• Korean Journal of Mathematics
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• 제6권2호
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• pp.243-257
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• 1998

The purpose of this paper is to give some characterizations of $n$-dimensional CR-submanifolds with ($n-1$) CR-dimension immersed in a complex projective space $CP^{\frac{n+2}{2}}$, in terms of the Riemannian curvature tensor R.

• 대한수학회보
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• 제40권1호
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• pp.17-28
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• 2003

Let X be a closed, oriented, Riemannian 4-manifold with ${{b_2}^+}(x)\;>\;1$ and of simple type. Suppose that ${\sigma}\;:\;X\;{\rightarrow}\;X$ is an involution preserving orientation with an oriented, connected, compact 2-dimensional submanifold $\Sigma$ as a fixed point set with ${\Sigma\cdot\Sigma}\;{\geq}\;0\;and\;[\Sigma]\;{\neq}\;0\;{\in}\;H_2(X;\mathbb{Z})$. We show that if _X(\Sigma)\;+\;{\Sigma\cdots\Sigma}\;{\neq}\;0$ then the $Spin^{C}$ bundle $\={P}$ is not $\mathbb{Z}_2-equivariant$, where det $\={P}\;=\;L$ is a basic class with $c_1(L)[\Sigma]\;=\;0$.

• 대한수학회지
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• 제42권5호
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• pp.1101-1110
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• 2005

Recently, Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. In the present paper, we obtain sharp estimates for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for contact CR-warped products isometrically immersed in Kenmotsu space forms. The equality case is considered. Some applications are derived.

• 대한수학회지
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• 제51권5호
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• pp.881-895
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• 2014

In a recent paper [10] we introduced the notion of Levi harmonic map f from an almost contact semi-Riemannian manifold (M, ${\varphi}$, ${\xi}$, ${\eta}$, g) into a semi-Riemannian manifold $M^{\prime}$. In particular, we compute the tension field ${\tau}_H(f)$ for a CR map f between two almost contact semi-Riemannian manifolds satisfying the so-called ${\varphi}$-condition, where $H=Ker({\eta})$ is the Levi distribution. In the present paper we show that the condition (A) of Rawnsley [17] is related to the ${\varphi}$-condition. Then, we compute the tension field ${\tau}_H(f)$ for a CR map between two arbitrary almost contact semi-Riemannian manifolds, and we study the concept of Levi pluriharmonicity. Moreover, we study the harmonicity on quasicosymplectic manifolds.