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SURFACES OF REVOLUTION WITH POINTWISE 1-TYPE GAUSS MAP
CHEN BANG-YEN,CHOI MIEKYUNG,KIM YOUNG HO Korean Mathematical Society 2005 대한수학회지 Vol.42 No.3
In this article, we introduce the notion of pointwise 1-type Gauss map of the first and second kinds and study surfaces of revolution with such Gauss map. Our main results state that surfaces of revolution with pointwise 1-type Gauss map of the first kind coincide with surfaces of revolution with constant mean curvature; and the right cones are the only rational surfaces of revolution with pointwise 1-type Gauss map of the second kind.
SPHERICAL SUBMANIFOLDS WITH FINITE TYPE SPHERICAL GAUSS MAP
Chen, Bang-Yen,Lue, Huei-Shyong Korean Mathematical Society 2007 대한수학회지 Vol.44 No.2
The study of Euclidean submanifolds with finite type "classical" Gauss map was initiated by B.-Y. Chen and P. Piccinni in [11]. On the other hand, it was believed that for spherical sub manifolds the concept of spherical Gauss map is more relevant than the classical one (see [20]). Thus the purpose of this article is to initiate the study of spherical submanifolds with finite type spherical Gauss map. We obtain several fundamental results in this respect. In particular, spherical submanifolds with 1-type spherical Gauss map are classified. From which we conclude that all isoparametric hypersurfaces of $S^{n+1}$ have 1-type spherical Gauss map. Among others, we also prove that Veronese surface and equilateral minimal torus are the only minimal spherical surfaces with 2-type spherical Gauss map.
SOME RESULTS ON CONCIRCULAR VECTOR FIELDS AND THEIR APPLICATIONS TO RICCI SOLITONS
CHEN, BANG-YEN Korean Mathematical Society 2015 대한수학회보 Vol.52 No.5
A vector field on a Riemannian manifold (M, g) is called concircular if it satisfies ${\nabla}X^v={\mu}X$ for any vector X tangent to M, where ${\nabla}$ is the Levi-Civita connection and ${\mu}$ is a non-trivial function on M. A smooth vector field ${\xi}$ on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: $$\frac{1}{2}L_{\xi}g+Ric={\lambda}g$$, where $L_{\xi}g$ is the Lie-derivative of the metric tensor g with respect to ${\xi}$, Ric is the Ricci tensor of (M, g) and ${\lambda}$ is a constant. A Ricci soliton (M, g, ${\xi}$, ${\lambda}$) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field is a concircular vector field. In the first part of this paper we determine Riemannian manifolds which admit a concircular vector field. In the second part we classify Ricci solitons with concircular potential field. In the last part we prove some important properties of Ricci solitons on submanifolds of a Riemannian manifold equipped with a concircular vector field.
SURFACES OF REVOLUTION WITH POINTWISE 1-TYPE GAUSS MAP
Bang-Yen Chen,최미경,Young Ho Kim 대한수학회 2005 대한수학회지 Vol.42 No.3
In this article, we introduce the notion of pointwise 1- type Gauss map of the ¯rst and second kinds and study surfaces of revolution with such Gauss map. Our main results state that surfaces of revolution with pointwise 1-type Gauss map of the ¯rst kind coincide with surfaces of revolution with constant mean curva- ture; and the right cones are the only rational surfaces of revolution with pointwise 1-type Gauss map of the second kind.
RIEMANNIAN SUBMANIFOLDS WITH CONCIRCULAR CANONICAL FIELD
Chen, Bang-Yen,Wei, Shihshu Walter Korean Mathematical Society 2019 대한수학회보 Vol.56 No.6
Let ${\tilde{M}}$ be a Riemannian manifold equipped with a concircular vector field ${\tilde{X}}$ and M a submanifold (with its induced metric) of ${\tilde{M}}$. Denote by X the restriction of ${\tilde{X}}$ on M and by $X^T$ the tangential component of X, called the canonical field of M. In this article we study submanifolds of ${\tilde{M}}$ whose canonical field $X^T$ is also concircular. Several characterizations and classification results in this respect are obtained.
Euclidean submanifolds with conformal canonical vector field
Bang-Yen Chen,Sharief Deshmukh 대한수학회 2018 대한수학회보 Vol.55 No.6
The position vector field $\hbox{\bf x}$ 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 $\hbox{\bf x}^T$ of the position vector field is the most natural vector field tangent to the Euclidean submanifold $M$. We simply call the vector field $\hbox{\bf x}^T$ the \textit{canonical vector field} of the Euclidean submanifold $M$. In earlier articles \cite{C16,C17a,C17e,CV17,CW17}, we investigated Euclidean submanifolds whose canonical vector fields are concurrent, concircular, torse-forming, conservative or incompressible. 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.
EUCLIDEAN SUBMANIFOLDS WITH CONFORMAL CANONICAL VECTOR FIELD
Chen, Bang-Yen,Deshmukh, Sharief Korean Mathematical Society 2018 대한수학회보 Vol.55 No.6
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 canonical vector field of the Euclidean submanifold M. In earlier articles [4,5,9,11,12], we investigated Euclidean submanifolds whose canonical vector fields are concurrent, concircular, torse-forming, conservative or incompressible. 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.