• Communications of the Korean Mathematical Society
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• v.11 no.2
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• pp.445-455
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• 1996

In this paper we study the Reidemeister number $R(f_G)$ for a self-map $f_G : (X, G) \to (X, G)$ of the transformation group (X,G), as an extenstion of the Reidemeister number R(f) for a self-map $f : X \to X$ of a topological space X.

• Journal of the Korean Mathematical Society
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• v.54 no.2
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• pp.381-397
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• 2017

In this paper, we study surfaces in 3-dimensional Minkowski space in terms of certain type of their Gauss map. We give several results on these surfaces whose Gauss map G satisfies ${\square}G=f(G+C)$ for a smooth function f and a constant vector C, where ${\square}$ denotes the ChengYau operator. In particular, we obtain classification theorems on the rotational surfaces in ${\mathbb{E}}^3_1$ with space-like axis of rotation in terms of type of their Gauss map concerning the Cheng-Yau operator.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.3
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• pp.245-259
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• 2007

We define and study a concept of $T^f$-space for a map, which is a generalized one of a T-space, in terms of the Gottlieb set for a map. We show that X is a $T_f$-space if and only if $G({\Sigma}B;A,f,X)=[{\Sigma}B,X]$ for any space B. For a principal fibration $E_k{\rightarrow}X$ induced by $k:X{\rightarrow}X^{\prime}$ from ${\epsilon}:PX^{\prime}{\rightarrow}X^{\prime}$, we obtain a sufficient condition to having a lifting $T^{\bar{f}}$-structure on $E_k$ of a $T^f$-structure on X. Also, we define and study a concept of co-$T^g$-space for a map, which is a dual one of $T^f$-space for a map. We obtain a dual result for a principal cofibration $i_r:X{\rightarrow}C_r$ induced by $r:X^{\prime}{\rightarrow}X$ from ${\iota}:X^{\prime}{\rightarrow}cX^{\prime}$.

• The Pure and Applied Mathematics
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• v.5 no.1
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• pp.55-63
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• 1998

A Gauss map of m-dimensional distribution on a Riemannian manifold M is called a harmonic Gauss map if it is a harmonic map from the manifold into its Grassmann bundle $G_m$(TM) of m-dimensional tangent subspace. We calculate the tension field of the Gauss map of m-dimensional distribution and especially show that the Hopf fibrations on $S^{4n＋3}$ are the harmonic Gauss map of 3-dimensional distribution.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.709-715
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• 1997

For a fibration (E,B,p) with fiber F and a fiber map f, we show that if the inclusion $i : F \to E$ has a left homotopy inverse, then $G^f_n(E,F)$ is isomorphic to $G^f_n(F,E) \oplus \pi_n(B)$. In particular, by taking f as the identity map on E we have $G_n(E,F)$ is isomorphic to $G_n(F) \oplus \pi_n(B)$.

• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.429-437
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• 1994

Let $M^n$ be a connected hypersurface in Euclidean (n + 1)-space $E^{n+1}$, and let $G : G^n \longrightarrow S^n(1) \subset E^{n+1}$ be its Gauss map.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1345-1356
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• 2013

In this paper, we study rotational and helicoidal surfaces in Euclidean 3-space in terms of their Gauss map. We obtain a complete classification of these type of surfaces whose Gauss maps G satisfy $L_1G=f(G+C)$ for some constant vector $C{\in}\mathbb{E}^3$ and smooth function $f$, where $L_1$ denotes the Cheng-Yau operator.

• Bulletin of the Korean Mathematical Society
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• v.36 no.4
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• pp.649-654
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• 1999

We consider the forgetful map from the group of equivariant self equivalences to the group of non-equivariant self equivalences. A sufficient condition for this forgetful map being a monomorphism is obtained. Several examples are given.

• Bulletin of the Korean Mathematical Society
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• v.42 no.1
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• pp.23-27
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• 2005

In this paper, we study rotation surfaces in a Euclidean space with pointwise 1-type Gauss map and obtain by the use of the concept of pointwise finite type Gauss map, a characterization theorem for rotation surfaces of constant mean curvature.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.4
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• pp.831-841
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• 2009

For a map f : A $\rightarrow$ X, we define and study a concept of $G^f$-space for a map, which is a generalized one of a G-space. Any G-space is a $G^f$-space, but the converse does not hold. In fact, $S^2$ is a $G^{\eta}$-space, but not G-space. We show that X is a $G^f$-space if and only if $G_n$(A, f,X) = $\pi_n(X)$ for all n. It is clear that any $H^f$-space is a $G^f$-space and any $G^f$-space is a $W^f$-space. We can also obtain some results about $G^f$-spaces in Postnikov systems for spaces, which are generalization of Haslam's results about G-spaces.