• Kyungpook Mathematical Journal
• /
• v.63 no.2
• /
• pp.287-311
• /
• 2023

In this paper, we introduce the notion of cyclic structure Jacobi semi-symmetric real hypersurfaces in the complex hyperbolic quadric Q^m* = SO⁰_2,m/SO₂SO_m. We give a classifiction of when real hypersurfaces in the complex hyperbolic quadric Q^m* having 𝔄-principal or 𝔄-isotropic unit normal vector fields have cyclic structure Jacobi semi-symmetric tensor.

• Bulletin of the Korean Mathematical Society
• /
• v.60 no.4
• /
• pp.849-861
• /
• 2023

In this paper, we introduce the notion of semi-symmetric structure Jacobi operator for Hopf real hypersufaces in the complex quadric Q^m = SO_m+2/SO_mSO₂. Next we prove that there does not exist any Hopf real hypersurface in the complex quadric Q^m = SO_m+2/SO_mSO₂ with semi-symmetric structure Jacobi operator. As a corollary, we also get a non-existence property of Hopf real hypersurfaces in the complex quadric Q^m with either symmetric (parallel), or recurrent structure Jacobi operator.

• Journal of the Korean Mathematical Society
• /
• v.58 no.4
• /
• pp.895-920
• /
• 2021

In this paper, first we introduce the full expression of the Riemannian curvature tensor of a real hypersurface M in the complex quadric Q^m from the equation of Gauss and some important formulas for the structure Jacobi operator R_ξ and its derivatives ∇R_ξ under the Levi-Civita connection ∇ of M. Next we give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, ∇_ξR_ξ = 0, in the complex quadric Q^m for m ≥ 3. In addition, we also consider a new notion of 𝒞-parallel structure Jacobi operator of M and give a nonexistence theorem for Hopf real hypersurfaces with 𝒞-parallel structure Jacobi operator in Q^m, for m ≥ 3.

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

• Kyungpook Mathematical Journal
• /
• v.57 no.4
• /
• pp.683-699
• /
• 2017

We introduce the notion of Killing shape operator for real hypersurfaces in the complex hyperbolic quadric $Q^{m*}=SO_{m,2}/SO_mSO_2$. The Killing shape operator implies that the unit normal vector field N becomes A-principal or A-isotropic. Then according to each case, we give a complete classification of real hypersurfaces in $Q^{m*}=SO_{m,2}/SO_mSO_2$ with Killing shape operator.

• Kyungpook Mathematical Journal
• /
• v.60 no.3
• /
• pp.551-570
• /
• 2020

We introduce the notion of Lie invariant normal Jacobi operators for real hypersurfaces in the complex hyperbolic quadric Q^m∗ = SO^o_m,2/SO_mSO₂. The invariant normal Jacobi operator implies that the unit normal vector field N becomes 𝕬-principal or 𝕬-isotropic. Then in each case, we give a complete classification of real hypersurfaces in Q^m∗ = SO^o_m,2/SO_mSO₂ with Lie invariant normal Jacobi operators.

• East Asian mathematical journal
• /
• v.32 no.1
• /
• pp.93-100
• /
• 2016

The generalized Gaussian image of a spacelike surface in $L^n$ lies in the indefinite positive quadric ${\mathbb{Q}}_+^{n-2}$ in the open submanifold ${\mathbb{C}}P_+^{n-1}$ of the complex projective space ${\mathbb{C}}P^{n-1}$. The purpose of this paper is to find out detailed information about ${\mathbb{Q}}_+^{n-2}{\subset}{\mathbb{C}}P_+^{n-1}$.

• Proceedings of the IEEK Conference
• /
• 2000.11c
• /
• pp.109-112
• /
• 2000

Many applications in computer graphics require highly detailed complex models. However, the level of detail may vary considerably according to applications. It is often desirable to use approximations in place of excessively detailed models. We have developed a surface simplification algorithm which uses iterative contractions of edges to simplify models and maintains surface error approximations using a quadric metric. In this paper, we present an improved quadric error metric for simplifying meshes. The new metric, based on subdivided edge classification, results in more accurate simplified meshes. We show that a subdivided edge classification captures discontinuities efficiently. The new scheme is demonstrated on a variety of meshes.

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.2
• /
• pp.327-332
• /
• 2002

In this paper we prove that if M is a J-invariant sub-manifold of codimension 2 in a complex projective space $P_{n+1}(C)$, and the second fundamental tensor is cyclic-parallel or M has harmonic curvature, then M is locally complex quadric Q$_n$(C) or P$_n$(C).

• Bulletin of the Korean Space Science Society
• /
• 2004.10b
• /
• pp.347-350
• /
• 2004

A space radiation analysis has been used to evaluate an ability of electronic equipment boxes or spacecrafts to endure various radiation effects, so it helps design thicknesses of structure and allocate components to meet the radiation requirements. A comparison study of space radiation dose analysis programs SPENVIS Sectoring Tool (SST) and SIGMA II is conducted through some structure cases, simple sphere shell, box and representative satellite configurations. The results and a discussion of comparison will be given. A general comparison will be shown for understanding those programs. The both programs use the same strategy, solid angle sectoring with ray-tracing method to produce an approximate dose at points in representative simple and complex models of spacecraft structures. Also the particle environment data corresponding to mission specification and radiation transport data are used as input data. But there are distinctions between them. The specification of geometry model and its input scheme, the assignment of dose point and the numbers, the prerequisite programs and ways of representing results will be discussed. SST is a web-based interactive program for sectoring analysis of complex geometries. It may be useful for a preliminary dose assessment with user-friendly interfaces and a package approach. SIGMA II is able to obtain from RSICC (Radiation Safety Information Computational Center) as a FOR-TRAN 77 source code. It may be suitable for either parametric preliminary design or detailed final design, e.g. a manned flight or radiation-sensitive component configuration design. It needs some debugs, recompiling and a tedious work to make geometrical quadric surfaces for actual spacecraft configuration, and has poor documentation. It is recommend to vist RSICC homepage and GEANT4/SSAT homepage.