Some curvature pinching theorems for compact or complete totally real minimal submanifolds in a quaternion projective space are given,so that the corresponding results due to B. Y.Chen and C. S. Houh as well as Y. B. ...Some curvature pinching theorems for compact or complete totally real minimal submanifolds in a quaternion projective space are given,so that the corresponding results due to B. Y.Chen and C. S. Houh as well as Y. B. Shen are improved and generalized.展开更多
The notion of finite type submanifolds was introduced by B. Y. Chen. In this paper we consider the characteristics and the classifications of finite type non-minimal submanifolds. The characteristic theorems of 2-type...The notion of finite type submanifolds was introduced by B. Y. Chen. In this paper we consider the characteristics and the classifications of finite type non-minimal submanifolds. The characteristic theorems of 2-type Chen submanifolds,mass-symmetrie hypersurfaces and Dupin hypersurfaces in E_3~m are obtained. The classification theorems of 3-type hypersurfaces and null 2-type curves in E_3~m are also proved.展开更多
In this paper, we discuss the compact minimal submanifolds in locally symmetric Riemannian manifolds. Two Pinching theorems are obtained and two corresponding results of Chern, S. S. and Yau S. T. are generalized.
In this paper,we extend two important theorem in[1],[2]to the minimal submanifolds in a Locally symmetric and conformally flat Riemannian mainfold N^(+p).When N^(+p)is a space S_(1)^(+p) of constant curvature,our theo...In this paper,we extend two important theorem in[1],[2]to the minimal submanifolds in a Locally symmetric and conformally flat Riemannian mainfold N^(+p).When N^(+p)is a space S_(1)^(+p) of constant curvature,our theorems reduce to the theorems of[1],[2].展开更多
Tang and Zhang(2020)and Choe and Hoppe(2018)showed independently that one can produce minimal submanifolds in spheres via the Clifford type minimal product of minimal submanifolds.In this paper,we show that the minima...Tang and Zhang(2020)and Choe and Hoppe(2018)showed independently that one can produce minimal submanifolds in spheres via the Clifford type minimal product of minimal submanifolds.In this paper,we show that the minimal product is immersed by its first eigenfunctions(of its Laplacian)if and only if the two beginning minimal submanifolds are immersed by their first eigenfunctions.Moreover,we give the estimates of the Morse index and the nullity of the minimal product.In particular,we show that the Clifford minimal submanifold(√n1/nS^(n1).....,√nk/nS^(nk)■S^(n+k-1))has the index(k-1)(n+k+1)and the nullity(k-1)∑_(1≤i<j≤k)(n_(i)+1)(nj+1)(with n=∑n_(j)).展开更多
Let Mn be an n-dimensional compact minimal submanifolds in Sin(1)× R. We prove two pinching theorems by the Ricci curvature and the sectional curvature pinching conditions respectively. In fact, we characterize...Let Mn be an n-dimensional compact minimal submanifolds in Sin(1)× R. We prove two pinching theorems by the Ricci curvature and the sectional curvature pinching conditions respectively. In fact, we characterize the Clifford tori and Veronese submanifolds by our pinching conditions respectively.展开更多
In this paper we study the C3 compactness for minimal submanifolds in the unit sphere. We obtain two compactness theorems. As an application, we prove that there is a positive number δ(n), such that if the square of ...In this paper we study the C3 compactness for minimal submanifolds in the unit sphere. We obtain two compactness theorems. As an application, we prove that there is a positive number δ(n), such that if the square of the length of the second fundamental form of a minimal subrnanifold in the unit sphere is less than 2n/3+δ(n), it must be totally geodesic or diffeomorphic to a Veronese surface.展开更多
The compact minimal submanifold in a locally symetric and conformally flat Riemann manifold is discussed in this paper.We get the Pinching constant for scalar curvature.The result of Li[2]is generallied,but the method...The compact minimal submanifold in a locally symetric and conformally flat Riemann manifold is discussed in this paper.We get the Pinching constant for scalar curvature.The result of Li[2]is generallied,but the method is completely different. Meanwhile,we get better conclusion than that of [3].We also research the Pinching problem for sectional curvature on compact minimal submanifolds in a unit sphere, partially improving the results of S.T.Yan[4].展开更多
The authors derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau’s results and Ecker-Huisken’s results are generalized ...The authors derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau’s results and Ecker-Huisken’s results are generalized to higher codimension. In this way, Hildebrandt-Jost-Widman’s result for the Bernstein type theorem is improved.展开更多
In this paper,the minimal conjecture on the Veronese Generating submanifolds in S(1)5 is generalized to minmal submanifolds in Sm. Two classes of the Generating submanifolds of spherical minimal submanifolds in Minkow...In this paper,the minimal conjecture on the Veronese Generating submanifolds in S(1)5 is generalized to minmal submanifolds in Sm. Two classes of the Generating submanifolds of spherical minimal submanifolds in Minkowski space are respectively space-like Minimal submanifolds of Hyperbolic space and pseudo-Rie-mannian Minimal submanifolds of pseudo-Riemannian sphere and they are of 1-type submanifolds in Minkowski space is proved.展开更多
