Bang-Yen Chen

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• Doctor of Philosophy
• University Distinguished Professor Emeritus at Michigan State University

By applying Kodaira's classification theorem of algebraic surfaces and using methods in algebraic geometry, we completely classify compact complex surfaces which admit a Bochner-Kaehler metric (or equivalently, admit a self-dual Kaehler metric).

A new geometric invariant, called two-number, was introduced, studied and determined on compact symmetric spaces. Further, we establish several relations between two-numbers with topology and compact Lie groups. In particular, we provide a solution of a problem in group theory raised by A. Borel and J.-P. Serre.

In economics, homothetic functions are production functions whose marginal technical rate of substitution is homogeneous of degree zero. In this paper we classify homothetic functions satisfying the homogeneous Monge-Ampere equation. Several applications to production theory in economics will also be given.

In 1990s, B.-Y. Chen introduced the notion of δ-invariants and established shapr inequalities involving δ-invariants and the squared mean curvature |H|^2 for submanifolds in real space forms. In this article, for an n-dimensional Lagrangian submanifold of a complex space form, we prove a pointwise inequality δ(n_1,\ldots,n_k) ≤ a(n,k,n_1,...,n_k) |...

We introduce a new type of Riemannian curvature invariants and show that these new invariants have interesting applications to several areas of mathematics; in particular, we provide new obstructions to minimal and Lagrangian isometric immersions. Moreover, these new invariants enable us to introduce and to study the notion of ideal immersions (i.e...

This article examines characterizations of generalized quasi-Einstein space-times in f(R)−gravity. It is established that a Ricci symmetric generalized quasi-Einstein space-time is either Einstein or quasi-Einstein or static. It is demonstrated that a perfect fluid generalized quasi-Einstein space-time represents radiation era under certain conditi...

Geometric inequalities play a crucial role in various branches of science and engineering, providing foundational tools for theoretical development and problem-solving. This book encompasses a wide array of topics on geometric inequalities, including inequalities related to geometric solitons, generalized Ricci-Yamabe solitons on 3-dimensional Lie...

The well-known Hitchin-Thorpe inequality is an important geometric inequality which states that if a closed, oriented 4-manifold M admits an Einstein metric, then the Euler characteristic χ(M) and the signature τ(M) of M satisfy χ(M) ≥ (3/2) |τ(M)|. In 1974, N. Hitchin further established a complete characterization of the equality case. In additio...

A submanifold φ: M → E^{m} is called "biharmonic" if it satisfies Δ^{2}φ=0 identically, according to the author. On the other hand, G.-Y. Jiang studied biharmonic maps between Riemannian manifolds as critical points of the bienergy functional, and proved that biharmonic maps φ are characterized by vanishing of bitension τ_{2} of φ. During last thr...

This book attempts to present a comprehensive survey of the geometry of CR-submanifolds. The theory of submanifolds is one of the most interesting topics in differential geometry. The topic is introduced by Aurel Bejancu as a generalization of holomorphic and totally real submanifolds of almost Hermitian manifolds, in 1978. Afterward, the study of...

Metallic structures, introduced by V. de Spinadel in 2002, opened a new avenue in differential geometry. Building upon this concept, C. E. Hreţcanu and M. Crasmareanu laid the foundation for metallic Riemannian manifolds in 2013. The field's rich potential and diverse applications have since attracted significant research efforts, leading to a weal...

The study of biharmonic submanifolds in Euclidean spaces was introduced in the middle of the 1980s by the author in his program studying finite-type submanifolds. He defined biharmonic submanifolds in Euclidean spaces as submanifolds whose position vector field (x) satisfies the biharmonic equation, i.e., Δ2x=0. A well-known conjecture proposed by...

Fundamentally, duality gives two different points of view of looking at the same object. It appears in many subjects in mathematics (geometry, algebra, analysis, PDEs, Geometric Measure Theory, etc.) and in physics. For example, Connections on Fiber Bundles in mathematics, and Gauge Fields in physics are exactly the same. In n-dimensional geometry,...

