ON CONSTRUCTIONS OF MINIMAL SURFACES

Dae Won Yoon 충청수학회 2021 충청수학회지 Vol.34 No.1

In the recent papers, Sanchez-Reyes [Appl. Math. Model. 40 (2016), 1676{1682] described the method for nding a minimal surface through a geodesic, and Li et al. [Appl. Math. Model. 37 (2013), 6415{6424] studied the approximation of minimal surfaces with a geodesic from Dirichlet function. In the present article, we consider an isoparametric surface generated by Frenet frame of a curve introduced by Wang et al. [Comput. Aided Des. 36 (2004), 447-459], and give the necessary and sucient condition to satisfy both geodesic of the curve and minimality of the surface. From this, we construct minimal surfaces in terms of constant curvature and torsion of the curve. As a result, we present a new approach for constructions of the minimal surfaces from a prescribed closed geodesic and unclosed geodesic, and show some new examples of minimal surfaces with a circle and a helix as a geodesic. Our approach can be used in design of minimal surfaces from geodesics.

H-V-SEMI-SLANT SUBMERSIONS FROM ALMOST QUATERNIONIC HERMITIAN MANIFOLDS

Park, Kwang-Soon Korean Mathematical Society 2016 대한수학회보 Vol.53 No.2

We introduce the notions of h-v-semi-slant submersions and almost h-v-semi-slant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We obtain characterizations, investigate the integrability of distributions, the geometry of foliations, and a decomposition theorem. We find a condition for such submersions to be totally geodesic. We also obtain an inequality of a h-v-semi-slant submersion in terms of squared mean curvature, scalar curvature, and h-v-semi-slant angle. Finally, we give examples of such maps.

일방향 지오데식을 활용한 곡면 형상의 패널링 : 복합 곡면을 중심으로

홍지학(Hong, Ji-Hak),성우제(Sung, Woo-Jae) 한국BIM학회 2021 KIBIM Magazine Vol.11 No.4

Paneling building facades is one of the essential procedures in building construction. Traditionally, it has been an easy task of simply projecting paneling patterns drawn in drawing boards onto 3d building facades. However, as many organic or curved building shapes are designed and constructed in modern architectural practices, the traditional one-to-one projection is becoming obsolete for the building types of the kind. That is primarily because of the geometrical discrepancies between 2d drawing boards and 3d curved building surfaces. In addition, curved compound surfaces are often utilized to accommodate the complicated spatial programs, building codes, and zoning regulations or to achieve harmonious geometrical relationships with neighboring buildings in highly developed urban contexts. The use of the compound surface apparently makes the traditional paneling pattern projection more challenging. Various mapping technics have been introduced to deal with the inabilities of the projection methods for curved facades. The mapping methods translate geometries on a 2d surface into a 3d building façade at the same topological locations rather than relying on Euclidean or Affine projection. However, due to the intrinsic differences of the planar 2d and curved 3d surfaces, the mapping often comes with noticeable distortions of the paneling patterns. Thus, this paper proposes a practical method of drawing paneling patterns directly on a curved compound surface utilizing Geodesic, which is faithful to any curved surface, to minimize unnecessary distortions.

NON-INVARIANT HYPERSURFACES OF A (ε,δ)-TRANS SASAKIAN MANIFOLDS

Toukeer Khan,Sheeba Rizvi 경남대학교 수학교육과 2021 Nonlinear Functional Analysis and Applications Vol.26 No.5

The object of this paper is to study non-invariant hypersurface of a (ε,δ)-trans Sasakian manifolds equipped with (f,g,u,v,λ)-structure. Some properties obeyed by this structure are obtained. The necessary and sufficient conditions also have been obtained for totally umbilical non-invariant hypersurface with (f, g, u, v, λ)-structure of a (ε, δ)-trans Sasakian manifolds to be totally geodesic. The second fundamental form of a non-invariant hypersurface of a (ε, δ)-trans Sasakian manifolds with (f, g, u, v, λ)-structure has been traced under the condition when f is parallel.

H-v-semi-slant submersions from almost quaternionic Hermitian manifolds

Kwang-Soon Park 대한수학회 2016 대한수학회보 Vol.53 No.2

We introduce the notions of h-v-semi-slant submersions and almost h-v-semi-slant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We obtain characterizations, investigate the integrability of distributions, the geometry of foliations, and a decomposition theorem. We find a condition for such submersions to be totally geodesic. We also obtain an inequality of a h-v-semi-slant submersion in terms of squared mean curvature, scalar curvature, and h-v-semi-slant angle. Finally, we give examples of such maps.

Vortex Motion on Riemann Surfaces

김선철 한국물리학회 2011 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.59 No.1

Vortex motion of an incompressible inviscid flow on a infinitely thin curved shell specified as a Riemann surface is studied. The intrinsic dynamical equations are derived and compared on two conformally equivalent Riemann surfaces. In particular, the relations for the streamfunction, velocity, and vorticity are found by using explicit computations. Then, the result is applied to the problem of motion of point vortex dipoles on Riemann surfaces whose trajectories in time turn out to be a geodesic. Also, the case of more complicated surfaces with a boundary, is discussed with an example.

