• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.175-177
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• 1989

The following is the basic problem about the stability in Riemannian Geometry; given a Riemannian manifold N, find all stable complete minimal submanifolds of N. As answers of this problem, do Carmo-Peng [1] and Fischer-Colbrie and Schoen [3] showed that the stable minimal surfaces in R$^{3}$ are planes and Schoen-Yau [5] and Fischer-Colbrie and Schoen [3] gave a solution for the case where the ambient space is a three dimensional manifold with nonnegative scalar curvature. In this paper we will remove the assumption of finite absolute total curvature in [3, Theorem 3].

• Bulletin of the Korean Mathematical Society
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• v.47 no.3
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• pp.541-549
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• 2010

In this paper we derive an integral formula on an (n + 1)-dimensional, compact, minimal contact CR-submanifold M of (n - 1) contact CR-dimension immersed in a unit (2m+1)-sphere $S^{2m+1}$. Using this integral formula, we give a sufficient condition concerning with the scalar curvature of M in order that such a submanifold M is to be a generalized Clifford torus.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.109-116
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• 2007

We study an (n+1)($(n{\geq}3)$-dimensional contact CR-submanifold of (n-1) contact CR-dimension in a (2m+1)-unit sphere $S^{2m+1}$, and to determine such sub manifolds under conditions concerning the second fundamental form and the induced almost contact structure.

• 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 $\xi$.

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.327-332
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• 2002

In this paper we prove that if M is a J-invariant sub-manifold of codimension 2 in a complex projective space $P_{n+1}(C)$, and the second fundamental tensor is cyclic-parallel or M has harmonic curvature, then M is locally complex quadric Q$_n$(C) or P$_n$(C).

• Communications of the Korean Mathematical Society
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• v.19 no.1
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• pp.129-134
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• 2004

It is well known that the warped product $L\;{\times}\;{_f}\;F$ of a line L and a Kaehler manifold F is an almost contact Riemannian manifold which is characterized by some tensor equations appeared in (1.7) and (1.8). In this paper we determine contact slant submanifolds tangent to the structure vector field of $L\;{\times}\;{_f}\;F$.

• Bulletin of the Korean Mathematical Society
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• v.25 no.2
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• pp.171-174
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• 1988

Let M be an n-dimensional compact connected and oriented Riemannian manifold isometrically immersed in an (n+2)-dimensional Euclidean space $R^{n+2}$. Moore [5] proved that if M is of positive curvature, then M is a homotopy sphere. This result is generalized by Baldin and Mercuri [2], Baik and Shin [1] to the case of non-negative curvature, which is stated as follows: If M of non-negative curvature, then M is either a homotopy sphere or diffeomorphic to a product of two spheres. In particular, if there is a point at which the curvature operator is positive, then M is homeomorphic to a sphere.e.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.1049-1060
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• 2020

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension two surfaces, which is different from the conditions of Risa and Sinestrari in [26] and we also remove the condition that the second fundamental form is uniformly bounded when t ∈ (-∞, -1).

• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.109-123
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• 1999

In this paper we prove a reduction theorem of the codimension for real submanifold of quaternionic projective space as a quaternionic analogue corresponding to those in Cecil [4], Erbacher [5] and Okumura [9], and apply the theorem to quaternionic CR- submanifold of quaternionic projective space.

• 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.