First we derive some sharp inequalities for statistical submersions. Then we study Ricci-Bourguignon solitons on statistical submersions with parallel vertical or horizontal distribution. Finally, we study Ricci-Bourguignon solitons on statistical submersions with conformal or gradient potential vector field.

A Ricci–Bourguignon soliton is a self-similar solution to the Ricci–Bourguignon flow equation, and a Ricci–Bourguignon soliton is called trivial if its potential field is zero or killing. Each trivial Ricci–Bourguignon soliton is an Einstein manifold. The main purpose of this paper is to discover geometric conditions on compact Ricci–Bourguignon so...

This article investigates the impact of the semiconformal curvature tensor's symmetry on the base and fiber manifolds of a warped product manifold. It establishes that the fiber manifold of a warped product manifold has a constant sectional curvature, whereas the base manifold is semiconformally symmetric. Furthermore, the article derives the speci...

We provide conditions for a Riemannian manifold with a nontrivial closed affine conformal Killing vector field to be isometric to a Euclidean sphere or to the Euclidean space. Also, we formulate some triviality results for almost Ricci solitons with affine conformal Killing potential vector field.

A disaffinity vector on a Riemannian manifold is a vector field whose affinity tensor vanishes. In this paper, we prove that every disaffinity vector on a compact Riemannian manifold is an incompressible vector field. Then, we discover a sufficient condition for an incompressible vector field to be disaffinity. Next, we study trans-Sasakian 3-manif...

A disaffinity vector on a Riemannian manifold (M,g) is a vector field whose affinity tensor vanishes. In this paper, we observe that nontrivial disaffinity functions offer obstruction to the topology of M and show that the existence of a nontrivial disaffinity function on M does not allow M to be compact. A characterization of the Euclidean space i...

Let x be an isometric immersion of a Riemannian n-manifold M into a Euclidean n+1-space En+1 which does not pass through the origin of En+1. Then, the tangential part of the position vector field x of x is called the canonical vector field, and the normal part gives rise to a scalar function called the support function. Using the canonical vector f...

In differential geometry, the concept of golden structure represents a compelling area with wide-ranging applications. The exploration of golden Riemannian manifolds was initiated by C. E. Hretcanu and M. Crasmareanu in 2008, following the principles of the golden structure. Subsequently, numerous researchers have contributed significant insights w...

In Riemannian geometry, Ricci soliton inequalities are an important field of study that provide profound insights into the geometric and analytic characteristics of Riemannian manifolds. An extensive study of Ricci soliton inequalities is given in this review article, which also summarizes their historical evolution, core ideas, important findings,...

For a compact Riemannian m-manifold (Mm,g),m>1, endowed with a nontrivial conformal vector field ζ with a conformal factor σ, there is an associated skew-symmetric tensor φ called the associated tensor, and also, ζ admits the Hodge decomposition ζ=ζ¯+∇ρ, where ζ¯ satisfies divζ¯=0, which is called the Hodge vector, and ρ is the Hodge potential of ζ...

In differential geometry, the concept of golden structure, initially proposed by S. I. Goldberg and K. Yano in 1970, presents a compelling area with wide-ranging applications. The exploration of golden Riemannian manifolds was initiated by C. E. Hretcanu and M. Crasmareanu in 2008, following the principles of the golden structure. Subsequently, num...

The theory of designs is an important branch of combinatorial mathematics. It is well-known in the theory of designs that a finite subset of a sphere is a tight spherical 1-design if and only if it is a pair of antipodal points. On the other hand, antipodal sets and 2-number for a Riemannian manifold are introduced by B.-Y. Chen and T. Nagano in 19...

This work investigates the effects on the factor manifolds of a singly warped product manifold resulting from the presence of a quasi-conformally flat, quasi-conformally symmetric, or divergence-free quasi-conformal curvature tensor. Quasi-conformally flat warped product manifolds exhibit three distinct scenarios: in one scenario, the base manifold...

