In this article, it is proved that there doesn’t exist any nonsingular holomorphic sphere in complex Grassmann manifold G（2, 5） with constant curvature k = 4/7, 1/2, 4/9. Thus, from [7] it follows that if φ ： S2 ...In this article, it is proved that there doesn’t exist any nonsingular holomorphic sphere in complex Grassmann manifold G（2, 5） with constant curvature k = 4/7, 1/2, 4/9. Thus, from [7] it follows that if φ ： S2 → G（2, 5） is a nonsingular holomorphic curve with constant curvature K, then, K = 4, 2, 4/3, 1 or 4/5.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.
Based on the relationship between symplectic group Sp（2） and （2）, we provide an intuitive explanation （model） of the 3-dimensional Lagrangian Grassmann manifold （2）, the singular cycles of （2）, and the speci...Based on the relationship between symplectic group Sp（2） and （2）, we provide an intuitive explanation （model） of the 3-dimensional Lagrangian Grassmann manifold （2）, the singular cycles of （2）, and the special Lagrangian Grassmann manifold S（2）. Under this model, we give a formula of the rotation paths defined by Arnold.展开更多
We prove that if φ is a homogeneous harmonic map from a Riemann surface M into a complex Grassmann manifold G（k, n）, then the maps of the harmonic sequences generated by φ are all homogeneous.
We derive the Hessian estimate of the w-function defined on the pseudo-Grassmann manifold Gnmm, which is convex by the estimate. As an application, we give a direct proof of the Bernstein-type theorem due to Y. Xin [M...We derive the Hessian estimate of the w-function defined on the pseudo-Grassmann manifold Gnmm, which is convex by the estimate. As an application, we give a direct proof of the Bernstein-type theorem due to Y. Xin [Mannscripta Math., 2000, 103： 191-202]. We also estimate the squared norm of the second fundamental form of a complete spacelike submanifold in pseudoEuclidean space in terms of the w-function and the mean curvature.展开更多
A generalized Frenet formula for holomorphic 2 spheres S 2 in Grassmann manifolds G(k,n) is introduced and a generalized Plücker formula is achieved. By use of these formulae some pinching theorems for Gaussian c...A generalized Frenet formula for holomorphic 2 spheres S 2 in Grassmann manifolds G(k,n) is introduced and a generalized Plücker formula is achieved. By use of these formulae some pinching theorems for Gaussian curvature are obtained which generalize the corresponding results about holomorphic and minimal S 2 in CP n-1 . Einsteinian holomorphic subbundle of the trivial bundle S 2×C n is investigated.展开更多
In the complex Grassmann manifold F(m,n), the space of complex \%n\%\|planes passes through the origin of C \%m+n\% ; the local coordinate of the space can be arranged into an \%m×n\% matrix \%Z.\% It is proved t...In the complex Grassmann manifold F(m,n), the space of complex \%n\%\|planes passes through the origin of C \%m+n\% ; the local coordinate of the space can be arranged into an \%m×n\% matrix \%Z.\% It is proved thatK=K(Z,\%d\%Z)=(I+ZZ ) -12 (I+ZZ ) 12 -(I+ZZ ) 12 ·(I+ZZ ) -12 is a \%U(m)\%\|connection of F\%(m,n)\% and its curvature formΩ\-1= d K+K∧Ksatisfies the Yang\|Mills equation. Moreover,B=B(Z, d Z)=K(Z, d Z)- tr (K(Z, d Z))mI (m) is an SU(m) \|connection and its curvature formΩ\-2= d B+B∧Bsatisfies the Yang\|Mills equation.展开更多
1 Introduction Let G<sub>m,n</sub> be the Grassmann manifold of the unoriented, m-dimensional vector subspaces of the vector space R<sup>m+n</sup>. Then G<sub>m,n</sub> is a closed,...1 Introduction Let G<sub>m,n</sub> be the Grassmann manifold of the unoriented, m-dimensional vector subspaces of the vector space R<sup>m+n</sup>. Then G<sub>m,n</sub> is a closed, smooth manifold of dimension mn. G<sub>m,n</sub>≌G<sub>n,m</sub>, and G<sub>1,n</sub>≌G<sub>n,1</sub> is just the real projective space RP<sup>n</sup>. For convenience, we assume that m≤n. Suppose M<sup>p</sup>R<sup>q</sup> means that the p-dimensional manifold M<sup>p</sup> is immersed in q-展开更多
In this paper, we construct a class of homogeneous minimal 2-spheres in complex Grassmann manifolds by applying the irreducible unitary representations of SU （2）. Furthermore, we compute induced metrics, Gaussian cu...In this paper, we construct a class of homogeneous minimal 2-spheres in complex Grassmann manifolds by applying the irreducible unitary representations of SU （2）. Furthermore, we compute induced metrics, Gaussian curvatures, Khler angles and the square lengths of the second fundamental forms of these homogeneous minimal 2-spheres in G（2, n ＋ 1） by making use of Veronese sequence.展开更多
基金Supported by the National Natural Science Foundation of China (10531090)Knowledge Innovation Funds of CAS (KJCX3-SYW-S03)
文摘In this article, it is proved that there doesn’t exist any nonsingular holomorphic sphere in complex Grassmann manifold G（2, 5） with constant curvature k = 4/7, 1/2, 4/9. Thus, from [7] it follows that if φ ： S2 → G（2, 5） is a nonsingular holomorphic curve with constant curvature K, then, K = 4, 2, 4/3, 1 or 4/5.
