Classification of compact transformation groups on cohomology complex projective spaces with codimension one orbits

Uchida, Fuichi

doi:10.4099/math1924.3.141

Cited by 32 publications

(24 citation statements)

References 12 publications

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“…Hence, together with the fact from transformation group theory (see, e.g., [Br,Theorem 8.2 in Chap. IV] or [Uc,Lemma 1.2.1]), we obtain the following lemma:…”

Section: Torus Manifoldmentioning

confidence: 99%

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Classification of torus manifolds with codimension one extended actions

Kuroki

2011

Transformation Groups

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The purpose of this paper is to classify torus manifolds (M 2n , T n ) with codimension one extended G-actions (M 2n , G) up to essential isomorphism, where G is a compact, connected Lie group whose maximal torus is T n . For technical reasons, we do not assume torus manifolds are orientable. We prove that there are seven types of such manifolds. As a corollary, if a nonsingular toric variety or a quasitoric manifold has a codimension one extended action then such manifold is a complex projective bundle over a product of complex projective spaces.

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“…Hence, together with the fact from transformation group theory (see, e.g., [Br,Theorem 8.2 in Chap. IV] or [Uc,Lemma 1.2.1]), we obtain the following lemma:…”

Section: Torus Manifoldmentioning

confidence: 99%

“…In order to check whether M (f ) and M (f ) are equivariantly diffeomorphic, the following lemma is useful (see [Uc,Lemma 5.3.1]).…”

Section: Attaching Mapsmentioning

confidence: 99%

Classification of torus manifolds with codimension one extended actions

Kuroki

2011

Transformation Groups

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“…In this section we will list all isometric cohomogeneity one actions on compact simply connected symmetric spaces of dimension seven or less. Hsiang and Lawson [1971] classified cohomogeneity one actions on symmetric spheres in (see [Straume 1996] for correction) and later Uchida [1977] did the same for complex projective spaces. Kollross [2002] generalized these results to a classification of cohomogeneity one actions on irreducible symmetric spaces of compact type.…”

Section: Identifying Some Actionsmentioning

confidence: 99%

Classification of cohomogeneity one manifolds in low dimensions

Hoelscher¹

2010

Pacific J. Math.

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A cohomogeneity one manifold is a manifold whose quotient by the action of a compact Lie group is one-dimensional. Such manifolds are of interest in Riemannian geometry in the context of nonnegative sectional curvature, as well as in other areas of geometry and in physics. We classify compact simply connected cohomogeneity one manifolds in dimensions 5, 6 and 7. We also show that all such manifolds admit metrics of nonnegative sectional curvature, with the possible exception of two families of manifolds.

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“…However in many cases, e.g. if N (K) is connected, any glueing map can be extended to an equivariant diffeomorphism of at least one of the two disk bundles, and so all maps give the same manifold up to equivariant diffeomorphism (see [21] for details).…”

Section: Cohomogeneity Onementioning

confidence: 99%

Cohomogeneity One Einstein-Sasaki 5-Manifolds

Conti

2007

Commun. Math. Phys.

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Abstract. We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use these equations to classify Einstein-Sasaki 5-manifolds of cohomogeneity one.

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Classification of compact transformation groups on cohomology complex projective spaces with codimension one orbits

Cited by 32 publications

References 12 publications

Classification of torus manifolds with codimension one extended actions

Classification of torus manifolds with codimension one extended actions

Classification of cohomogeneity one manifolds in low dimensions

Cohomogeneity One Einstein-Sasaki 5-Manifolds

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