By studying cohomology classes that are related with n-harmonic morphisms and F-harmonic maps, we augment and extend several results on F-harmonic maps, harmonic maps in [1, 3, 14], p-harmonic morphisms in [17], and also revisit our previous results in [9, 10, 21] on Riemannian submersions and n-harmonic morphisms which are submersions. The results...

First, we derive various curvature inequalities involving Ricci and scalar curvatures of horizontal and vertical distributions of a quasi bi-slant Riemannian submersion from complex space forms onto a Riemannian manifold. Then we establish our main results which are the Chen–Ricci inequalities for such quasi bi-slant Riemannian submersions. Finally...

Antipodal sets is a fundamental concept in differential geometry. In this talk I will present some applications or links of this concept to several important areas in mathematics; in particular, to topology and group theory.

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. For this respect, the first author established in 1996 a basic inequality involving the Ricci curvature and the squared mean curvature o...

This book features chapters written by renowned scientists from various parts of the world, providing an up-to-date survey of submanifold theory, spanning diverse topics and applications. The book covers a wide range of topics such as Chen–Ricci inequalities in differential geometry, optimal inequalities for Casorati curvatures in quaternion geomet...

Preface and Contents of the book "Geometry of Submanifolds and Applications".

In this paper, we study different solitons on multiply warped product manifolds and realize the geometry of base manifold and fiber manifolds. We also study the base manifolds and fiber manifolds when the multiply warped product manifold is either concircularly flat or conharmonically flat.

We provide conditions for a Riemannian manifold with a nontrivial closed affine conformal Killing vector field to be isometric to a Euclidean sphere or to the Euclidean space. Also, we formulate some triviality results for almost Ricci solitons with affine conformal Killing potential vector field.

We establish that if the soliton vector field is the Reeb vector field, then a hypersurface of a nearly Kählerr manifold is a quasi-Yamabe soliton if and only if it is a Yamabe soliton. We prove that if a hypersurface of an arbitrary nearly Kähler manifold admits a (quasi)-Yamabe soliton with the Reeb vector field as a soliton vector field, then it...

In this article we study geometric properties of generalized solitons. In particular, we establish some relations between harmonicity and generalized solitons. For a generalized soliton (g,ξ,η,ß,γ,δ), we provide necessary and sufficient conditions for the dual 1-form $ξ^{\flat}$ of the potential vector field ξ to be a solution of the Schrödinger-Ri...

Differential geometry studies the geometry of curves, surfaces and higher dimensional smooth manifolds. For submanifolds in Euclidean spaces, the position vector is the most natural geometric object. Position vectors find applications throughout mathematics, engineerings and natural sciences. The purpose of this survey article is to present six res...

Maximal antipodal sets of Riemannian manifolds were introduced by the author and T. Nagano in [Un invariant g\'eom\'etrique riemannien, C. R. Acad. Sci. Paris S\'er. I Math. 295 (1982), no. 5, 389--391]. Since then maximal antipodal sets have been studied by many mathematicians and they shown that maximal antipodal sets are related to several impor...

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 c...

Curvature invariants of both intrinsic and extrinsic nature play a significant role in elucidating the geometry of a spacetime. In particular, these invariants are useful in detecting event horizon of black holes. Notable examples of spacetimes are provided by the generalized Robertson-Walker (GRW) models. An $(m+1)$-dimensional GRW spacetime is a...

In this paper, we consider PMC surfaces in complex space forms, and study the interaction between the notions of PMC, totally real and biconservative. We first consider PMC surfaces in a non-flat complex space form and prove that they are biconservative if and only if totally real. Then, we find a Simons-type formula for a well-chosen vector field...

In this article we consider PMC surfaces in complex space forms, and study the interaction between the notions of PMC, totally real and biconservative. We first consider PMC surfaces in non-flat complex space forms and prove that they are biconservative if and only if totally real. Then, we find a Simons type formula for a well-chosen vector field...

