• Honam Mathematical Journal
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• v.28 no.1
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• pp.141-164
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• 2006

Let $G=O(2){\times}O(2){\times}O(2)$ and let $M^4$ be a closed G-invariant minimal hypersurface with constant scalar curvature in $S^5$. In this paper, we prove a property on $M^4$.

• Honam Mathematical Journal
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• v.31 no.3
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• pp.381-398
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• 2009

Let $G\;=\;O(k){\times}O(k){\times}O(q)$ and let $M^n$ be a closed G-invariant minimal hypersurface with constant scalar curvature in $S^{n+1}$. Then we obtain a theorem: If $M^n$ has 2 distinct principal curvatures at some point p, then the square norm of the second fundamental form of $M^n$, S = n.

• Communications of the Korean Mathematical Society
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• v.17 no.2
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• pp.261-278
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• 2002

Let G = O(2) $\times$ O(2) $\times$O(2) and let M$^4$be closed G-invariant minimal hypersurface with constant scalar curvature in S$^{5}$ . If M$^4$has 2 distinct principal curvatures at some point, then S = 4. Moreover, if S > 4, then M$^4$does not have simple principal curvatures everywhere.

• Kyungpook Mathematical Journal
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• v.53 no.4
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• pp.515-540
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• 2013

Let $G=O(2){\times}O(2){\times}O(2)$. Then a closed G-invariant minimal hypersurface with constant scalar curvature in $S^5$ is a product of spheres, i.e., the square norm of its second fundamental form, S = 4.

• Nonlinear Functional Analysis and Applications
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• v.26 no.5
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• pp.985-994
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• 2021

The object of this paper is to study non-invariant hypersurface of a (𝜖, 𝛿)-trans Sasakian manifolds equipped with (f, g, u, v, λ)-structure. Some properties obeyed by this structure are obtained. The necessary and sufficient conditions also have been obtained for totally umbilical non-invariant hypersurface with (f, g, u, v, λ)-structure of a (𝜖, 𝛿)-trans Sasakian manifolds to be totally geodesic. The second fundamental form of a non-invariant hypersurface of a (𝜖, 𝛿)-trans Sasakian manifolds with (f, g, u, v, λ)-structure has been traced under the condition when f is parallel.

• Honam Mathematical Journal
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• v.23 no.1
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• pp.113-132
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• 2001

We prove an equality that holds on G-invariant minimal hypersurfaces with constant scalar curvatures in $S^5$.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.229-257
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• 2022

Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (𝜙, 𝜉, 𝜂, g) in a complex space form M_n+1(c), c ≠ 0. We denote by A and R_𝜉 the shape operator in the direction of distinguished normal vector field and the structure Jacobi operator with respect to the structure vector 𝜉, respectively. Suppose that the third fundamental form t satisfies dt(X, Y) = 2𝜃g(𝜙X, Y) for a scalar 𝜃(< 2c) and any vector fields X and Y on M. In this paper, we prove that if it satisfies R_𝜉A = AR_𝜉 and at the same time ∇_𝜉R_𝜉 = 0 on M, then M is a Hopf hypersurface of type (A) provided that the scalar curvature s of M holds s - 2(n - 1)c ≤ 0.

• 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 𝔏_ξ∇ = 0 in a nonflat complex space form, where ${\nabla}$ denotes the Riemannian connection induced on the submanifold, and 𝔏_ξ is the operator of the Lie derivative with respect to the structure vector field ${\xi}$.

• 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 $\xi$.

• 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).