• Communications of the Korean Mathematical Society
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• v.13 no.4
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• pp.825-838
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• 1998

We prove that if a real hypersurface with constant mean curvature of a complex space form satisfying ▽$_{ξ/}$S = 0 and Sξ = $\sigma$ξ for a smooth function $\sigma$, then the structure vector field ξ is principal, where S denotes the Ricci tensor of the hypersurface.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.819-835
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• 1998

The main purpose of this paper is to prove that if a real hypersurfaces M of a complex space form satisfies ${\nabla}_{\xi}S$=0 and $S{\xi}=\sigma\xi$ for some constant on $\sigma$ on M, then the structure vector field $\xi$ is principal, where S denotes the Ricci tensors of M.

• Bulletin of the Korean Mathematical Society
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• v.42 no.2
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• pp.279-294
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• 2005

We study a real hypersurface M satisfying $L_{\xi}S=0\;and\;R_{\xi}S=SR_{\xi}$ in a complex hyperbolic space $H_n\mathbb{C}$, where S is the Ricci tensor of type (1,1) on M, $L_{\xi}\;and\;R_{\xi}$ denotes the operator of the Lie derivative and the Jacobi operator with respect to the structure vector field e respectively.

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.935-949
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• 2013

In this paper, we study surfaces in $\mathb{E}^3$ whose Gauss map G satisfies the equation ${\Box}G=f(G+C)$ for a smooth function $f$ and a constant vector C, where ${\Box}$ stands for the Cheng-Yau operator. We focus on surfaces with constant Gaussian curvature, constant mean curvature and constant principal curvature with such a property. We obtain some classification and characterization theorems for these kinds of surfaces. Finally, we give a characterization of surfaces whose Gauss map G satisfies the equation ${\Box}G={\lambda}(G+C)$ for a constant ${\lambda}$ and a constant vector C.

• Bulletin of the Korean Mathematical Society
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• v.34 no.2
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• pp.173-184
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• 1997

An n-dimensional complex space form $M_n(c)$ is a Kaehlerian manifold of constant holomorphic sectional curvature c. As is well known, complete and simply connected complex space forms are a complex projective space $P_n C$, a complex Euclidean space $C_n$ or a complex hyperbolic space $H_n C$ according as c > 0, c = 0 or c < 0.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.737-767
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• 2016

This paper concerns closed hypersurfaces of dimension $n{\geq}2$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature evolving in direction of its normal vector, where the speed equals a power ${\beta}{\geq}1$ of the mean curvature. The main result is that if the initial closed, weakly h-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to 1, depending only on n and ${\beta}$, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in ${\mathbb{H}}_{\kappa}^{n+1}$ in a maximal finite time, and when rescaling appropriately, the evolving hypersurfaces exponential convergence to a unit geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$.

• Communications of the Korean Mathematical Society
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• v.13 no.3
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• pp.545-560
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• 1998

Let ø and A be denoted by the structure tensor field of type (1,1) and by the shape operator of a real hypersurface in a complex space form $M_{n}$ (c), c $\neq$ 0 respectively. The main purpose of this paper is to prove that if a real hypersurface in $M_{n}$ (c) satisfies $R_{ξ}$ øA = $AøR_{ξ}$, then the structure vector field ξ is principal, where $R_{ξ}$ / is the Jacobi operator with respect to ξ.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.531-538
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• 2014

Concentric hyperspheres in the n-dimensional Euclidean space $\mathbb{R}^n$ are the level hypersurfaces of a radial function f : $\mathbb{R}^n{\rightarrow}\mathbb{R}$. The magnitude $||{\nabla}f||$ of the gradient of such a radial function f : $\mathbb{R}^n{\rightarrow}\mathbb{R}$ is a function of the function f. We are interested in the converse problem. As a result, we show that if the magnitude of the gradient of a function f : $\mathbb{R}^n{\rightarrow}\mathbb{R}$ with isolated critical points is a function of f itself, then f is either a radial function or a function of a linear function. That is, the level hypersurfaces are either concentric hyperspheres or parallel hyperplanes. As a corollary, we see that if the magnitude of a conservative vector field with isolated singularities on $\mathbb{R}^n$ is a function of its scalar potential, then either it is a central vector field or it has constant direction.

• Journal of the Korean Mathematical Society
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• v.61 no.2
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• pp.309-339
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• 2024

Let M be a real hypersurface in the complex hyperbolic quadric Q^m*, m ≥ 3. The Riemannian curvature tensor field R of M allows us to define a symmetric Jacobi operator with respect to the Reeb vector field ξ, which is called the structure Jacobi operator R_ξ = R( · , ξ)ξ ∈ End(TM). On the other hand, in [20], Semmelmann showed that the cyclic parallelism is equivalent to the Killing property regarding any symmetric tensor. Motivated by his result above, in this paper we consider the cyclic parallelism of the structure Jacobi operator R_ξ for a real hypersurface M in the complex hyperbolic quadric Q^m*. Furthermore, we give a complete classification of Hopf real hypersurfaces in Q^m* with such a property.

• Journal of the HCI Society of Korea
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• v.1 no.1
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• pp.37-44
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• 2006

Specular reflections provide the visual feedback that describes the material type of an object, its local shape, and lighting environment. In photorealistic rendering, there have been a number of research available to render specular reflections effectively based on a local reflection model. In traditional cel animations and cartoons, specular reflections plays important role in representing artistic intentions for an object and its related environment reflections, so the shapes of highlights are quite stylistic. In this paper, we present a method to render and control stylized specular reflections using projective textures based on principal curvature analysis. Specifying a texture as a pattern of a highlight and projecting the texture on the specular region of a given 3D model, we can obtain a stylized representation of specular reflections. For a given polygonal model, a view point, and a light source, we first find the maximum specular intensity point, and then locate the texture projector along the line parallel to the normal vector and passing through the point. The orientation of the projector is determined by the principal directions at the point. Finally, the size of the projection frustum is determined by the principal curvatures corresponding to the principal directions. The proposed method can control the position, orientation, and size of the specular reflection efficiently by translating the projector along the principal directions, rotating the projector about the normal vector, and scaling the principal curvatures, respectively. The method is be applicable to real-time applications such as cartoon style 3D games. We implement the method by Microsoft DirectX 9.0c SDK and programmable vertex/pixel shaders on Nvidia GeForce FX 7800 graphics subsystems. According to our experimental results, we can render and control the stylized specular reflections for a 3D model of several ten thousands of triangles in real-time.