• Title, Summary, Keyword: Mean curvature

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NEW RELATIONSHIPS INVOLVING THE MEAN CURVATURE OF SLANT SUBMANIFOLDS IN S-SPACE-FORMS

  • Fernandez, Luis M.;Hans-Uber, Maria Belen
    • Journal of the Korean Mathematical Society
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    • v.44 no.3
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    • pp.647-659
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    • 2007
  • Relationships between the Ricci curvature and the squared mean curvature and between the shape operator associated with the mean curvature vector and the sectional curvature function for slant submanifolds of an S-space-form are proved, particularizing them to invariant and anti-invariant submanifolds tangent to the structure vector fields.

RICCI CURVATURE OF SUBMANIFOLDS IN A QUATERNION PROJECTIVE SPACE

  • Liu, Ximin;Dai, Wanji
    • Communications of the Korean Mathematical Society
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    • v.17 no.4
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    • pp.625-633
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    • 2002
  • Recently, Chen establishes sharp relationship between the k-Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. In this paper, we establish sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in quaternion projective spaces.

ANOTHER CHARACTERIZATION OF ROUND SPHERES

  • Lee, Seung-Won;Koh, Sung-Eun
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.4
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    • pp.701-706
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    • 1999
  • A characterization of geodesic spheres in the simply connected space forms in terms of the ratio of the Gauss-Kronecker curvature and the (usual) mean curvature is given: An immersion of n dimensional compact oriented manifold without boundary into the n + 1 dimensional Euclidean space, hyperbolic space or open half sphere is a totally umbilicimmersion if the mean curvature $H_1$ does not vanish and the ratio $H_n$/$H_1$ of the Gauss-Kronecker curvature $H_n$ and $H_1$ is constant.

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TUBES OF WEINGARTEN TYPES IN A EUCLIDEAN 3-SPACE

  • Ro, Jin Suk;Yoon, Dae Won
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.3
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    • pp.359-366
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    • 2009
  • In this paper, we study a tube in a Euclidean 3-space satisfying some equation in terms of the Gaussian curvature, the mean curvature and the second Gaussian curvature.

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ON SOME GEOMETRIC PROPERTIES OF QUADRIC SURFACES IN EUCLIDEAN SPACE

  • Ali, Ahmad T.;Aziz, H.S. Abdel;Sorour, Adel H.
    • Honam Mathematical Journal
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    • v.38 no.3
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    • pp.593-611
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    • 2016
  • This paper is concerned with the classifications of quadric surfaces of first and second kinds in Euclidean 3-space satisfying the Jacobi condition with respect to their curvatures, the Gaussian curvature K, the mean curvature H, second mean curvature $H_{II}$ and second Gaussian curvature $K_{II}$. Also, we study the zero and non-zero constant curvatures of these surfaces. Furthermore, we investigated the (A, B)-Weingarten, (A, B)-linear Weingarten as well as some special ($C^2$, K) and $(C^2,\;K{\sqrt{K}})$-nonlinear Weingarten quadric surfaces in $E^3$, where $A{\neq}B$, A, $B{\in}{K,H,H_{II},K_{II}}$ and $C{\in}{H,H_{II},K_{II}}$. Finally, some important new lemmas are presented.

SOME INEQUALITIES ON TOTALLY REAL SUBMANIFOLDS IN LOCALLY CONFORMAL KAEHLER SPACE FORMS

  • Alfonso, Carriazo;Kim, Young-Ho;Yoon, Dae-Won
    • Journal of the Korean Mathematical Society
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    • v.41 no.5
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    • pp.795-808
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    • 2004
  • In this article, we establish sharp relations between the sectional curvature and the shape operator and also between the k-Ricci curvature and the shape operator for a totally real submanifold in a locally conformal Kaehler space form of constant holomorphic sectional curvature with arbitrary codimension. mean curvature, sectional curvature, shape operator, k-Ricci curvature, locally conformal Kaehler space form, totally real submanifold.

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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