Recently, we studied CR-slant warped products B1×fM⊥, where B1=MT×Mθ is the Riemannian product of holomorphic and proper slant submanifolds and M⊥ is a totally real submanifold in a nearly Kaehler manifold. In the continuation, in this paper, we study B2×fMθ, where B2=MT×M⊥ is a CR-product of a nearly Kaehler manifold and establish Chen’s inequalit...

First, we derive some sharp inequalities for statistical submersions. Then we study Ricci-Bourguignon solitons on statistical submersions with parallel vertical or horizontal distribution. Finally, we study Ricci-Bourguignon solitons on statistical submersions with conformal or gradient potential vector field.

The theory of finite type submanifolds was introduced by the first author in late 1970s and it has become a useful tool for investigation of submanifolds. Later, the first author and P. Piccinni extended the notion of finite type submanifolds to finite type maps of submanifolds; in particular, to submanifolds with finite type Gauss map. Since then,...

By studying cohomology classes that are related with $p$-harmonic morphisms, $F$-harmonic maps, and $f$-harmonic maps, we extend several of our previous results on Riemannian submersions and $p$-harmonic morphisms to $F$-harmonic maps, and $f$-harmonic maps which are Riemannian submersions.

The purpose of this talk is to present six research topics in differential geometry in which the position vector field play important roles.

In this paper we characterize weakly Ricci-symmetric spacetimes (WRS)_{n} endowed with the Gray decomposition. We provide, several interesting results of (WRS)_{n} in Gray's decomposition. In addition we discuss some results based on weakly Ricci-symmetric Generalized Robertson—Walker spacetime (GRW). Moreover, we study (WRS)_{n} spacetimes which s...

The book gathers a wide range of topics such as warped product semi-slant submanifolds, slant submersions, semi-slant ξ^⊥-, hemi-slant ξ^⊥-Riemannian submersions, quasi hemi-slant submanifolds, slant submanifolds of metric f-manifolds, slant lightlike submanifolds, geometric inequalities for slant submanifolds, 3-slant submanifolds, and semi-slant...

This is the front matter of the book "Contact Geometry of Slant Submanifolds" published by Springer, 2022. The front matter contains Preface, Contents and brief informations about the editors.

Developable surfaces are surfaces in three-dimensional Euclidean space with zero Gaussian curvature. If these surfaces are explicitly defined in the functional form z=f(x,y), then f is nothing but a solution of the homogeneous Monge-Ampère equation. The main aim of this paper is to classify developable surfaces defined as graphs of weighted-homogen...

This is the front matter of the book "Complex Geometry of Slant Submanifolds" published by Springer, 2022. The front matter contains Preface, Contents and brief informations about the editors.

The purpose of this chapter is to present a comprehensive survey on the main results on proper slant surfaces in Kaehler manifolds and some related results on slant submanifolds obtained during the last three decades. In the last section, we present an open problem and three conjectures on slant surfaces in complex space forms.

In this chapter, we survey important results on CR-products, CR-warped products, bi-slant warped products, hemi-slant warped products, semi-slant warped products, and CR-slant warped products in Kaehler and nearly Kaehler manifolds. In the last two sections, we present related results on slant submanifolds in generalized complex space forms and in...

The main purpose of this chapter is to provide a detailed survey of recent results in complex slant geometry of Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds.

This book contains an up-to-date survey and self-contained chapters on complex slant submanifolds and geometry, authored by internationally renowned researchers. The book discusses a wide range of topics, including slant surfaces, slant submersions, nearly Kaehler, locally conformal Kaehler, and quaternion Kaehler manifolds. It provides several cla...

Professor Tadashi Nagano (1930-2017) was one of the great Japanese differential geometers, whose fundamental and seminal work still attracts much interest today. This volume, consisting of 13 contributed chapters, is inspired by his work and his legacy and, while reminding historical results obtained in the past, presents recent developments in the...

