In this paper,we revisit the Kahler structures on the affine quadrics M1={z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=1}in the paper by Bo Yang and Fang-Yang Zheng.We found that the Kahler structures on the complex surface are more...In this paper,we revisit the Kahler structures on the affine quadrics M1={z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=1}in the paper by Bo Yang and Fang-Yang Zheng.We found that the Kahler structures on the complex surface are more complicated than what they have thought.We shall also give some detail calculations and found that our results fit quite well with earlier papers of the first author,one of them with X.X.Chen.展开更多
Let E be a simply connected rational homology sphere. A pair of disjoint closed submanifolds M＋, M_ C ＋ are called dual to each other if the complement ~ - M＋ strongly homotopy retracts onto M- or vice-versa. In th...Let E be a simply connected rational homology sphere. A pair of disjoint closed submanifolds M＋, M_ C ＋ are called dual to each other if the complement ~ - M＋ strongly homotopy retracts onto M- or vice-versa. In this paper, we are concerned with the basic problem of which integral triples （n; m＋, m-） E Na can appear, where n ： dime - 1 and m＋ = codimM~ - 1. The problem is motivated by several fundamental aspects in differential geometry. （i） The theory of isoparametric/I）upin hypersurfaces in the unit sphere Sn＋l initiated by ]~lie Cartan, where M=t= are the focal manifolds of the isoparametric/Dupin hypersurface M C Snq-1, and m~= coincide with the multiplicities of principal curvatures of M. （ii） The Grove-Ziller construction of non-negatively curved Riemannian metrics on the Milnor exotic spheres ~, i.e., total spaces of smooth S3-bundles over $4 homeomorphic but not diffeomorphic to S7, where M~ P~ ~so（4） $3, P -＋ $4 the principal SO（4）-bundle of ~ and P~ the singular orbits of a cohomogeneity one SO（4） ~ SO（3）-action on P which are both of codimension 2. Based on the important result of Grove-Halperin, we provide a surprisingly simple answer, namely, if and only if one of the following holds true：m＋ =m- =n; 1 {1, 2,4, 8}; m＋=m_=1/3n∈ {1,2}; m＋=m_=1/4n∈{1,2}; m＋=m_=1/6n∈{1,2}; n In addition, if E is a homotopy sphere and the ratio n/m＋m-2 （for simplicity let us assume 2 ≤ m_ 〈 m＋）, we observe that the work of Stolz on the multiplicities of isoparametric hypersurfaces applies almost identically to conclude that, the pair can be realized if and only if, either （m＋, m_） = （5, 4） or m＋ ＋ m- ＋ 1 is divisible by the integer 5（m_） （see the table on Page 1551）, which is equivalent to the existence of （m- - 1） linearly independent vector fields on the sphere Sin＋＋m- by Adams＇ celebrated work. In contrast, infinitely many counterexamples are given if E is a rational homology sphere.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.12171140).
文摘In this paper,we revisit the Kahler structures on the affine quadrics M1={z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=1}in the paper by Bo Yang and Fang-Yang Zheng.We found that the Kahler structures on the complex surface are more complicated than what they have thought.We shall also give some detail calculations and found that our results fit quite well with earlier papers of the first author,one of them with X.X.Chen.
基金supported by National Natural Science Foundation of China(Grant No.11431009)the Ministry of Education in China,and the Municipal Administration of Beijing
文摘Let E be a simply connected rational homology sphere. A pair of disjoint closed submanifolds M＋, M_ C ＋ are called dual to each other if the complement ~ - M＋ strongly homotopy retracts onto M- or vice-versa. In this paper, we are concerned with the basic problem of which integral triples （n; m＋, m-） E Na can appear, where n ： dime - 1 and m＋ = codimM~ - 1. The problem is motivated by several fundamental aspects in differential geometry. （i） The theory of isoparametric/I）upin hypersurfaces in the unit sphere Sn＋l initiated by ]~lie Cartan, where M=t= are the focal manifolds of the isoparametric/Dupin hypersurface M C Snq-1, and m~= coincide with the multiplicities of principal curvatures of M. （ii） The Grove-Ziller construction of non-negatively curved Riemannian metrics on the Milnor exotic spheres ~, i.e., total spaces of smooth S3-bundles over $4 homeomorphic but not diffeomorphic to S7, where M~ P~ ~so（4） $3, P -＋ $4 the principal SO（4）-bundle of ~ and P~ the singular orbits of a cohomogeneity one SO（4） ~ SO（3）-action on P which are both of codimension 2. Based on the important result of Grove-Halperin, we provide a surprisingly simple answer, namely, if and only if one of the following holds true：m＋ =m- =n; 1 {1, 2,4, 8}; m＋=m_=1/3n∈ {1,2}; m＋=m_=1/4n∈{1,2}; m＋=m_=1/6n∈{1,2}; n In addition, if E is a homotopy sphere and the ratio n/m＋m-2 （for simplicity let us assume 2 ≤ m_ 〈 m＋）, we observe that the work of Stolz on the multiplicities of isoparametric hypersurfaces applies almost identically to conclude that, the pair can be realized if and only if, either （m＋, m_） = （5, 4） or m＋ ＋ m- ＋ 1 is divisible by the integer 5（m_） （see the table on Page 1551）, which is equivalent to the existence of （m- - 1） linearly independent vector fields on the sphere Sin＋＋m- by Adams＇ celebrated work. In contrast, infinitely many counterexamples are given if E is a rational homology sphere.
