Riemannian foliations on simply connected manifolds and actions of tori on orbifolds

Haefliger, André; Salem, Eliane

doi:10.1215/ijm/1255988064

Cited by 10 publications

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“…Proof of Theorem B. To deform any Killing foliation F with compact leaf closures into a closed foliation, whilst maintaining some properties of its transverse geometry, we will use the following theorem by A. Haefliger and E. Salem [22,Theorem 3.4]. Suppose that M is compact and F is a Killing foliation of M. We can deform F using this result as follows.…”

Section: Deformations Of Killing Foliationsmentioning

confidence: 99%

“…Moreover, since there is a surjective homomorphism π 1 M → π 1 H F (see [37, Section 1.11]), we have that H F is 1-connected. We can now apply [22,Theorem 3.7] to conclude that there exists a Riemannian foliation F ′ of a simply-connected compact manifold M ′ with (T ′ , H F ′ ) equivalent to ( T , H F ). In particular, F ′ is Killing and also satisfies sec F ′ ≥ c > 0, since it is endowed with the transverse metric (g T ) ′ on T ′ induced by ρ * (g T ) via the equivalence.…”

Section: A Closed Leaf Theoremmentioning

confidence: 99%

“…The proof of Theorem B is based on a result by A. Haefliger and E. Salem [22] that expresses F as the pullback of a homogeneous foliation of an orbifold. As Theorem A exemplifies, with this technique we are able to reduce results about the transverse geometry of Killing foliations to theorems about Riemannian orbifolds.…”

Section: Introductionmentioning

confidence: 99%

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Positively curved Killing foliations via deformations

Caramello

Töben

2019

Trans. Amer. Math. Soc.

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We show that a compact manifold admitting a Killing foliation with positive transverse curvature fibers over finite quotients of spheres or weighted complex projective spaces, provided that the singular foliation defined by the closures of the leaves has maximal dimension. This result is obtained by deforming the foliation into a closed one while maintaining transverse geometric properties, which allows us to apply results from the Riemannian geometry of orbifolds to the space of leaves. We also show that the basic Euler characteristic is preserved by such deformations. Using this fact we prove that a Riemannian foliation of a compact manifold with finite fundamental group and nonvanishing Euler characteristic is closed. As another application we obtain that, for a positively curved Killing foliation of a compact manifold, if the structural algebra has sufficiently large dimension then the basic Euler characteristic is positive.

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Section: Deformations Of Killing Foliationsmentioning

confidence: 99%

Section: A Closed Leaf Theoremmentioning

confidence: 99%

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Positively curved Killing foliations via deformations

Caramello

Töben

2019

Trans. Amer. Math. Soc.

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“…Transverse structure of Killing foliations. The transverse structure of a Killing foliation coincides with that of an (Abelian) homogeneous foliation on an orbifold, as established by A. Haefliger and E. Salem in [36]. More precisely, by comparing the local models of the transverse structure of a Killing foliation on a neighborhood of a leaf closure and the local model of an orbit of a torus action on an orbifold, the authors obtain the following.…”

Section: 1mentioning

confidence: 55%

Leaf closures of Riemannian foliations: a survey on topological and geometric aspects of Killing foliations

Alexandrino¹,

Caramello²

2020

Preprint

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A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply connected manifold, or more generally of a Killing foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the main objects appearing in this theory, such as pseudogroups, holonomy and basic cohomology. We then review Molino's structural theory for Riemannian foliations and present its transverse counterpart in the theory of complete pseudogroups of isometries, emphasizing the connections between these topics. We also survey some classical results and recent developments in the theory of Killing foliations. Finally, we review some topics in the theory of singular Riemannian foliations and discuss singular Killing foliations, also proposing a new approach to them via holonomy metric pseudogroups and the theory of blow-ups, which possibly opens up a new area of interest. Contents2010 Mathematics Subject Classification. 53C12.

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“…2 of [11] as a general reference. Other sources of background material on orbifolds are [20][21][22]25,34]. But note that the notation differs slightly from source to source.…”

Section: Gkm Orbifoldsmentioning

confidence: 99%

Non-negatively curved GKM orbifolds

Goertsches

Wiemeler²

2021

Math. Z.

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In this paper we study non-negatively curved and rationally elliptic GKM$$_4$$ 4 manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifolds. Moreover, we give a simplified proof of a characterisation of products of simplices among orbit spaces of locally standard torus manifolds. This characterisation was originally proved in Wiemeler (J Lond Math Soc 91(3): 667–692, 2015) and was used there to obtain a classification of non-negatively curved torus manifolds.

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Riemannian foliations on simply connected manifolds and actions of tori on orbifolds

Cited by 10 publications

References 0 publications

Positively curved Killing foliations via deformations

Positively curved Killing foliations via deformations

Leaf closures of Riemannian foliations: a survey on topological and geometric aspects of Killing foliations

Non-negatively curved GKM orbifolds

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