM and SKM algorithms, respectively, we analyze the advantages and disadvantages of different methods. We also test the three algorithms on a UCI ML dataset. The simulation results show the advantages of the SKM algorithm.