给出了 Dupin 超曲面的基本公式,由此研究了 Dupin 超曲面成为等参超曲面的条件。设 M 是(C)的具常中曲率 H 的 Dupin 超曲面,λ_1<λ_2<…<λ_g 是 M 所有不同的主曲率,如果λ_η为常数(η≥3)且 C+λ_1λ_η≥0,或者,M具...给出了 Dupin 超曲面的基本公式,由此研究了 Dupin 超曲面成为等参超曲面的条件。设 M 是(C)的具常中曲率 H 的 Dupin 超曲面,λ_1<λ_2<…<λ_g 是 M 所有不同的主曲率,如果λ_η为常数(η≥3)且 C+λ_1λ_η≥0,或者,M具有三个不同主曲率,且λ_1为常数(λ_1≠H)则 M 是等参超曲面。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
文摘给出了 Dupin 超曲面的基本公式,由此研究了 Dupin 超曲面成为等参超曲面的条件。设 M 是(C)的具常中曲率 H 的 Dupin 超曲面,λ_1<λ_2<…<λ_g 是 M 所有不同的主曲率,如果λ_η为常数(η≥3)且 C+λ_1λ_η≥0,或者,M具有三个不同主曲率,且λ_1为常数(λ_1≠H)则 M 是等参超曲面。
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.
