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
Let M be a concircularly fiat totally real minimal submanifold in CP4. The infimum Vm of the volume V （M） of M is obtained, also the necessary and sufficient conditions of ＂V（M）=Vm＂ is given.
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
In this paper, we discuss the relations between the 2-harmornic totally real submsnifold and the minimal totall real submanifold in the complex protective spsace, and obtain the pinching conductions for the second fu...In this paper, we discuss the relations between the 2-harmornic totally real submsnifold and the minimal totall real submanifold in the complex protective spsace, and obtain the pinching conductions for the second fundamental form and the Rieci curature of the 2-harmornic totally real submanifold in the complex projective space.展开更多
In terms of the almost complex affine connection and moving unitary frames, all totally rael minimal immersions from R-2 into the nearly Kahler S-6 axe determine explicitly. Moreover, the complete flat almost complex ...In terms of the almost complex affine connection and moving unitary frames, all totally rael minimal immersions from R-2 into the nearly Kahler S-6 axe determine explicitly. Moreover, the complete flat almost complex curves in the nearly Kahler S-6 are determined completely.展开更多
Our purpose is to study the minimal tori in the hyperquadric Q2. Firstly, we obtain a necessary and sufficient condition for the minimal surface in Qn which is also minimal in CPn＋1. Next, we show that this kind of m...Our purpose is to study the minimal tori in the hyperquadric Q2. Firstly, we obtain a necessary and sufficient condition for the minimal surface in Qn which is also minimal in CPn＋1. Next, we show that this kind of minimal surface （neither holomorphic nor anti-holomorphic） with constant curvature in Q2 is part of a flat totally real torus. Finally, we prove that totally real minimal fiat tori in Q2 must be totally geodesic, and we classify all the totally geodesic closed surfaces in Q2.展开更多
Let HPn be the quaternionic projective space with constant quaternionic sectional curvature 4. Then locally there exists a tripe {I, J, K} of complex structures on HPn satisfying U = -JI = K,JK = -KJ = /, KI = -IK = J...Let HPn be the quaternionic projective space with constant quaternionic sectional curvature 4. Then locally there exists a tripe {I, J, K} of complex structures on HPn satisfying U = -JI = K,JK = -KJ = /, KI = -IK = J. A surface M(?) HPn is called totally real, if at each point p ∈M the tangent plane TPM is perpendicular to I(TPM), J(TPM) and K(TPM). It is known that any surface M(?)RPn(?) HPn is totally real, where RPn (?) HPn is the standard embedding of real projective space in HPn induced by the inclusion R in H, and that there are totally real surfaces in HPn which don't come from this way. In this paper we show that any totally real minimal 2-sphere in HPn is isometric to a full minimal 2-sphere in Rp2m (?) RPn(?) HPn with 2m≤n. As a consequence we show that the Veronese sequences in KP2m (m≥1) are the only totally real minimal 2-spheres with constant curvature in the quaternionic projective space.展开更多
Given a harmonic map φfrom a Riemann surface M into the complex projectivespace CP^n, by using the -transform associated to the map φ, Chern et al. obtained the se-quence of harmonic maps. We call it a harmonic sequ...Given a harmonic map φfrom a Riemann surface M into the complex projectivespace CP^n, by using the -transform associated to the map φ, Chern et al. obtained the se-quence of harmonic maps. We call it a harmonic sequence. We will say that a harmonicmap φ: M→CP^n is k-orthogonal if k consecutive maps in the harmonic sequence of φ aremutually orthogonal. In particular, φis conformal if and only if φ is 3-orthogonal and, inall cases, φ is at most (n+1)-orthogonal. There are two possible ways in which the展开更多
Let CP^n be a complex projective n-space with the Fubini-Study metric of constantholomorphic sectional curvature c,and M be an n-dimensional compact totally real minimalsubmanifold in CP^n. It is known from refs. [1-3...Let CP^n be a complex projective n-space with the Fubini-Study metric of constantholomorphic sectional curvature c,and M be an n-dimensional compact totally real minimalsubmanifold in CP^n. It is known from refs. [1-3] that if the scalar curvature ρ≥n^2(n-2)c/2(2n-1) for M, then M is either totally geodesic in CP^n or n=2 and ρ=0, and M is a finiteRiemannian covering of the unique flat torus minimally imbedded in CP^2 with the展开更多
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 NSF of Education Department of Henan Province(20021100002)Supported by the NSF of Education Department of Henan Province(200510475038)
文摘Let M be a concircularly fiat totally real minimal submanifold in CP4. The infimum Vm of the volume V （M） of M is obtained, also the necessary and sufficient conditions of ＂V（M）=Vm＂ is given.