The notion of finite type submanifolds was introduced by B. Y. Chen. In this paper the conjectures on scalar curvature of Veronese generating submanifolds in E~σ and the minimal conjecture on Veronese space-like subm...The notion of finite type submanifolds was introduced by B. Y. Chen. In this paper the conjectures on scalar curvature of Veronese generating submanifolds in E~σ and the minimal conjecture on Veronese space-like submanifold Σ and Veronese pseudo-Riemannian submanifold in E_1~σ are proved. We have Σ is minimal in H^5. is minimal in S_1~5, Σ and are of 1-type in E_1~σ.展开更多
In this paper,the higher dimensional conjecture on Veronese generating subman-ifolds proposed by Prof. SUN Zhen-zu is generalized to Pseudo-Euclidean Space L1(m+1), it is proved that in the higher dimentional Lorentz ...In this paper,the higher dimensional conjecture on Veronese generating subman-ifolds proposed by Prof. SUN Zhen-zu is generalized to Pseudo-Euclidean Space L1(m+1), it is proved that in the higher dimentional Lorentz Space L1(m+1), the generating submanifolds of an n dimentional submanifold of Pseudo-Riemannian unit sphere S1m is an n+1 dimentional minimal submanifold of S1(m+1) in L1(m+2) and is of 1-type in L1(m+2).展开更多
This paper studies the relationship between the pseudo-umbilical totally real submanifolds and the minimal totally real submanifolds in a complex projective space. Two theo- rems which claim that some types of pseudo-...This paper studies the relationship between the pseudo-umbilical totally real submanifolds and the minimal totally real submanifolds in a complex projective space. Two theo- rems which claim that some types of pseudo-umbilical totally real submanifolds must be minimal submanifolds are proved.展开更多
Proves that cosymplectic hypersurfaces of six dimensional Hermitian submanifolds of the octave algebra are ruled manifolds and establishes a necessary and sufficient condition for a cosymplectic hypersurface of Hermit...Proves that cosymplectic hypersurfaces of six dimensional Hermitian submanifolds of the octave algebra are ruled manifolds and establishes a necessary and sufficient condition for a cosymplectic hypersurface of Hermitian M 6 O to be a minimal submanifold of M 6.展开更多
Let N n+p be an (n+p)-dimensional locally symmetric and conformally flat Riemannian manifold and Mn be an n-dimensional compact submanifold minimally immersed in N n+p . Instead of (n+p)-dimensional unit sphere, we ge...Let N n+p be an (n+p)-dimensional locally symmetric and conformally flat Riemannian manifold and Mn be an n-dimensional compact submanifold minimally immersed in N n+p . Instead of (n+p)-dimensional unit sphere, we generalize Pinching Theorems about submanifolds in unit sphere and get theorems about submanifolds in locally symmetric and conformally flat Riemannian manifold.展开更多
We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angl...We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angle functions encoding the geometry of the Lagrangian submanifold at hand.We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface.We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions,respectively all but one,coincide.展开更多
The equivariant Lagrangian minimal immersion of Sa into C.Pa is studied. The complete classification and analytic expression for such kinds of immersions are given.
文摘Some curvature pinching theorems for compact or complete totally real minimal submanifolds in a quaternion projective space are given,so that the corresponding results due to B. Y.Chen and C. S. Houh as well as Y. B. Shen are improved and generalized.
基金Supported by the National Fundations of Natural sciences. Supported by the Henan Fundations of Scientific Committee.
文摘The notion of finite type submanifolds was introduced by B. Y. Chen. In this paper we consider the characteristics and the classifications of finite type non-minimal submanifolds. The characteristic theorems of 2-type Chen submanifolds,mass-symmetrie hypersurfaces and Dupin hypersurfaces in E_3~m are obtained. The classification theorems of 3-type hypersurfaces and null 2-type curves in E_3~m are also proved.
文摘In this paper, we discuss the compact minimal submanifolds in locally symmetric Riemannian manifolds. Two Pinching theorems are obtained and two corresponding results of Chern, S. S. and Yau S. T. are generalized.
文摘In this paper,we extend two important theorem in[1],[2]to the minimal submanifolds in a Locally symmetric and conformally flat Riemannian mainfold N^(+p).When N^(+p)is a space S_(1)^(+p) of constant curvature,our theorems reduce to the theorems of[1],[2].
基金supported by National Natural Science Foundation of China(Grant No.11831005)supported by National Natural Science Foundation of China(Grant No.11971107)。
文摘Tang and Zhang(2020)and Choe and Hoppe(2018)showed independently that one can produce minimal submanifolds in spheres via the Clifford type minimal product of minimal submanifolds.In this paper,we show that the minimal product is immersed by its first eigenfunctions(of its Laplacian)if and only if the two beginning minimal submanifolds are immersed by their first eigenfunctions.Moreover,we give the estimates of the Morse index and the nullity of the minimal product.In particular,we show that the Clifford minimal submanifold(√n1/nS^(n1).....,√nk/nS^(nk)■S^(n+k-1))has the index(k-1)(n+k+1)and the nullity(k-1)∑_(1≤i<j≤k)(n_(i)+1)(nj+1)(with n=∑n_(j)).
