• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.925-934
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• 2010

A crystallization of a closed connected PL manifold M is a special edge-colored graph representing M via a contracted triangulation. The regular genus of M is the minimum genus of a closed connected surface into which a crystallization of M regularly embeds. We disprove a conjecture on the regular genus of $\mathbb{S}\;{\times}\;\mathbb{S}^n$, $n\;{\geq}\;3$, stated in [J. Korean Math. Soc. 41 (2004), no. 3, p. 420].

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.273-291
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• 2018

In this paper we investigate the geometry of doubly twisted product manifolds and we study the harmonicity and biharmonicity of maps between doubly twisted product Riemannian manifold. Also we characterize the conformal biharmonic maps and construct some new proper biharmonic maps.

• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.1037-1049
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• 2015

It is well-known that the spectrum of a $spin^{\mathbb{C}}$ Dirac operator on a closed Riemannian $spin^{\mathbb{C}}$ manifold $M^{2k}$ of dimension 2k for $k{\in}{\mathbb{N}}$ is symmetric. In this article, we prove that over an odd-dimensional Riemannian product $M^{2p}_1{\times}M^{2q+1}_2$ with a product $spin^{\mathbb{C}}$ structure for $p{\geq}1$, $q{\geq}0$, the spectrum of a $spin^{\mathbb{C}}$ Dirac operator given by a product connection is symmetric if and only if either the $spin^{\mathbb{C}}$ Dirac spectrum of $M^{2q+1}_2$ is symmetric or $(e^{{\frac{1}{2}}c_1(L_1)}{\hat{A}}(M_1))[M_1]=0$, where $L_1$ is the associated line bundle for the given $spin^{\mathbb{C}}$ structure of $M_1$.

• Bulletin of the Korean Mathematical Society
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• v.49 no.2
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• pp.435-443
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• 2012

The Yamabe invariant is a topological invariant of a smooth closed manifold, which contains information about possible scalar curvature on it. It is well-known that a product manifold $T^m{\times}B$ where $T^m$ is the m-dimensional torus, and B is a closed spin manifold with nonzero $\^{A}$-genus has zero Yamabe invariant. We generalize this to various T-structured manifolds, for example $T^m$-bundles over such B whose transition functions take values in Sp(m, $\mathbb{Z}$) (or Sp(m - 1, $\mathbb{Z}$) ${\oplus}\;{{\pm}1}$ for odd m).

• Journal of the Chungcheong Mathematical Society
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• v.28 no.4
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• pp.591-602
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• 2015

In this paper, we prove the existence and the nonexistence of warping functions on Riemannian warped product manifolds with some prescribed scalar curvatures according to the fiber manifolds of class (A).

• Journal of the Chungcheong Mathematical Society
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• v.26 no.3
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• pp.525-532
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• 2013

In this paper, when N is a compact Riemannian manifold of class (A), we consider the existence of some warping functions on Riemannian warped product manifolds $M=[a,{\infty}){\times}_fN$ with prescribed scalar curvatures.

• Journal of applied mathematics & informatics
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• v.7 no.3
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• pp.1039-1044
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• 2000

We investigate the projectively flat warped product manifolds and study the geometric structure of the base space and its fibre. Specifically we find the conditions that the scalar curvature of the base space (B,g) vanishes if and only if f is harmonic on (B,g) and the fibre (F,$\bar{g}$) is a space of constant curvature.

• Communications of the Korean Mathematical Society
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• v.18 no.4
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• pp.687-693
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• 2003

In this paper, when N is a compact Riemannian manifold, using warped products, we discuss the conformal deformation of a warped product metric on $M\;\;[a,\;\infty)\;\times\;_f\;N$ with specific warping functions.

• Bulletin of the Korean Mathematical Society
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• v.48 no.3
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• pp.601-609
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• 2011

Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle $c_1$ centered at origin with an Euclidean planar curve $c_2$ has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle $c_1$ centered at origin with an Euclidean planar curve $c_2$ to have pointwise 1-type Gauss map.

• Bulletin of the Korean Mathematical Society
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• v.53 no.1
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• pp.83-90
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• 2016

We show that any orthogonal almost complex structure on a warped product Riemannian manifold of an oriented closed surface with nonnegative Gaussian curvature and a round 4-sphere is never integrable. This provides a partial answer to a question raised by E. Calabi.