In this paper we study the cohomogeneity one de Sitter space S1 n. We consider the actions in both proper and non-proper cases. In the first case we characterize the acting groups and orbits and we prove that the orbi...In this paper we study the cohomogeneity one de Sitter space S1 n. We consider the actions in both proper and non-proper cases. In the first case we characterize the acting groups and orbits and we prove that the orbit space is homeomorphic to R. In the latter case we determine the groups and consequently the orbits in some different cases and prove that the orbit space is not Hausdorff.展开更多
In this paper, we study flat Riemannian manifolds which have codimension two orbits, under the action of a closed and connected Lie group G of isometries. We assume that G has fixed points, then characterize M and orb...In this paper, we study flat Riemannian manifolds which have codimension two orbits, under the action of a closed and connected Lie group G of isometries. We assume that G has fixed points, then characterize M and orbits of M.展开更多
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
In this paper,we study a class of Finsler metrics of cohomogeneity two on R×R~n.They are called weakly orthogonally invariant Finsler metrics.These metrics not only contain spherically symmetric Finsler metrics a...In this paper,we study a class of Finsler metrics of cohomogeneity two on R×R~n.They are called weakly orthogonally invariant Finsler metrics.These metrics not only contain spherically symmetric Finsler metrics and Marcal-Shen's warped product metrics but also partly contain another"warping"introduced by Chen-Shen-Zhao.We obtain differential equations that characterize weakly orthogonally invariant Finsler metrics with vanishing Douglas curvature,and therefore we provide a unifying frame work for Douglas equations due to Liu-Mo,Mo-Solórzano-Tenenblat and Solórzano.As an application,we obtain a lot of new examples of weakly orthogonally invariant Douglas metrics.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 Iranian Presidential Office (Grant No. 83211)
文摘In this paper we study the cohomogeneity one de Sitter space S1 n. We consider the actions in both proper and non-proper cases. In the first case we characterize the acting groups and orbits and we prove that the orbit space is homeomorphic to R. In the latter case we determine the groups and consequently the orbits in some different cases and prove that the orbit space is not Hausdorff.
文摘In this paper, we study flat Riemannian manifolds which have codimension two orbits, under the action of a closed and connected Lie group G of isometries. We assume that G has fixed points, then characterize M and orbits of M.
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 the National Natural Science Foundation of China(Grant Nos.12101022,12071228,12171005,11771020)the Science and Technology Project of Beijing Municipal Education Commission(Grant No.KM202010005026)。
文摘In this paper,we study a class of Finsler metrics of cohomogeneity two on R×R~n.They are called weakly orthogonally invariant Finsler metrics.These metrics not only contain spherically symmetric Finsler metrics and Marcal-Shen's warped product metrics but also partly contain another"warping"introduced by Chen-Shen-Zhao.We obtain differential equations that characterize weakly orthogonally invariant Finsler metrics with vanishing Douglas curvature,and therefore we provide a unifying frame work for Douglas equations due to Liu-Mo,Mo-Solórzano-Tenenblat and Solórzano.As an application,we obtain a lot of new examples of weakly orthogonally invariant Douglas metrics.
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.