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 author is grateful to Professor Yiming Long for his interest and Professor Xijun Hu for many useful advises and patient guidance. Also, the author would like to convey thanks to the anonymous referees for useful comments and suggestions. Finally, the author won＇t forget his beloved friends and family members, for their understanding and endless love through the duration of his studies. This work was partially supported by the National Natural Science Foundation of China （Grant No. 11425105）.
文摘Based on the relationship between symplectic group Sp（2） and （2）, we provide an intuitive explanation （model） of the 3-dimensional Lagrangian Grassmann manifold （2）, the singular cycles of （2）, and the special Lagrangian Grassmann manifold S（2）. Under this model, we give a formula of the rotation paths defined by Arnold.
基金This work was supported in part by the National Natural Science Foundation of China （Grant No. 11401481） and the Research Development Fund of XJTLU （Grant No. RDF13-01-14）.
文摘We prove that if φ is a homogeneous harmonic map from a Riemann surface M into a complex Grassmann manifold G（k, n）, then the maps of the harmonic sequences generated by φ are all homogeneous.
文摘We derive the Hessian estimate of the w-function defined on the pseudo-Grassmann manifold Gnmm, which is convex by the estimate. As an application, we give a direct proof of the Bernstein-type theorem due to Y. Xin [Mannscripta Math., 2000, 103： 191-202]. We also estimate the squared norm of the second fundamental form of a complete spacelike submanifold in pseudoEuclidean space in terms of the w-function and the mean curvature.
文摘A generalized Frenet formula for holomorphic 2 spheres S 2 in Grassmann manifolds G(k,n) is introduced and a generalized Plücker formula is achieved. By use of these formulae some pinching theorems for Gaussian curvature are obtained which generalize the corresponding results about holomorphic and minimal S 2 in CP n-1 . Einsteinian holomorphic subbundle of the trivial bundle S 2×C n is investigated.
文摘In the complex Grassmann manifold F(m,n), the space of complex \%n\%\|planes passes through the origin of C \%m+n\% ; the local coordinate of the space can be arranged into an \%m×n\% matrix \%Z.\% It is proved thatK=K(Z,\%d\%Z)=(I+ZZ ) -12 (I+ZZ ) 12 -(I+ZZ ) 12 ·(I+ZZ ) -12 is a \%U(m)\%\|connection of F\%(m,n)\% and its curvature formΩ\-1= d K+K∧Ksatisfies the Yang\|Mills equation. Moreover,B=B(Z, d Z)=K(Z, d Z)- tr (K(Z, d Z))mI (m) is an SU(m) \|connection and its curvature formΩ\-2= d B+B∧Bsatisfies the Yang\|Mills equation.
文摘1 Introduction Let G<sub>m,n</sub> be the Grassmann manifold of the unoriented, m-dimensional vector subspaces of the vector space R<sup>m+n</sup>. Then G<sub>m,n</sub> is a closed, smooth manifold of dimension mn. G<sub>m,n</sub>≌G<sub>n,m</sub>, and G<sub>1,n</sub>≌G<sub>n,1</sub> is just the real projective space RP<sup>n</sup>. For convenience, we assume that m≤n. Suppose M<sup>p</sup>R<sup>q</sup> means that the p-dimensional manifold M<sup>p</sup> is immersed in q-
基金supported by the NSFC (11071248, 11071249)supported by the Fundamental Research Funds for the Central Universities(USTC)
文摘In this paper, we construct a class of homogeneous minimal 2-spheres in complex Grassmann manifolds by applying the irreducible unitary representations of SU （2）. Furthermore, we compute induced metrics, Gaussian curvatures, Khler angles and the square lengths of the second fundamental forms of these homogeneous minimal 2-spheres in G（2, n ＋ 1） by making use of Veronese sequence.