Professor Tadashi Nagano (January 9, 1930 – February 1, 2017) is one of the great Japanese differential geometers. All three authors were extremely fortunate to have Professor Tadashi Nagano as a supervisor during their graduate school years. It is their intention and their hope with this essay to share their knowledge of and appreciation of Profes...

The (M+,M_)-method of compact symmetric spaces was introduced by the author and T. Nagano in [Duke Math. J. 45 (1978), 405-425]. Since then this theory has been studied and applied to several important areas in mathematics by various authors. The main purpose of this article is to provide a comprehensive survey on the (M+,M_)-method and its applica...

A bi-warped product of the form: $M=N_T \times_{f_1}N^{n_{1}}_\perp\times_{f_2} N^{n_{2}}_\theta$ in a contact metric manifold is called a CRS bi-warped product, where $N_T,\, N^{n_{1}}_\perp$ and $N^{n_{2}}_θ$ are invariant, anti-invariant and proper pointwise slant submanifolds, respectively. First, we prove that there are no proper CRS bi-warpe...

One of the fundamental problems in the theory of submanifolds is to establish optimal relationships between intrinsic and extrinsic invariants for submanifolds. In order to establish such relations, the first author introduced in the 1990s the notion of δ-invariants for Riemannian manifolds, which are different in nature from the classical curvatur...

In this paper, we study the geometry of pointwise semi-slant warped products in a locally conformal Kaehler manifold. In particular, we obtain several results which extend Chen's inequality for CR-warped product submanifolds in Kaehler manifolds. Also, we study the corresponding equality cases. Several related results on pointwise semi-slant warped...

Associated with a Frenet curve \(\alpha \) in Euclidean 3-space \(\mathbb {E} ^{3} \), there exists the notion of natural mate \(\beta \) of \(\alpha \). In this article, we extend the natural mate \(\beta \) to sequential natural mates \(\{ \alpha _{1},\alpha _{2},\ldots ,\alpha _{n_{\alpha }}\}\) with \( \alpha _{1}=\beta \). We call each curve \...

An isometric immersion f : Mn ? ?Mm from an n-dimensional Riemannian manifold Mn into an almost Hermitian manifold ?Mm of complex dimension m is called pointwise slant if its Wirtinger angles define a function defined on Mn. In this paper we establish the Existence and Uniqueness Theorems for pointwise slant immersions of Riemannian manifolds Mn in...

The purpose of this article is to establish some inequalities concerning the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of totally real spacelike submanifolds in statistical manifolds of the type para-Kähler space form. Moreover, this study is focused on the equality cases in these inequal...

The notions of harmonic metrics and harmonic tensors were introduced by the author and T. Nagano in [Harmonic metrics, harmonic tensors, and Gauss maps, J. Math. Soc. Japan 36 (1984), 295--313]. Since then, it was shown that there are links between harmonic metrics with several important research areas with many applications. In this article, we pr...

The constant elasticity of substitution (CES for short) is a basic property widely used in neoclassical theory of production, but also in some other areas of economics, that involves a system of second-order nonlinear partial differential equations. One of the most remarkable results in economic analysis states that under homogeneity condition - i....

Warped products are the most natural and fruitful generalization of Riemannian products. Such products play very important roles in differential geometry and in general relativity. After Bishop and O’Neill’s 1969 article, there have been many works done on warped products from intrinsic point of view during the last fifty years. In contrast, the st...

We provide necessary and sufficient conditions for some couples (g,∇) of pseudo-Riemannian metrics and affine connections to be statistical structures if we have gradient almost Einstein, almost Ricci, almost Yamabe solitons, or a more general type of solitons on the manifold. In particular cases, we establish a formula for the volume of the manifo...

P. Wintgen proved in [Sur l'inegalite de Chen-Willmore, C. R. Acad. Sci. Paris, 288 (1979), 993--995] that the Gauss curvature G and the normal curvature K^D of a surface in the Euclidean 4-space E^4 satisfy G + |K^D| ≤ ||H||^2, where ||H||^2 is the squared mean curvature. A surface M in E^4 is called a Wintgen ideal surface if it satisfies the equ...

