Riemannian Foliations

Molino, Pierre

doi:10.1007/978-1-4684-8670-4

Cited by 432 publications

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“…Furthermore [13], if J is given by (2.5), we get 10) which yields the classical tensors of a generalized Hermitian structure:…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover, if the structure of N is integrable, i.e., J ± are integrable and (2.14) holds, the projected structure has the same properties. Indeed, the Lie bracket of projectable vector fields is projectable to the corresponding Lie bracket and the Levi-Civita connection of G projects to that of the projected metric (for details, a text on foliations and metrics, e.g., [10], should be consulted). The conclusion is that a submersion sends a projectable generalized Kähler structure to a generalized Kähler structure.…”

Section: Submanifolds With Induced Generalized Kähler Structurementioning

confidence: 99%

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A note on submanifolds and mappings in generalized complex geometry

Vaisman

2015

Monatsh Math

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A note on submanifolds and mappings in generalized complex geometry by Izu VaismanABSTRACT. In generalized complex geometry, we revisit linear subspaces and submanifolds that have an induced generalized complex structure. We give an expression of the induced structure that allows us to deduce a smoothness criterion, we dualize the results to submersions and we make a few comments on generalized complex mappings. Then, we discuss submanifolds of generalized Kähler manifolds that have an induced generalized Kähler structure. These turn out to be the common invariant submanifolds of the two classical complex structures of the generalized Kähler manifold. IntroductionInduced generalized complex structures of submanifolds were introduced and studied in [3] and the study was continued in [2,6,12], etc. After a preliminary Section 2, where we recall known results, in Section 3, we give a general expression of the induced structure and derive a smoothness criterion. The results are dualized to submersions and a few remarks on more general mappings are made, including a proposed, new definition of generalized complex mappings. In Section 4, we consider the generalized Hermitian and generalized Kähler case, which is known to be given by a 1-1 correspondence with quadruples (γ, ψ, J ± ), where γ is a metric, ψ is a 2-form and J ± are classical γ-compatible complex structures. Then, we define the notion of induced structure in a way that is compatible with the generalized complex case and is such that existence of the induced structure is equivalent to J ± -invariance. We work in the C ∞ -category and use the classical notation of Differential Geometry [8].* 2000 Mathematics Subject Classification: 53C56 .

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“…Furthermore [13], if J is given by (2.5), we get 10) which yields the classical tensors of a generalized Hermitian structure:…”

Section: Preliminariesmentioning

confidence: 99%

Section: Submanifolds With Induced Generalized Kähler Structurementioning

confidence: 99%

A note on submanifolds and mappings in generalized complex geometry

Vaisman

2015

Monatsh Math

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“…A surface foliation without sphere or projective plane leaves is typical iff the fundamental group π 1 (λ) of each leaf injects into the holonomy groupoid of F . For a surface foliation of a 3-manifold, the holonomy groupoid Γ is a groupoid of germs of homeomorphisms of R. If F is C ∞ , the holonomy groupoid Γ is a groupoid of germs of diffeomorphisms of R. For a more leisurely discussion of holonomy groupoids, see [9]. Here BΓ is the classifying space of the groupoid Γ.…”

Section: Typical Foliationsmentioning

confidence: 99%

Every orientable 3–manifold is a BΓ

Calegari

2002

Algebr. Geom. Topol.

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We show that every orientable 3-manifold is a classifying space BΓ where Γ is a groupoid of germs of homeomorphisms of R. This follows by showing that every orientable 3-manifold M admits a codimension one foliation F such that the holonomy cover of every leaf is contractible. The F we construct can be taken to be C 1 but not C 2 . The existence of such an F answers positively a question posed by Tsuboi (see [21]), but leaves open the question of whether M = BΓ for some C ∞ groupoid Γ.

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“…Além de de nir tais objetos, serão generalizadas algumas propriedades das folheações Riemannianas singulares para o contexto Finsleriano, obtendo a estrutura in nitesimal. Como referência para o estudo das folheações Riemannianas singulares veja [1], [3], [5], [6], [21].…”

Section: Capítulo 2 Folheações Finslerianas Singulares De Nições E Punclassified

“…Apesar destas di culdades, serão generalizados: o resultado que garante a existência de vizinhanças tubulares para algumas subvariedades mergulhadas (Teorema 1.13.1), a caracterização das folheações Finslerianas singulares via estruturas tubulares (Proposição 2.6.1), o Lema de Homotetia (Proposição 2.6.2), algumas propriedades de estrati cação e o Teorema do Slice (Teorema 2.0.2). Para o estudo de tais resultados no contexto Riemanniano consulte [1], [3], [5], [21].…”

Section: Introductionunclassified

Sobre folheações Finslerianas singulares

Alves¹

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Riemannian Foliations

Cited by 432 publications

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A note on submanifolds and mappings in generalized complex geometry

A note on submanifolds and mappings in generalized complex geometry

Every orientable 3–manifold is a BΓ

Sobre folheações Finslerianas singulares

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