• 제목/요약/키워드: linear Weingarten surface

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TIMELIKE TUBULAR SURFACES OF WEINGARTEN TYPES AND LINEAR WEINGARTEN TYPES IN MINKOWSKI 3-SPACE

  • Chenghong He;He-jun Sun
    • 대한수학회보
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    • 제61권2호
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    • pp.401-419
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    • 2024
  • Let K, H, KII and HII be the Gaussian curvature, the mean curvature, the second Gaussian curvature and the second mean curvature of a timelike tubular surface Tγ(α) with the radius γ along a timelike curve α(s) in Minkowski 3-space E31. We prove that Tγ(α) must be a (K, H)-Weingarten surface and a (K, H)-linear Weingarten surface. We also show that Tγ(α) is (X, Y)-Weingarten type if and only if its central curve is a circle or a helix, where (X, Y) ∈ {(K, KII), (K, HII), (H, KII), (H, HII), (KII , HII)}. Furthermore, we prove that there exist no timelike tubular surfaces of (X, Y)-linear Weingarten type, (X, Y, Z)-linear Weingarten type and (K, H, KII, HII)-linear Weingarten type along a timelike curve in E31, where (X, Y, Z) ∈ {(K, H, KII), (K, H, HII), (K, KII, HII), (H, KII, HII)}.

Elliptic Linear Weingarten Surfaces

  • Kim, Young Ho
    • Kyungpook Mathematical Journal
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    • 제58권3호
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    • pp.547-557
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    • 2018
  • We establish some characterizations of isoparametric surfaces in the three-dimensional Euclidean space, which are associated with the Laplacian operator defined by the so-called II-metric on surfaces with non-degenerate second fundamental form and the elliptic linear Weingarten metric on surfaces in the three-dimensional Euclidean space. We also study a Ricci soliton associated with the elliptic linear Weingarten metric.

GEOMETRIC CHARACTERIZATIONS OF CANAL SURFACES IN MINKOWSKI 3-SPACE I

  • Fu, Xueshan;Jung, Seoung Dal;Qian, Jinhua;Su, Mengfei
    • 대한수학회보
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    • 제56권4호
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    • pp.867-883
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    • 2019
  • The canal surfaces foliated by pseudo spheres $\mathbb{S}_1^2$ along a space curve in Minkowski 3-space are studied. The geometric properties of such surfaces are shown by classifying the linear Weingarten canal surfaces, the developable, minimal and umbilical canal surfaces are discussed at the same time.

SOME CHARACTERIZATIONS OF CANAL SURFACES

  • Kim, Young Ho;Liu, Huili;Qian, Jinhua
    • 대한수학회보
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    • 제53권2호
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    • pp.461-477
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    • 2016
  • This work considers a particular type of swept surface named canal surfaces in Euclidean 3-space. For such a kind of surfaces, some interesting and important relations about the Gaussian curvature, the mean curvature and the second Gaussian curvature are found. Based on these relations, some canal surfaces are characterized.

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

  • Aleksieva, Yana;Ganchev, Georgi;Milousheva, Velichka
    • 대한수학회지
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    • 제53권5호
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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.