• Title, Summary, Keyword: semi-invariant submanifold

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ON SEMI-INVARIANT SUBMANIFOLDS OF LORENTZIAN ALMOST PARACONTACT MANIFOLDS

  • Tripathi, Mukut-Mani
    • The Pure and Applied Mathematics
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    • v.8 no.1
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    • pp.1-8
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    • 2001
  • Semi-invariant submanifolds of Lorentzian almost paracontact mani-folds are studied. Integrability of certain distributions on the submanifold are in vestigated. It has been proved that a LP-Sasakian manifold does not admit a proper semi-invariant submanifold.

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RICCI CURVATURE OF 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.46 no.5
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    • pp.979-998
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    • 2009
  • Involving the Ricci curvature and the squared mean curvature, we obtain a basic inequality for a submanifold of an S-space form tangent to structure vector fields. Equality cases are also discussed. As applications we find corresponding results for almost semi-invariant submanifolds, $\theta$-slant submanifolds, anti-invariant submanifold and invariant submanifolds. A necessary and sufficient condition for a totally umbilical invariant submanifold of an S-space form to be Einstein is obtained. The inequalities for scalar curvature and a Riemannian invariant $\Theta_k$ of different kind of submanifolds of a S-space form $\tilde{M}(c)$ are obtained.

ON A SEMI-INVARIANT SUBMANIFOLD OF CODIMENSION 3 WITH CONSTANT MEAN CURVATURE IN A COMPLEX PROJECTIVE SPACE

  • Lee, Seong-Baek
    • Communications of the Korean Mathematical Society
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    • v.18 no.1
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    • pp.75-85
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    • 2003
  • Let M be 3 Semi-invariant submanifold of codimension 3 with lift-flat normal connection in a complex projective space. Further, if the mean curvature of M is constant, then we prove that M is a real hypersurface of a complex projective space of codimension 2 in the ambient space.

SEMI-INVARIANT MINIMAL SUBMANIFOLDS OF CONDIMENSION 3 IN A COMPLEX SPACE FORM

  • Lee, Seong-Cheol;Han, Seung-Gook;Ki, U-Hang
    • Communications of the Korean Mathematical Society
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    • v.15 no.4
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    • pp.649-668
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    • 2000
  • In this paper we prove the following : Let M be a real (2n-1)-dimensional compact minimal semi-invariant submanifold in a complex projective space P(sub)n+1C. If the scalar curvature $\geq$2(n-1)(2n+1), then m is a homogeneous type $A_1$ or $A_2$. Next suppose that the third fundamental form n satisfies dn = 2$\theta\omega$ for a certain scalar $\theta$$\neq$c/2 and $\theta$$\neq$c/4 (4n-1)/(2n-1), where $\omega$(X,Y) = g(X,øY) for any vectors X and Y on a semi-invariant submanifold of codimension 3 in a complex space form M(sub)n+1 (c). Then we prove that M has constant principal curvatures corresponding the shape operator in the direction of the distingusihed normal and the structure vector ξ is an eigenvector of A if and only if M is locally congruent to a homogeneous minimal real hypersurface of M(sub)n (c).

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SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 SATISFYING 𝔏ξ∇ = 0 IN A NONFLAT COMPLEX SPACE FORM

  • AHN, SEONG-SOO;LEE, SEONG-BAEK;LEE, AN-AYE
    • Honam Mathematical Journal
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    • v.23 no.1
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    • pp.133-143
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    • 2001
  • In this paper, we characterize some semi-invariant submanifolds of codimension 3 with almost contact metric structure (${\phi}$, ${\xi}$, g) satisfying 𝔏ξ∇ = 0 in a nonflat complex space form, where ${\nabla}$ denotes the Riemannian connection induced on the submanifold, and 𝔏ξ is the operator of the Lie derivative with respect to the structure vector field ${\xi}$.

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SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 IN A COMPLEX HYPERBOLIC SPACE

  • KI, U-HANG;LEE, SEONG-BAEK;LEE, AN-AYE
    • Honam Mathematical Journal
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    • v.23 no.1
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    • pp.91-111
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    • 2001
  • In this paper we prove the following : Let M be a semi-invariant submanifold with almost contact metric structure (${\phi}$, ${\xi}$, g) of codimension 3 in a complex hyperbolic space $H_{n+1}{\mathbb{C}}$. Suppose that the third fundamental form n satisfies $dn=2{\theta}{\omega}$ for a certain scalar ${\theta}({\leq}{\frac{c}{2}})$, where ${\omega}(X,\;Y)=g(X,\;{\phi}Y)$ for any vectors X and Y on M. Then M has constant eigenvalues correponding the shape operator A in the direction of the distinguished normal and the structure vector ${\xi}$ is an eigenvector of A if and only if M is locally congruent to one of the type $A_0$, $A_1$, $A_2$ or B in $H_n{\mathbb{C}}$.

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ON SOME SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 IN A COMPLEX PROJECTIVE SPACE

  • Lee, Seong-Baek;Kim, Soo-Jin
    • Communications of the Korean Mathematical Society
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    • v.18 no.2
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    • pp.309-323
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    • 2003
  • In this paper, We characterize a semi-invariant sub-manifold of codimension 3 satisfying ∇$\varepsilon$A = 0 in a complex projective space CP$\^$n+1/, where ∇$\varepsilon$A is the covariant derivative of the shape operator A in the direction of the distinguished normal with respect to the structure vector field $\varepsilon$.

SEMI-INVARIANT SUBMANIFOLDS OF (LCS)n-MANIFOLD

  • Bagewadi, Channabasappa Shanthappa;Nirmala, Dharmanaik;Siddesha, Mallannara Siddalingappa
    • Communications of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.1331-1339
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    • 2018
  • In this paper the decomposition of basic equations of $(LCS)_n$-manifolds is carried out into horizontal and vertical projections. Further, we study the integrability of the distributions $D,D{\oplus}[{\xi}]$ and $D^{\perp}$ totally geodesic of semi-invariant submanifolds of $(LCS)_n$-manifold.

SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 OF A COMPLEX PROJECTIVE SPACE IN TERMS OF THE JACOBI OPERATOR

  • HER, JONG-IM;KI, U-HANG;LEE, SEONG-BAEK
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.1
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    • pp.93-119
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    • 2005
  • In this paper, we characterize some semi-invariant sub-manifolds of codimension 3 with almost contact metric structure ($\phi$, $\xi$, g) in a complex projective space $CP^{n+1}$ in terms of the structure tensor $\phi$, the Ricci tensor S and the Jacobi operator $R_\xi$ with respect to the structure vector $\xi$.

STRUCTURE JACOBI OPERATOR OF SEMI-INVARINAT SUBMANIFOLDS IN COMPLEX SPACE FORMS

  • KI, U-HANG;KIM, SOO JIN
    • East Asian mathematical journal
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    • v.36 no.3
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    • pp.389-415
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    • 2020
  • Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (��, ξ, η, g) in a complex space form Mn+1(c), c ≠ 0. We denote by Rξ and R'X be the structure Jacobi operator with respect to the structure vector ξ and be R'X = (∇XR)(·, X)X for any unit vector field X on M, respectively. Suppose that the third fundamental form t satisfies dt(X, Y) = 2��g(��X, Y) for a scalar ��(≠ 2c) and any vector fields X and Y on M. In this paper, we prove that if it satisfies Rξ�� = ��Rξ and at the same time R'ξ = 0, then M is a Hopf real hypersurfaces of type (A), provided that the scalar curvature ${\bar{r}}$ of M holds ${\bar{r}}-2(n-1)c{\leq}0$.