• Title, Summary, Keyword: parallel vector field

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GENERIC SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE VECTOR OF A SASAKIAN SPACE FORM

  • Ahn, Seong-Soo;Ki, U-Hang
    • Bulletin of the Korean Mathematical Society
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    • v.31 no.2
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    • pp.215-236
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    • 1994
  • The purpose of the present paper is to study generic submanifolds of a Sasakian space form with nonvanishing parallel mean curvature vector field such that the shape operator in the direction of the mean curvature vector field commutes with the structure tensor field induced on the submanifold. In .cint. 1 we state general formulas on generic submanifolds of a Sasakian manifold, especially those of a Sasakian space form. .cint.2 is devoted to the study a generic submanifold of a Sasakian manifold, which is not tangent to the structure vector. In .cint.3 we investigate generic submanifolds, not tangent to the structure vector, of a Sasakian space form with nonvanishing parallel mean curvature vactor field. In .cint.4 we discuss generic submanifolds tangent to the structure vector of a Sasakian space form and compute the restricted Laplacian for the shape operator in the direction of the mean curvature vector field. As a applications of these, in the last .cint.5 we prove our main results.

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C-parallel Mean Curvature Vector Fields along Slant Curves in Sasakian 3-manifolds

  • Lee, Ji-Eun;Suh, Young-Jin;Lee, Hyun-Jin
    • Kyungpook Mathematical Journal
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    • v.52 no.1
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    • pp.49-59
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    • 2012
  • In this article, using the example of C. Camci([7]) we reconfirm necessary sufficient condition for a slant curve. Next, we find some necessary and sufficient conditions for a slant curve in a Sasakian 3-manifold to have: (i) a $C$-parallel mean curvature vector field; (ii) a $C$-proper mean curvature vector field (in the normal bundle).

ON 3-DIMENSIONAL LORENTZIAN CONCIRCULAR STRUCTURE MANIFOLDS

  • Chaubey, Sudhakar Kumar;Shaikh, Absos Ali
    • Communications of the Korean Mathematical Society
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    • v.34 no.1
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    • pp.303-319
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    • 2019
  • The aim of the present paper is to study the Eisenhart problems of finding the properties of second order parallel tensors (symmetric and skew-symmetric) on a 3-dimensional LCS-manifold. We also investigate the properties of Ricci solitons, Ricci semisymmetric, locally ${\phi}$-symmetric, ${\eta}$-parallel Ricci tensor and a non-null concircular vector field on $(LCS)_3$-manifolds.

ON THE THEORY OF LORENTZ SURFACES WITH PARALLEL NORMALIZED MEAN CURVATURE VECTOR FIELD IN PSEUDO-EUCLIDEAN 4-SPACE

  • Aleksieva, Yana;Ganchev, Georgi;Milousheva, Velichka
    • Journal of the Korean Mathematical Society
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    • v.53 no.5
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    • pp.1077-1100
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    • 2016
  • We develop an invariant local theory of Lorentz surfaces in pseudo-Euclidean 4-space by use of a linear map of Weingarten type. We find a geometrically determined moving frame field at each point of the surface and obtain a system of geometric functions. We prove a fundamental existence and uniqueness theorem in terms of these functions. On any Lorentz surface with parallel normalized mean curvature vector field we introduce special geometric (canonical) parameters and prove that any such surface is determined up to a rigid motion by three invariant functions satisfying three natural partial differential equations. In this way we minimize the number of functions and the number of partial differential equations determining the surface, which solves the Lund-Regge problem for this class of surfaces.

Field Oriented Control in Parallel Operation System of Induction Motors (유도전동기(誘導電動機)의 병렬운전(竝列運轉) System에서의 벡터제어(制御))

  • Kim, Sang-Hoon
    • Journal of Industrial Technology
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    • v.18
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    • pp.149-155
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    • 1998
  • This paper describes a reference flux angle selection for a vector control in the parallel operation system that consists of a inverter and several induction motors. In particular, this paper suggests which flux angle of motors prefers for the vector control in the train drive system that diameters of wheels are different. Through simulation for a 210[kW] induction motor drive system, it is clear that the vector control by using of the flux angle of a motor having a minimum wheel diameter leads to a minimum torque difference. However, it requires too many current sensors. So, it is shown that the vector control by a average flux angle of motors is preferable.

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SPACE-LIKE SUBMANIFOLDS WITH CONSTANT SCALAR CURVATURE IN THE DE SITTER SPACES

  • Liu, Ximin
    • Journal of the Korean Mathematical Society
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    • v.38 no.1
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    • pp.135-146
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    • 2001
  • Let M(sup)n be a space-ike submanifold in a de Sitter space M(sub)p(sup)n+p (c) with constant scalar curvature. We firstly extend Cheng-Yau's Technique to higher codimensional cases. Then we study the rigidity problem for M(sup)n with parallel normalized mean curvature vector field.

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A CHARACTERIZATION OF CONCENTRIC HYPERSPHERES IN ℝn

  • Kim, Dong-Soo;Kim, Young Ho
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.531-538
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    • 2014
  • Concentric hyperspheres in the n-dimensional Euclidean space $\mathbb{R}^n$ are the level hypersurfaces of a radial function f : $\mathbb{R}^n{\rightarrow}\mathbb{R}$. The magnitude $||{\nabla}f||$ of the gradient of such a radial function f : $\mathbb{R}^n{\rightarrow}\mathbb{R}$ is a function of the function f. We are interested in the converse problem. As a result, we show that if the magnitude of the gradient of a function f : $\mathbb{R}^n{\rightarrow}\mathbb{R}$ with isolated critical points is a function of f itself, then f is either a radial function or a function of a linear function. That is, the level hypersurfaces are either concentric hyperspheres or parallel hyperplanes. As a corollary, we see that if the magnitude of a conservative vector field with isolated singularities on $\mathbb{R}^n$ is a function of its scalar potential, then either it is a central vector field or it has constant direction.