Homogeneous Codimension One Foliations on Noncompact Symmetric Spaces

Berndt, Jürgen; Tamaru, Hiroshi

doi:10.4310/jdg/1080835656

Cited by 48 publications

(67 citation statements)

References 18 publications

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“…Cohomogeneity one actions with no singular orbits, i.e. giving rise to homogeneous regular Riemannian foliations, were classified in [16]; they correspond to case (1) in Theorem 5.2. Note that they are induced by subgroups of AN .…”

Section: 2mentioning

confidence: 99%

Submanifold geometry in symmetric spaces of noncompact type

Díaz-Ramos

Dominguez-Vazquez

Sanmartín-López

2019

São Paulo J. Math. Sci.

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Section: 2mentioning

confidence: 99%

Submanifold geometry in symmetric spaces of noncompact type

Díaz-Ramos

Dominguez-Vazquez

Sanmartín-López

2019

São Paulo J. Math. Sci.

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“…In [4] it was shown that there exist only two such actions without a singular orbit. The first one is given by the action of the nilpotent group N in an Iwasawa decomposition G = KAN of G = I o (FH n ), and the orbits form a horosphere foliation.…”

Section: The Classificationmentioning

confidence: 99%

“…In [4] we obtained the classification, up to orbit equivalence, of all cohomogeneity one actions on irreducible symmetric spaces of noncompact type that induce a Riemannian foliation, that is, have no singular orbit. A surprising consequence of this result is that the moduli space of all such actions just depends on the rank of the symmetric space and possible duality or triality principles on the space.…”

Section: Introductionmentioning

confidence: 99%

Cohomogeneity one actions on noncompact symmetric spaces of rank one

Berndt

Tamaru

2007

Trans. Amer. Math. Soc.

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Abstract. We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces CH n , n ≥ 3. For the quaternionic hyperbolic spaces HH n , n ≥ 3, we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved byÉlie Cartan.

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“…By imitating the proof of Lemma 5.1 of [3], it is shown that a ′ is a maximal abelian subspace of p ′ because the S ′ -action has flat section. There exists g ∈ G satisfying Ad(g)(f a, where we note that…”

Section: Complex Equifocal Submanifoldsmentioning

confidence: 99%

“…Next we give some examples of a complex hyperpolar action without singular orbit as the free actions of solvable groups contained in AN (see Examples 1 and 2 of Section 3), which contain examples of cohomogeneity one actions without singular orbit constructed by J. Berndt and H. Tamaru [3] as special cases (see also [1]). Among these examples, we find complex hyperpolar actions all of whose orbits are non-proper complex equifocal or non-curvature-adapted.…”

Section: All Isoparametric Submanifolds With Flat Section In a Symmetmentioning

confidence: 99%

Examples of a complex hyperpolar action without singular orbit

Koike¹

2010

Cubo

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The notion of a complex hyperpolar action on a symmetric space of non-compact type has recently been introduced as a counterpart to the hyperpolar action on a symmetric space of compact type. As examples of a complex hyperpolar action, we have Hermann type actions, which admit a totally geodesic singular orbit (or a fixed point) except for one example. All principal orbits of Hermann type actions are curvature-adapted and proper complex equifocal. In this paper, we give some examples of a complex hyperpolar action without singular orbit as solvable group free actions and find complex hyperpolar actions all of whose orbits are non-curvature-adapted or non-proper complex equifocal among the examples. Also, we show that some of the examples possess the only minimal orbit. RESUMENLa noción de una acción hiperpolar compleja sobre un espacio simétrico de tipo no compacto fue recientemente introducida como el análogo de la acción hiperpolar sobre un espacio simétrico de tipo compacto. Como ejemplos de una acción hiperpolar complejas, nosotros tenemos acciones de tipo Hermann, las cuales admiten una orbita (o un punto fijo) singular totalmente geodesica excepto para un ejemplo. Todas las orbitas principales de acciones de tipo Hermann son curvatura-adaptadas y unifocales complejas propias. En 12, 2 (2010) este artículo, nosotros damos algunos ejemplos de una acción hiperpolar compleja sin orbitas singulares como grupo soluble de acciones libres y encontramos acciones complejas hiperpolares cuyas orbitas son no curvatura-adaptadas o no propias unifocales complejas. También, mostramos que algunos de los ejemplos poseen solamente orbitas minimales.

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Homogeneous Codimension One Foliations on Noncompact Symmetric Spaces

Cited by 48 publications

References 18 publications

Submanifold geometry in symmetric spaces of noncompact type

Submanifold geometry in symmetric spaces of noncompact type

Cohomogeneity one actions on noncompact symmetric spaces of rank one

Examples of a complex hyperpolar action without singular orbit

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