In this paper, a rigidity theorem of hypersurface in real space form will be given. In addition, we obtain rigidity theorems of submanifold in sphere which improve the result of Hou and Xu.
The Frenet-Serret formula is used to characterize the constant angle ruled surfaces in R3. When the surfaces are the tangent developmental and normal surfaces, that is, r（s, v） = tr（s） ＋v（cosα（s） . t（s） ＋s...The Frenet-Serret formula is used to characterize the constant angle ruled surfaces in R3. When the surfaces are the tangent developmental and normal surfaces, that is, r（s, v） = tr（s） ＋v（cosα（s） . t（s） ＋sina（s） . n（s））, it is shown that each of these surfaces is locally isometric to a piece of a plane or a certain special surface. When the surfaces are normal and binormal surfaces, that is, r （ s, v ） = σ （ s ） ＋ v （ cosa （ s ） . n（s） ＋ since（s） . b（s））, it is shown that each of these surfaces is locally isometric to a piece of a plane or a cylindrical surface.展开更多
The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^（n＋1）（ 1） is considered. Let A = ∑i,j,k h（ijk）~2（ λi＋ nH）~2,B = ∑i,j,k h（ijk）~2（ λi＋ nH） ·（ ...The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^（n＋1）（ 1） is considered. Let A = ∑i,j,k h（ijk）~2（ λi＋ nH）~2,B = ∑i,j,k h（ijk）~2（ λi＋ nH） ·（ λj＋ nH）,S = ∑i（ λi＋ nH）~2, where h（ij）= λiδ（ij）. Utilizing Lagrange＇s method, a sharper pointwise estimation of 3（A- 2B） in terms of S and ｜▽h｜~2 is obtained, here ｜▽h｜~2= ∑i,j,k h（ijk）~2. Then, with the help of this, it is proved that Mis isometric to the Clifford hypersurface if the square norm of the second fundamental form of Msatisfies certain conditions. Hence, the pinching result of the minimal hypersurface is extended to the hypersurface with constant mean curvature case.展开更多
In this paper, some construction theorems of pluriharmonic maps into complex Grassmann manifolds axe obtained. By these, there exists a characterization of strongly isotropic pluriharmonic maps.
The rigidity of spacelike hypersurface Mn immersed in locally symmetric space M1n+1 is investigated,where the(normalized)scalar curvature R and mean curvature H of Mn satisfy R=aH+b,and a,b are real constants.First,an...The rigidity of spacelike hypersurface Mn immersed in locally symmetric space M1n+1 is investigated,where the(normalized)scalar curvature R and mean curvature H of Mn satisfy R=aH+b,and a,b are real constants.First,an estimate of the upper bound of the function L(nH)is given,where L is a second-order differential operator.Then,under the assumption that the square norm of the second fundamental form is bounded by a given positive constant,it is proved that Mn must be either totally umbilical or contain two distinct principle curvatures,one of which is simple.Moreover,a similar result is obtained for complete noncompact spacelike hypersurfaces in locally symmetric Einstein spacetime.Hence,some known rigidity results for hypersurface with constant scalar curvature are extended for the linear Weingarten case.展开更多
文摘In this paper, a rigidity theorem of hypersurface in real space form will be given. In addition, we obtain rigidity theorems of submanifold in sphere which improve the result of Hou and Xu.
基金The National Natural Science Foundation of China(No.10971029,11101078,11171064)the Natural Science Foundation of Jiangsu Province(No.BK2011583)
文摘The Frenet-Serret formula is used to characterize the constant angle ruled surfaces in R3. When the surfaces are the tangent developmental and normal surfaces, that is, r（s, v） = tr（s） ＋v（cosα（s） . t（s） ＋sina（s） . n（s））, it is shown that each of these surfaces is locally isometric to a piece of a plane or a certain special surface. When the surfaces are normal and binormal surfaces, that is, r （ s, v ） = σ （ s ） ＋ v （ cosa （ s ） . n（s） ＋ since（s） . b（s））, it is shown that each of these surfaces is locally isometric to a piece of a plane or a cylindrical surface.
文摘The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^（n＋1）（ 1） is considered. Let A = ∑i,j,k h（ijk）~2（ λi＋ nH）~2,B = ∑i,j,k h（ijk）~2（ λi＋ nH） ·（ λj＋ nH）,S = ∑i（ λi＋ nH）~2, where h（ij）= λiδ（ij）. Utilizing Lagrange＇s method, a sharper pointwise estimation of 3（A- 2B） in terms of S and ｜▽h｜~2 is obtained, here ｜▽h｜~2= ∑i,j,k h（ijk）~2. Then, with the help of this, it is proved that Mis isometric to the Clifford hypersurface if the square norm of the second fundamental form of Msatisfies certain conditions. Hence, the pinching result of the minimal hypersurface is extended to the hypersurface with constant mean curvature case.
基金Research supported by National Nature Science Foundation of China(10171012),Tian Yuan Foundation 10226001 and Foundation of Southeast University
文摘 In this paper, some construction theorems of pluriharmonic maps into complex Grassmann manifolds axe obtained. By these, there exists a characterization of strongly isotropic pluriharmonic maps.
基金The Natural Science Foundation of Jiangsu Province(No.BK20161412)the Fundamental Research Funds for the Central Universitiesthe Scientific Innovation Research of College Graduates in Jiangsu Province(No.KYCX17_0041)
文摘The rigidity of spacelike hypersurface Mn immersed in locally symmetric space M1n+1 is investigated,where the(normalized)scalar curvature R and mean curvature H of Mn satisfy R=aH+b,and a,b are real constants.First,an estimate of the upper bound of the function L(nH)is given,where L is a second-order differential operator.Then,under the assumption that the square norm of the second fundamental form is bounded by a given positive constant,it is proved that Mn must be either totally umbilical or contain two distinct principle curvatures,one of which is simple.Moreover,a similar result is obtained for complete noncompact spacelike hypersurfaces in locally symmetric Einstein spacetime.Hence,some known rigidity results for hypersurface with constant scalar curvature are extended for the linear Weingarten case.
