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TUBULAR HYPERSURFACES IN A NEARLY KA¨HLER 6-SPHERE
FUNABASHI, SHOICHI,PAK, JIN SUK TOPOLOGY AND GEOMETRY RESEARCH CENTER 1999 Proceedings of the Topology and Geometry Research Vol.10 No.-
In this paper we study 3-dimensional submanifolds of a nearly Khler 6-sphere as orbits of the symplectic group SP(1) of order 1 and obtain some geometric characterization of the tubues of those submanifolds
$F$-traceless component of the conformal curvature tensor on K\"ahler manifold
Shoichi Funabashi,김향숙,김영미,박진석 대한수학회 2007 대한수학회보 Vol.44 No.4
We investigate F-traceless component of the conformal cur-vature tensor dened by (3.6) in K¨ahler manifolds of dimension 4,and show that the F-traceless component is invariant under concircu-lar change. In particular, we determine K¨ahler manifolds with parallelF-traceless component and improve some theorems, provided in the pre-vious paper ([2]), which are concerned with the traceless component of theconformal curvature tensor and the spectrum of the Laplacian acting onp (0 p 2)-forms on the manifold by using the F-traceless component.
F-TRACELESS COMPONENT OF THE CONFORMAL CURVATURE TENSOR ON KÄHLER MANIFOLD
Funabashi, Shoichi,Kim, Hang-Sook,Kim, Young-Mi,Pak, Jin-Suk Korean Mathematical Society 2007 대한수학회보 Vol.44 No.4
We investigate F-traceless component of the conformal curvature tensor defined by (3.6) in $K\ddot{a}hler$ manifolds of dimension ${\geq}4$, and show that the F-traceless component is invariant under concircular change. In particular, we determine $K\ddot{a}hler$ manifolds with parallel F-traceless component and improve some theorems, provided in the previous paper([2]), which are concerned with the traceless component of the conformal curvature tensor and the spectrum of the Laplacian acting on $p(0{\leq}p{\leq}2)$-forms on the manifold by using the F-traceless component.
HYPERSURFACES IN A 6-DIMENSIONAL SPHERE
Hashimoto, Hideya,Funabashi, Shoichi Korean Mathematical Society 1997 대한수학회지 Vol.34 No.1
A 6-dimensional sphere considered as a homogeneous space $G_2/SU(3)$ where $G_2$ is the group of automorphism of the octonians O. From this representation, we can define an almost comlex structure on a 6-dimensional sphere by making use of the vector cross product of the octonians. Also it is known that a homogeneous space $G_2/U(2)$ coincides with the Grassmann manifold of oriented 2-planes of a 7-dimensional Euclidean space.
SP(1)-orbits in a nearly Kaeher 6-sphere
Pak, Jin Suk,FUNABASHI, Shoichi 경북대학교 위상수학 기하학연구센터 1999 硏究論文集 Vol.8 No.-
In this paper, we study the SP(1)-orbits defined by f(q)=x + ??y ∈ for q ∈ SP(1) and their tubular hypersurfaces in a nearly Kaehler 6- sphere S^(6). Every orbit of an action of SP(1) is realized essentially by the 1-parameter representation. Let M_(n) 0≤r≤1, be the SP(1)-orbits in S^(6). Then M_(n) 0<r<1, is a 3-dimensional totally umbilical submanifold of S^(6) with constant mean curvature ??/r and its tubular hypersurface is not a Hopf hypersurface of S^(6). The orbit M_(1) is a totally geodesic totally real submanifold of S^(6) and its tubular hypersurface is a Hopf hypersurface of the form S^3(cos τ)×S^2(sin τ), (0<τ<π/2), in S^(6). These all tubuar hypersurfaces have exactly two distinct principal curvatures at each point.