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In this paper, we discuss the relations between the 2-harmornic totally real submsnifold and the minimal totall real submanifold in the complex protective spsace, and obtain the pinching conductions for the second fundamental form and the Rieci curature of the 2-harmornic totally real submanifold in the complex projective space.
基金Project supported by the National Natural Science Foundation of China (10271106)
文摘In terms of the almost complex affine connection and moving unitary frames, all totally rael minimal immersions from R-2 into the nearly Kahler S-6 axe determine explicitly. Moreover, the complete flat almost complex curves in the nearly Kahler S-6 are determined completely.
基金supported by National Natural Science Foundation of China(Grant Nos.11071248 and 11226079)Program of Natural Science Research of Jiangsu Higher Education Institutions of China(Grant No.12KJD110004)
文摘Our purpose is to study the minimal tori in the hyperquadric Q2. Firstly, we obtain a necessary and sufficient condition for the minimal surface in Qn which is also minimal in CPn＋1. Next, we show that this kind of minimal surface （neither holomorphic nor anti-holomorphic） with constant curvature in Q2 is part of a flat totally real torus. Finally, we prove that totally real minimal fiat tori in Q2 must be totally geodesic, and we classify all the totally geodesic closed surfaces in Q2.
基金Acknowledgements We would like to thank Mr.Ma Xiang for his helpful discussion.This work was supported by RFDP,Qiushi Award,973 ProjectJiechu Grant of NSFC.
文摘Let HPn be the quaternionic projective space with constant quaternionic sectional curvature 4. Then locally there exists a tripe {I, J, K} of complex structures on HPn satisfying U = -JI = K,JK = -KJ = /, KI = -IK = J. A surface M(?) HPn is called totally real, if at each point p ∈M the tangent plane TPM is perpendicular to I(TPM), J(TPM) and K(TPM). It is known that any surface M(?)RPn(?) HPn is totally real, where RPn (?) HPn is the standard embedding of real projective space in HPn induced by the inclusion R in H, and that there are totally real surfaces in HPn which don't come from this way. In this paper we show that any totally real minimal 2-sphere in HPn is isometric to a full minimal 2-sphere in Rp2m (?) RPn(?) HPn with 2m≤n. As a consequence we show that the Veronese sequences in KP2m (m≥1) are the only totally real minimal 2-spheres with constant curvature in the quaternionic projective space.
文摘Given a harmonic map φfrom a Riemann surface M into the complex projectivespace CP^n, by using the -transform associated to the map φ, Chern et al. obtained the se-quence of harmonic maps. We call it a harmonic sequence. We will say that a harmonicmap φ: M→CP^n is k-orthogonal if k consecutive maps in the harmonic sequence of φ aremutually orthogonal. In particular, φis conformal if and only if φ is 3-orthogonal and, inall cases, φ is at most (n+1)-orthogonal. There are two possible ways in which the
基金Project supported by the National Natural Science Foundation of China and the Science Foundation of Zhejiang Province.
文摘Let CP^n be a complex projective n-space with the Fubini-Study metric of constantholomorphic sectional curvature c,and M be an n-dimensional compact totally real minimalsubmanifold in CP^n. It is known from refs. [1-3] that if the scalar curvature ρ≥n^2(n-2)c/2(2n-1) for M, then M is either totally geodesic in CP^n or n=2 and ρ=0, and M is a finiteRiemannian covering of the unique flat torus minimally imbedded in CP^2 with the