In this paper, we study statistical submersions between statistical manifolds. In particular, we establish Chen-Ricci inequalities of statistical submersions between statistical manifolds and a δ(2,2) Chen-type inequality for statistical submersions. Some applications are also given.

It was proved in [12, 13] that every Lagrangian submanifold M of a complex space form M^n(4c) with constant holomorphic sectional curvature 4c satisfies the following optimal inequality: (A) δ(2,...,2) ≤ n^2(2n-k-2 )H^2/(2(2n-k+4)) + (n^2-n-2k )c/2, where H^2 is the squared mean curvature, δ(2,...,2) is a δ-invariant introduced by the first author...

The constant elasticity of substitution (CES for short) is a basic property widely used in some areas of economics, that involves a system of second-order nonlinear partial differential equations. One of the most remarkable results in mathematical economics states that under homogeneity condition-i.e. the production function is a homogeneous functi...

The theory of δ-invariants, initiated by the author in the early 1990s, is a challenging topic in modern differential geometry, having a lot of applications. In the spirit of δ-invariants, S. Decu, S. Haesen and L. Verstraelen initiated the study of δ-Casorati curvatures in 2007. Since then there are many interesting results on δ-Casorati curvature...

The notions of harmonic metrics, harmonic tensors were introduced by B.-Y. Chen and T. Nagano in 1984. Since then, harmonic metrics and harmonic tensors have been studied by various authors and many interesting results were obtained. The main purpose of my talk is to provide a brief survey on harmonic metrics, harmonic tensors and identity maps fr...

In many of my earlier classification theorems in differential geometry and related subjects, I had to solve some ordinary and partial differential equations in order to find their exact solutions. The purpose of this article is to collect many of those exact solutions of ordinary and partial differential equations from my earlier articles in one pl...

The aim of this paper is to extend the classical DDVV inequality to CR-submanifolds of quaternionic Kähler manifolds of constant quaternionic sectional curvature. We first obtain a more general inequality involving the normalized scalar normal curvature ρ N (defined from the second fundamental form) and then derive a DDVV-type inequality involving...

In this article, we show that the presence of a concircular vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian and Kaehler manifolds. More precisely, we find new geometrical characterizations of spheres, Euclidean spaces as well as of complex Euclidean spaces using non-trivial concircular vector fields.

A brief description of author's education in differential geometry in Taiwan and U.S.A. and his academic interactions with his mathematical mentors Tadashi Nagano, Shiing-Shen Chern, Tominosuke Otsuki, and Kentaro Yano.

In this paper, we study the geometry of pointwise semi-slant warped products in a locally conformal Kaehler manifold. In particular, we obtain several results which extend Chen's inequality for CR-warped product submanifolds in Kaehler manifolds. Also, we study the corresponding equality cases. Several related results on pointwise semi-slant warped...

The original ``Steiner point'' introduced in 1838 by the Swiss mathematician Jakob Steiner (1796-1863), also known as the ``Steiner curvature centroid'', is the geometric centroid of the system obtained by placing a mass equal to the magnitude of the exterior angle at each vertex of a triangle. Steiner points have been studied and applied in networ...

An identity map id_M : M → M is a bijective map from a manifold M onto itself which carries each point of M return to the same point. To study the differential geometry of an identity map id_M : M → M, we usually assume that the domain M and the range M admit metrics g and g′, respectively. The main purpose of this paper is to provide a comprehensi...

An identity map idM:M→M is a bijective map from a manifold M onto itself which carries each point of M return to the same point. To study the differential geometry of an identity map idM:M→M, we usually assume that the domain M and the range M admit metrics g and g′, respectively. The main purpose of this paper is to provide a comprehensive survey...

In this article we survey recent results on warped product and CR-warped product submanifolds in Kaehler manifolds. Several closely related results will also be presented.

