arXiv subjects

Bang-Yen Chen

Publications and source records attributed to Bang-Yen Chen.

At least 37 records · Page 2Linked to original sources

arXiv2019

Bi-warped product submanifolds of nearly Kaehler manifolds

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 $M=M_T\times_{f_1}\! M_\perp\times_{f_2}\! M_θ$ in a nearly Kaehler manifold satisfies the following sharp inequality: $$\|h\|^2\geq 2p\|\nabla (\ln f_1)\|^2+4q\left(1+{\small \frac{10}{9}}\cot^2θ\right)\|\nabla(\ln f_2)\|^2,$$ where $p=\dim M_\perp$, $q=\frac{1}{2}\dim M_θ$, and $f_1,\,f_2$ are smooth positive functions on $M_T$. We also investigate the equality case of this inequality. Further, some applications of this inequality are also given.

math.DG↗

arXiv2019

On some geometric properties of quasi-product production models

In this article we obtain classification results on the quasi-product production functions in terms of the geometry of their associated graph hypersurfaces, generalizing in a new setting some recent results concerning basic production models. In particular, we obtain several results on the geometry of Spillman-Mitscherlich and transcendental production functions.

math.DG↗

arXiv2018

Conformal vector fields and Yamabe solitons

In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are of Yamabe metrics.

math.DG↗

arXiv2018

Sharp growth estimates for warping functions in multiply warped product manifolds

By applying an average method in PDE, we obtain a dichotomy between "constancy" and "infinity" of the warping functions on complete noncompact Riemannian manifolds for an appropriate isometric immersion of a multiply warped product manifold $N_1\times_{f_2} N_2 \times \cdots \times _{f_k} N_k\, $ into a Riemannian manifold. Generalizing the earlier work of the authors in [{Glasg. Math. J. 51 (2009) 579-592], we establish sharp inequalities between the mean curvature of the immersion and the sectional curvatures of the ambient manifold under the influence of quantities of a purely analytic nature (the growth of the warping functions). Several applications of our growth estimates are also presented.

math.DG↗

arXiv2018

Geometry of warped product and CR-warped product submanifolds in Kaehler manifolds: modified version

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 Kaehler manifold $\tilde M$ is called a $CR$-warped product if it is a warped product $M_T\times_f N_\perp$ of a complex submanifold $M_T$ and a totally real submanifold $M_\perp$ of $\tilde M$. 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.

math.DG↗

arXiv2018

Two-numbers and their applications - A survey

The notion of two-numbers of connected Riemannian manifolds was introduced about 35 years ago in [Un invariant geometrique riemannien, C. R. Acad. Sci. Paris Math. 295 (1982), 389--391] by B.-Y. Chen and T. Nagano. Later, two-numbers have been studied by a number of mathematicians and it was then proved that two-numbers related closely with several important areas in mathematics. The main purpose of this article is to survey on two-numbers and their applications. At the end of this survey, we present several open problems and conjectures on two-numbers.

math.DG↗

arXiv2018

Euclidean submanifolds with incompressible canonical vector field

For a submanifold M in a Euclidean space, the tangential component x^T of the position vector field x of M is the most natural vector field tangent to the Euclidean submanifold, called the canonical vector field of M. In this article, first we prove that the canonical vector field of every Euclidean submanifold is always conservative. Then we initiate the study of Euclidean submanifolds with incompressible canonical vector fields. In particular, we obtain the necessary and sufficient conditions for the canonical vector field of a Euclidean submanifold to be incompressible. Further, we provide examples of Euclidean submanifolds with incompressible canonical vector field. Moreover, we classify planar curves, surfaces of revolution and hypercylinders with incompressible canonical vector fields.

math.DG↗

arXiv2017

Euclidean submanifolds with conformal canonical vector field

The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold $M$. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component x^T of the position vector field is the most natural vector field tangent to the Euclidean submanifold $M$. We simply call the vector field x^T the \textit{canonical vector field} of the Euclidean submanifold M. In earlier articles, we investigated Euclidean submanifolds whose canonical vector fields are concurrent, concircular, or torse-forming. In this article we study Euclidean submanifolds with conformal canonical vector field. In particular, we characterize such submanifolds. Several applications are also given. In the last section we present three global results on complete Euclidean submanifolds with conformal canonical vector field.

math.DG↗

arXiv2017

Warped Product Pointwise Bi-slant Submanifolds of Kaehler Manifolds

Warped product manifolds have been studied for a long period of time. In contrast, the study of warped product submanifolds from extrinsic point of view was initiated by the first author around the beginning of this century in [7, 8]. Since then the study of warped product submanifolds has been investigated by many geometers. The notion of slant submanifolds of almost Hermitian manifolds was introduced in [5]. Bi-slant submanifolds in almost contact metric manifolds were defined in [4] by J. L. Cabrerizo et al. In [26], we studied bi-slant submanifolds and warped product bi-slant submanifolds in Kaehler manifolds. In this article, we investigate warped product pointwise bi-slant submanifolds of Kaehler manifolds. Our main results extend several important results on warped product slant submanifolds obtained in [7, 21-23, 27].