ON LIGHTLIKE HYPERSURFACES OF COSYMPLECTIC SPACE FORM

Ejaz Sabir Lone,Pankaj Pandey Korean Mathematical Society 2023 대한수학회논문집 Vol.38 No.1

The main purpose of this paper is to study the lightlike hypersurface (M, $\overline{g}$) of cosymplectic space form $\overline{M}$(c). In this paper, we computed the Gauss and Codazzi formulae of (M, $\overline{g}$) of cosymplectic manifold ($\overline{M}$, g). We showed that we can't obtain screen semi-invariant lightlike hypersurface (SCI-LH) of $\overline{M}$(c) with parallel second fundamental form h, parallel screen distribution and c ≠ 0. We showed that if second fundamental form h and local second fundamental form B are parallel, then (M, $\overline{g}$) is totally geodesic. Finally we showed that if (M, $\overline{g}$) is umbilical, then cosymplectic manifold ($\overline{M}$, g) is flat.

QUASI BI-SLANT SUBMANIFOLDS OF KENMOTSU MANIFOLDS

Rajendra Prasad,Abdul Haseeb,Pooja Gupta,AHMED HUSSEIN MSMALI 장전수학회 2022 Advanced Studies in Contemporary Mathematics Vol.32 No.2

The fundamental motivation behind the current paper is to define and study the notion of quasi bi-slant submanifolds of Kenmotsu manifolds as a generalization of slant, semi-slant, hemi-slant, bi-slant and quasi hemislant submanifolds. First and foremost, we obtain the necessary and suffcient condition for the integrability of distributions of quasi bi-slant submanifolds of Kenmotsu manifolds and afterwards, we investigate the conditions for quasi bi-slant submanifolds of Kenmotsu manifolds to be totally geodesic. At long last, we additionally provide some examples of such submanifolds.

3D Magic Wand: 하모닉 필드를 이용한 메쉬 분할 기법

문지혜,박상훈,윤승현 (사)한국컴퓨터그래픽스학회 2022 컴퓨터그래픽스학회논문지 Vol.28 No.1

In this paper we present a new method for interactive segmentation of a triangle mesh by using the concavity-sensitive harmonic field and anisotropic geodesic. The proposed method only requires a single vertex in a desired feature region, while most of existing methods need explicit information on segmentation boundary. From the user-clicked vertex, a candidate region which contains the desired feature region is defined and concavity-senstive harmonic field is constructed on the region by using appropriate boundary constraints. An initial isoline is chosen from the uniformly sampled isolines on the harmonic field and optimal points on the initial isoline are determined as interpolation points. Final segmentation boundary is then constructed by computing anisotropic geodesics passing through the interpolation points. In experimental results, we demonstrate the effectiveness of the proposed method by selecting several features in various 3D models. 본 논문에서는 특징 추출 하모닉 필드(harmonic field)와 비등방 측지선(anisotropic geodesic)을이용하여 메쉬의 특징 영역을분할하는 새로운 기법을 제안한다. 기존 대부분의 메쉬 분할 기법들은 경계 영역에 대한 사용자의 명시적인입력을 요구하지만, 제안된 기법에서는 사용자가 관심 영역의 임의의 정점을 선택하여 직관적이고 편리하게 특징 영역을 분할한다. 사용자가선택한 정점을 중심으로 오목한(concave) 영역에서 큰 변화를 갖는 하모닉 필드를 생성한다. 생성된 하모닉 필드에서 하나의등위선(isoline)을 선택하여 초기 분할 경계선을 정하고, 선택된 등위선에서 최적의 특징점을 추출하여 비등방 측지선으로연결함으로써 최종적인 분할 경계선을 생성한다. 다양한 실험을 통해 제안된 기법이 사용자의 입력에 민감하지 않으며, 특징영역 분할에 효과적으로 사용될 수 있음을 보인다.

SECOND ORDER TANGENT VECTORS IN RIEMANNIAN GEOMETRY

Kwon, Soon-Hak Korean Mathematical Society 1999 대한수학회지 Vol.36 No.5

This paper considers foundational issues related to connections in the tangent bundle of a manifold. The approach makes use of second order tangent vectors, i.e., vectors tangent to the tangent bundle. The resulting second order tangent bundle has certain properties, above and beyond those of a typical tangent bundle. In particular, it has a natural secondary vector bundle structure and a canonical involution that interchanges the two structures. The involution provides a nice way to understand the torsion of a connection. The latter parts of the paper deal with the Levi-Civita connection of a Riemannian manifold. The idea is to get at the connection by first finding its.spary. This is a second order vector field that encodes the second order differential equation for geodesics. The paper also develops some machinery involving lifts of vector fields form a manifold to its tangent bundle and uses a variational approach to produce the Riemannian spray.