We call a submanifold M of a Kaehler manifold $\tilde M$ a pointwise CR-Slant warped product if it is a warped product, B x_f N_θ of a CR-product B = N_T x N_⊥ and a proper pointwise slant submanifold N_θ with slant function θ, where N_T and N_⊥ are complex and totally real submanifolds of $\tilde M$. We prove that if a pointwise CR-Slant warped pr...

The aim of this paper is to extend the classical DDVV inequality to CR-submanifolds of quaternionic Kaehler manifolds of constant quaternionic sectional curvature. We first obtain a more general inequality involving the normalized scalar normal curvature (defined from the second fundamental form) and then derive a DDVV-type inequality involving the...

In this paper, we derive Chen inequality for statistical submanifold of statistical warped product manifolds [Formula: see text]. Further, we derive Chen inequality for Legendrian statistical submanifold in statistical warped product manifolds [Formula: see text]. We also provide some applications of derived inequalities in a statistical warped pro...

The study of biharmonic submanifolds of Euclidean spaces was initiated in the middle of the 1980s by the work of the second author of this book in his program to study the finite type submanifolds of Euclidean spaces. In fact, the second author defined biharmonic submanifolds of Euclidean spaces as those whose position vector field x satisfies the...

An isometric immersion f : M^n →M^m from an n-dimensional Riemannian manifold M^n into an almost Hermitian manifold M^m of complex dimension m is called pointwise slant if its Wirtinger angles define a function defined on M^n. In this paper we establish the existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds M^n...

We study bi-warped product submanifolds of nearly Kaehler manifolds which are the natural extension of warped products. We prove that every bi-warped product submanifold of the form $M=M_T\times_{f_1} M_\perp\times_{f_2} M_\theta$ in a nearly Kaehler manifold satisfies the following sharp inequality: $$||h|^2 ≥ 2p||grad(ln f_1)||^2+4q\left(1+{\smal...

The warped product $N_1\times_f N_2$ of two Riemannian manifolds $(N_1,g_1)$ and $(N_2,g_2)$ is the product manifold $N_1\times N_2$ equipped with the warped product metric $g=g_1+f^2 g_2$, where $f$ is a positive function on $N_1$. Warped products play very important roles in differential geometry as well as in physics. A submanifold $M$ of a Kaeh...

This talk, in memory of Professor T. Nagano, is a brief version of my survey article "Two-numbers and their applications - a survey, Bull. Belg. Math. Soc. Simon Stevin 25 (2018)" which is available at Researchgate web page: https://www.researchgate.net/publication/318310609_Two-numbers_and_their_applications_-_A_survey

Let N be a Riemannian manifold equipped with a concircular vector field Y and M a submanifold (with its induced metric) of N. Denote by X the restriction of Y on M and by X^T the tangential component of X, called the canonical field of M. In this article we study submanifolds of N whose canonical field X^T is also concircular. Several characteriza...

The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The position vector field plays important roles in physics, in particular, in mechanics. For instance in any equation of motion, the position vector x(t) is usually the most sought-after quantity because the position vector field defines the mo...

In this article, we study Jacobi-type vector fields on Riemannian manifolds. A Killing vector field is a Jacobi-type vector field while the converse is not true, leading to a natural question of finding conditions under which a Jacobi-type vector field is Killing. In this article, we first prove that every Jacobi-type vector field on a compact Riem...

A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden–Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of R...

A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden-Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of R...

In this article, we study Jacobi-type vector fields on Riemannian manifolds. A Killing vector field is a Jacobi-type vector field while the converse is not true, leading to a natural question of finding conditions under which a Jacobi-type vector field is Killing. In this article we first prove that every Jacobi-type vector field on a compact Riema...

Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. In this article, first we define and study statistical solitons on Ricci-symmetric statistical warped products R×fN2 and N1×fR. Second, we study statistical warped products as submanifolds of statistical manifolds. For...