Inhomogeneous spaces of positive curvature. (English) Zbl 0778.53033
The author analyses the geometry and the topology of a compact simply- connected positively-curved Riemannian 6-manifold \(F'\) which is closely related to the flag manifold \(F\) over \(\mathbb{C} P^ 2\), and of an infinite series of simply connected circle bundles over \(F'\) (having also positive sectional curvature). In this context he shows that all these spaces are biquotients of the Lie group \(SU(3)\), but are not homeomorphic to a homogeneous space of positive curvature.
Reviewer: C.Udrişte (Bucureşti)
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 55R25 | Sphere bundles and vector bundles in algebraic topology |
References:
| [1] | Aloff, S.; Walach, N. L., An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc., 81, 93-97 (1975) · Zbl 0362.53033 |
| [2] | Berard Bergery, L., Les variétés Riemanniennes homogènes simplement connexes de dimension impair à courbure strictement positive, J. Math. Pures Appl., 55, 47-68 (1976) · Zbl 0289.53037 |
| [3] | Berger, M., Les variétés Riemanniennes homogenès normales simplement connexes à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa, 15, 179-246 (1961) · Zbl 0101.14201 |
| [4] | Bott, R.; Tu, L. W., Differential Forms in Algebraic Topology (1982), Springer: Springer Berlin · Zbl 0496.55001 |
| [5] | Cheeger, J.; Ebin, D. G., Comparison Theorems in Riemannian Geometry (1975), North Holland: North Holland Amsterdam · Zbl 0309.53035 |
| [6] | Eliasson, H. I., Die Krümung des Raumes Sp(2)/SU(2) von Berger, Math. Ann., 164, 317-323 (1966) · Zbl 0141.19401 |
| [7] | Eschenburg, J. H., New examples of manifolds with strictly positive curvature, Invent. Math., 66, 469-480 (1982) · Zbl 0484.53031 |
| [8] | Eschenburg, J. H., Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümten Orbiträumen, Schr. Math. Inst. Univ. Münster, 32, 2 (1984) · Zbl 0551.53024 |
| [9] | J.H. Eschenburg, Cohomology of biquotients, Manuscripta Math., to appear.; J.H. Eschenburg, Cohomology of biquotients, Manuscripta Math., to appear. · Zbl 0769.53029 |
| [10] | Gromoll, D.; Meyer, W. T., An exotic sphere with nonnegative sectional curvature, Ann. of Math., 100, 401-406 (1974) · Zbl 0293.53015 |
| [11] | E. Heintze, The curvature of \(SU Sp S^1\); E. Heintze, The curvature of \(SU Sp S^1\) · Zbl 0252.53042 |
| [12] | O’Neill, B., The fundamental equations of a submersion, Michigan Math. J., 23, 459-469 (1966) · Zbl 0145.18602 |
| [13] | Samelson, H., On curvature and characteristic of homogeneous spaces, Michigan Math. J., 5, 13-18 (1958) · Zbl 0084.37407 |
| [14] | Walach, N. L., Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math., 96, 277-295 (1972) · Zbl 0261.53033 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
