• Title, Summary, Keyword: totally real submanifold

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RICCI CURVATURE OF INTEGRAL SUBMANIFOLDS OF AN S-SPACE FORM

  • Kim, Jeong-Sik;Dwivedi, Mohit Kumar;Tripathi, Mukut Mani
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.3
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    • pp.395-406
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    • 2007
  • Involving the Ricci curvature and the squared mean curvature, we obtain a basic inequality for an integral submanifold of an S-space form. By polarization, we get a basic inequality for Ricci tensor also. Equality cases are also discussed. By giving a very simple proof we show that if an integral submanifold of maximum dimension of an S-space form satisfies the equality case, then it must be minimal. These results are applied to get corresponding results for C-totally real submanifolds of a Sasakian space form and for totally real submanifolds of a complex space form.

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.

ON CURVATURE PINCHING FOR TOTALLY REAL SUBMANIFOLDS OF $H^n$(c)

  • Matsuyama, Yoshio
    • Journal of the Korean Mathematical Society
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    • v.34 no.2
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    • pp.321-336
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    • 1997
  • Let S be the Ricci curvature of an n-dimensional compact minimal totally real submanifold M of a quaternion projective space $HP^n (c)$ of quaternion sectional curvature c. We proved that if $S \leq \frac{16}{3(n -2)}c$, then either $S \equiv \frac{4}{n - 1}c$ (i.e. M is totally geodesic or $S \equiv \frac{16}{3(n - 2)}c$. All compact minimal totally real submanifolds of $HP^n (c)$ satisfy in $S \equiv \frac{16}{3(n - 2)}c$ are determined.

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KÄHLER SUBMANIFOLDS WITH LOWER BOUNDED TOTALLY REAL BISECTIONL CURVATURE TENSOR II

  • Pyo, Yong-Soo;Shin, Kyoung-Hwa
    • Communications of the Korean Mathematical Society
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    • v.17 no.2
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    • pp.279-293
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    • 2002
  • In this paper, we prove that if every totally real bisectional curvature of an n($\geq$3)-dimensional complete Kahler submanifold of a complex projective space of constant holomorphic sectional curvature c is greater than (equation omitted) (3n$^2$+2n-2), then it is totally geodesic and compact.

REAL HALF LIGHTLIKE SUBMANIFOLDS WITH TOTALLY UMBILICAL PROPERTIES

  • Jin, Dae-Ho
    • The Pure and Applied Mathematics
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    • v.17 no.1
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    • pp.51-63
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    • 2010
  • In this paper, we prove two characterization theorems for real half lightlike submanifold (M,g,S(TM)) of an indefinite Kaehler manifold $\bar{M}$ or an indefinite complex space form $\bar{M}$(c) subject to the conditions : (a) M is totally umbilical in $\bar{M}$, or (b) its screen distribution S(TM) is totally umbilical in M.

NOTE ON NORMAL EMBEDDING

  • Yi, Seung-Hun
    • Bulletin of the Korean Mathematical Society
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    • v.39 no.2
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    • pp.289-297
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    • 2002
  • It was shown by L. Polterovich ([3]) that if L is a totally real submanifold of a symplectic manifold $(M,\omega)$ and L is parallelizable then L is normal. So we try to find an answer to the question of whether there is a compatible almost complex structure J on the symplectic vector bundle $TM$\mid$_{L}$ such that $TL{\cap}JTL=0$ assuming L is normal and parallelizable. Although we could not reach an answer, we observed that the claim holds at the vector space level. And related to the question, we showed that for a symplectic vector bundle $(M,\omega)$ of rank 2n and $E=E_1{\bigoplus}E_2$, where $E=E_1,E_2$are Lagrangian subbundles of E, there is an almost complex structure J on E compatible with ${\omega}$ and $JE_1=E_2$. And finally we provide a necessary and sufficient condition for a given embedding into a symplectic manifold to be normal.

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.