基金supported by National Natural Science Foundation of China (Grant No.11271214)
文摘Let Mn be an n-dimensional compact minimal submanifolds in Sin(1)× R. We prove two pinching theorems by the Ricci curvature and the sectional curvature pinching conditions respectively. In fact, we characterize the Clifford tori and Veronese submanifolds by our pinching conditions respectively.
基金Supported by the National Natural Scieuce Foundation of China(19971081)
文摘In this paper we study the C3 compactness for minimal submanifolds in the unit sphere. We obtain two compactness theorems. As an application, we prove that there is a positive number δ(n), such that if the square of the length of the second fundamental form of a minimal subrnanifold in the unit sphere is less than 2n/3+δ(n), it must be totally geodesic or diffeomorphic to a Veronese surface.
文摘The compact minimal submanifold in a locally symetric and conformally flat Riemann manifold is discussed in this paper.We get the Pinching constant for scalar curvature.The result of Li[2]is generallied,but the method is completely different. Meanwhile,we get better conclusion than that of [3].We also research the Pinching problem for sectional curvature on compact minimal submanifolds in a unit sphere, partially improving the results of S.T.Yan[4].
基金supported by the National Natural Science Foundation of China (No. 10531090)the NaturalScience Foundation of the Ministry of Education of China
文摘The authors derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau’s results and Ecker-Huisken’s results are generalized to higher codimension. In this way, Hildebrandt-Jost-Widman’s result for the Bernstein type theorem is improved.
文摘In this paper,the minimal conjecture on the Veronese Generating submanifolds in S(1)5 is generalized to minmal submanifolds in Sm. Two classes of the Generating submanifolds of spherical minimal submanifolds in Minkowski space are respectively space-like Minimal submanifolds of Hyperbolic space and pseudo-Rie-mannian Minimal submanifolds of pseudo-Riemannian sphere and they are of 1-type submanifolds in Minkowski space is proved.
文摘The notion of finite type submanifolds was introduced by B. Y. Chen. In this paper the conjectures on scalar curvature of Veronese generating submanifolds in E~σ and the minimal conjecture on Veronese space-like submanifold Σ and Veronese pseudo-Riemannian submanifold in E_1~σ are proved. We have Σ is minimal in H^5. is minimal in S_1~5, Σ and are of 1-type in E_1~σ.
基金Supported by the Education Commission of Henan Province(20021100002) Supported by the NSF of Henan University(200110475028)
文摘In this paper,the higher dimensional conjecture on Veronese generating subman-ifolds proposed by Prof. SUN Zhen-zu is generalized to Pseudo-Euclidean Space L1(m+1), it is proved that in the higher dimentional Lorentz Space L1(m+1), the generating submanifolds of an n dimentional submanifold of Pseudo-Riemannian unit sphere S1m is an n+1 dimentional minimal submanifold of S1(m+1) in L1(m+2) and is of 1-type in L1(m+2).
基金the Natural Science Foundation of Education Committee of Anhui Province(2004kj166zd)Foundation for Younger Teachers of Anhui Normal University(2005xqn01).
文摘This paper studies the relationship between the pseudo-umbilical totally real submanifolds and the minimal totally real submanifolds in a complex projective space. Two theo- rems which claim that some types of pseudo-umbilical totally real submanifolds must be minimal submanifolds are proved.
文摘Proves that cosymplectic hypersurfaces of six dimensional Hermitian submanifolds of the octave algebra are ruled manifolds and establishes a necessary and sufficient condition for a cosymplectic hypersurface of Hermitian M 6 O to be a minimal submanifold of M 6.
文摘Let N n+p be an (n+p)-dimensional locally symmetric and conformally flat Riemannian manifold and Mn be an n-dimensional compact submanifold minimally immersed in N n+p . Instead of (n+p)-dimensional unit sphere, we generalize Pinching Theorems about submanifolds in unit sphere and get theorems about submanifolds in locally symmetric and conformally flat Riemannian manifold.
基金supported by the Tsinghua University-KU Leuven Bilateral Scientific Cooperation Fundcollaboration project funded by National Natural Science Foundation of China+6 种基金supported by National Natural Science Foundation of China(Grant Nos.11831005 and 11671224)supported byNational Natural Science Foundation of China(Grant Nos.11831005 and 11671223)supported by National Natural Science Foundation of China(Grant No.11571185)the Research Foundation Flanders(Grant No.11961131001)supported by the Excellence of Science Project of the Belgian Government(Grant No.GOH4518N)supported by the KU Leuven Research Fund(Grant No.3E160361)the Fundamental Research Funds for the Central Universities。
文摘We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angle functions encoding the geometry of the Lagrangian submanifold at hand.We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface.We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions,respectively all but one,coincide.
文摘The equivariant Lagrangian minimal immersion of Sa into C.Pa is studied. The complete classification and analytic expression for such kinds of immersions are given.