• Kyungpook Mathematical Journal
• /
• v.56 no.3
• /
• pp.965-977
• /
• 2016

The aim of this paper is to study the geometry of contact CR-warped product submanifolds in a cosymplectic manifold. We search several fundamental properties of contact CR-warped product submanifolds in a cosymplectic manifold. We also give necessary and sufficient conditions for a submanifold in a cosymplectic manifold to be contact CR-(warped) product submanifold. After then we establish a general inequality between the warping function and the second fundamental for a contact CR-warped product submanifold in a cosymplectic manifold and consider contact CR-warped product submanifold in a cosymplectic manifold which satisfy the equality case of the inequality and some new results are obtained.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.1
• /
• pp.215-222
• /
• 2015

We show that homology group on a contact CR-warped product submanifold in odd dimensional sphere is zero under certain conditions in terms of warping function and the dimension of the submanifold.

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

• Journal of the Korean Mathematical Society
• /
• v.44 no.2
• /
• pp.373-391
• /
• 2007

We study an (n+3)($n\;{\geq}\;7-dimensional$ real submanifold of a (4m+3)-unit sphere $S^{4m+3}$ with Sasakian 3-structure induced from the canonical quaternionic $K\"{a}hler$ structure of quaternionic (m+1)-number space $Q^{m+1}$, and especially determine contact three CR-submanifolds with (p-1) contact three CR-dimension under the equality conditions given in (4.1), where p = 4m - n denotes the codimension of the submanifold. Also we provide necessary conditions concerning sectional curvature in order that a compact contact three CR-submanifold of (p-1) contact three CR-dimension in $S^{4m+3}$ is the model space $S^{4n_1+3}(r_1){\times}S^{4n_2+3}(r_2)$ for some portion $(n_1,\;n_2)$ of (n-3)/4 and some $r_1,\;r_2$ with $r^{2}_{1}+r^{2}_{2}=1$.

• Bulletin of the Korean Mathematical Society
• /
• v.44 no.1
• /
• pp.109-116
• /
• 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
• /
• v.48 no.3
• /
• pp.585-600
• /
• 2011

In this paper we derive an integral formula on an (n + 3)-dimensional, compact, minimal contact three CR-submanifold M of (p-1) contact three CR-dimension immersed in a unit (4m+3)-sphere $S^{4m+3}$. Using this integral formula, we give a sufficient condition concerning 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
• /
• v.51 no.3
• /
• pp.863-881
• /
• 2014

Many differential geometric properties of a submanifold of a Kaehler manifold are conceived via canonical structure tensors T and F on the submanifold. For instance, a CR-submanifold of a Kaehler manifold is a CR-product if and only if T is parallel on the submanifold (c.f. [2]). Warped product submanifolds are generalized version of CR-product submanifolds. Therefore, it is natural to see how the non-triviality of the covariant derivatives of T and F gives rise to warped product submanifolds. In the present article, we have worked out characterizations in terms of T and F under which a contact CR- submanifold of a Kenmotsu manifold reduces to a warped product submanifold.

• The Pure and Applied Mathematics
• /
• v.5 no.1
• /
• pp.79-86
• /
• 1998

In this paper, we study an (n ＋ 1)-dimensional compact, orientable, minimal contact CR-submanifold of (n - 1) ontact CR-dimension in a (2m ＋ 1)-dimensional unit sphere $S^{2m＋1}$ in terms of integral formula.

• Communications of the Korean Mathematical Society
• /
• v.17 no.3
• /
• pp.519-529
• /
• 2002

In this paper. We Study (n + 1)-dimensional Contact CR-submanifolds of (n - 1) contact CR-dimension immersed in a Sasakian space form M$\^$2m+1/(c) (2m=n+p, p>0), and especially determine such submanifolds under additional condition concerning with shape operator.

• Communications of the Korean Mathematical Society
• /
• v.13 no.3
• /
• pp.561-577
• /
• 1998

We study (n＋3)-dimensional contact three CR submanifolds of a Riemannian manifold with Sasakian three structure and investigate some characterizations of $S^{4r＋3}$(a) $\times$ $S^{4s＋3}$(b) ($a^2$＋ $b^2$=1, 4(r ＋ s) = n - 3) as a contact three CR sub manifold of a (4m＋3)-dimensional unit sphere.