math.DG↗

arXiv2017

Yamabe and quasi-Yamabe solitons on Euclidean submanifolds

In this paper we initiate the study of Yamabe and quasi-Yamabe solitons on Euclidean submanifolds whose soliton fields are the tangential components of their position vector fields. Several fundamental results of such solitons were proved. In particular, we classify such Yamabe and quasi-Yamabe solitons on Euclidean hypersurfaces.

math.DG↗

arXiv2017

Covering maps and ideal embeddings of compact homogeneous spaces

The notion of ideal embeddings was introduced in [B.-Y. Chen, {Strings of Riemannian invariants, inequalities, ideal immersions and their applications.} The Third Pacific Rim Geometry Conference (Seoul, 1996), 7-60, Int. Press, Cambridge, MA, 1998]. Roughly speaking, an ideal embedding (or a best of living) is an isometrical embedding which receives the least possible amount of tension from the surrounding space at each point. In this article, we study ideal embeddings of irreducible compact homogenous spaces in Euclidean spaces via covering maps. Our main result states that $π: M\to N$ is a covering map between two irreducible compact homogeneous spaces and if $λ_1(M)\ne λ_1(N)$, then $N$ doesn't admit an ideal embedding in any Euclidean space; although $M$ could.

math.DG↗

arXiv2017

Classification of $δ(2,n-2)$-ideal Lagrangian submanifolds in $n$-dimensional complex space forms

It was proven in [B.-Y. Chen, F. Dillen, J. Van der Veken and L. Vrancken, Curvature inequalities for Lagrangian submanifolds: the final solution, Differ. Geom. Appl. 31 (2013), 808-819] that every Lagrangian submanifold $M$ of a complex space form $\tilde M^{n}(4c)$ of constant holomorphic sectional curvature $4c$ satisfies the following optimal inequality: \begin{align*} δ(2,n-2) \leq \frac{n^2(n-2)}{4(n-1)} H^2 + 2(n-2) c, \end{align*} where $H^2$ is the squared mean curvature and $δ(2,n-2)$ is a $δ$-invariant on $M$. In this paper we classify Lagrangian submanifolds of complex space forms $\tilde M^{n}(4c)$, $n \geq 5$, which satisfy the equality case of this inequality at every point.

math.DG↗

arXiv2016

Differential geometry of rectifying submanifolds

A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf 110} (2003), no. 2, 147-152]. In this present article, we introduce and study the notion of rectifying submanifolds in Euclidean spaces. In particular, we prove that a Euclidean submanifold is rectifying if and only if the tangential component of its position vector field is a concurrent vector field. Moreover, rectifying submanifolds with arbitrary codimension are completely determined.

math.DG↗

arXiv2014

$δ$(3)-ideal null 2-type hypersurfaces in Euclidean spaces

In the theory of finite type submanifolds, null 2-type submanifolds are the most simple ones, besides 1-type submanifolds (cf. e.g., [3, 12]). In particular, the classification problems of null 2-type hypersurfaces are quite interesting and of fundamentally important. In this paper, we prove that every $δ$(3)-ideal null 2-type hypersurface in a Euclidean space has constant mean curvature and constant scalar curvature.

math.DG↗

arXiv2014

A simple characterization of generalized Robertson-Walker spacetimes

A generalized Robertson-Walker spacetime is the warped product with base an open interval of the real line endowed with the opposite of its metric and base any Riemannian manifold. The family of generalized Robertson-Walker spacetimes widely extends the one of classical Robertson-Walker spacetimes. In this article we prove a very simple characterization of generalized Robertson-Walker spacetimes; namely, a Lorentzian manifold is a generalized Robertson-Walker spacetime if and only if it admits a timelike concircular vector field.

gr-qc↗

arXiv2014

Classification of Ricci solitons on Euclidean hypersurfaces

A Ricci soliton $(M,g,v,λ)$ on a Riemannian manifold $(M,g)$ is said to have concurrent potential field if its potential field $v$ is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in \cite{CD2}. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

math.DG↗

arXiv2014

Ricci solitons and concurrent vector fields

A Ricci soliton $(M^n,g,v,λ)$ on a Riemannian manifold $(M^n,g)$ is said to have concurrent potential field if its potential field $v$ is a concurrent vector field. In the first part of this paper we completely classify Ricci solitons with concurrent potential fields. In the second part we derive a necessary and sufficient condition for a submanifold to be a Ricci soliton in a Riemannian manifold equipped with a concurrent vector field. In the last part, we classify shrinking Ricci solitons with $λ=1$ on Euclidean hypersurfaces. Several applications of our results are also presented.

math.DG↗

arXiv2014

Notes on isotropic geometry of production models

The production function is one of the key concepts of mainstream neoclassical theories in economics. The study of the shape and properties of the production possibility frontier is a subject of great interest in economic analysis. In this respect, Cobb-Douglas and CES production functions with flat graph hypersurfaces in Euclidean spaces are first studied in [19,20]. Later, more general studies of production models were given in [4-10, 12, 21]. On the other hand, from visual-physical experiences, the second and third authors proposed in [14] to study production models via isotropic geometry. The purpose of this paper is thus to investigate important production models via isotropic geometry.

math.OC↗