Foliated and associated geometric structures on foliated manifolds

Wolak, Robert

doi:10.5802/afst.681

Cited by 30 publications

(43 citation statements)

References 6 publications

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“…Just as N (see above), P r (or better P r N ) is a functor between the categories F M q and F B. Let us mention that both functors are (prototypes of) foliated natural functors in the sense of [Wol89].…”

Section: Foliated Frame Bundlesmentioning

confidence: 99%

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A first approximation for quantization of singular spaces

Poncin

Radoux

Wolak

2009

Journal of Geometry and Physics

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Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces obtained as leaf closure spaces of regular Riemannian foliations on compact manifolds. These contain the orbit spaces of compact group actions and orbifolds. Our method uses foliation theory as a desingularization technique for such singular spaces. A quantization procedure on the orbit space of the symmetry group -that commutes with reduction -can be obtained from constructions which combine different geometries associated with foliations and new techniques originated in Equivariant Quantization. The present paper contains the first of two steps needed to achieve these just detailed goals. (2000) : 53D50, 53C12, 53B10, 53D20 Mathematics Subject Classification

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Section: Foliated Frame Bundlesmentioning

confidence: 99%

“…Hence, we refrain for instance to give a general description of adapted natural functors, see [Wol89], but provide examples of such functors.…”

Section: Adapted Differential Operators and Symbolsmentioning

confidence: 99%

A first approximation for quantization of singular spaces

Poncin

Radoux

Wolak

2009

Journal of Geometry and Physics

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“…We will only sketch this correspondence in the scope needed for this paper. A more general and exhaustive approach can be found, for example, in [22]. Eventually we shall see that Theorem 1 is a corollary of the following theorem.…”

Section: Introductionmentioning

confidence: 99%

A Remark on the Brylinski Conjecture For orbifolds

Bąk

Czarnecki

2011

J. Aust. Math. Soc.

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The paper presents a proof of the Brylinski conjecture for compact Kähler orbifolds. The result is a corollary of the foliated version of the Mathieu theorem on symplectic harmonic representations of de Rham cohomology classes. The proofs are based on the idea of representing an orbifold as the leaf space of a Riemannian foliation and on the correspondence between foliated and holonomy invariant objects for foliated manifolds.2010 Mathematics subject classification: primary 53D05; secondary 53C12, 57R18.

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“…[16], and if the connection D is foliated so the section R D . If the connection D is foliated the (foliated) curvature tensor field R D corresponds to the curvature tensor field of the induced connectionD on the transverse manifold.…”

Section: If (Mmentioning

confidence: 99%

Transversely Hessian foliations and information geometry

Boyom¹,

Wolak²

2015

AIP Conference Proceedings

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A family of probability distributions parametrized by an open domain Λ in R n defines the Fisher information matrix on this domain which is positive semi-definite. In information geometry the standard assumption has been that the Fisher information matrix tensor is positive definite defining in this way a Riemannian metric on Λ. If we replace the "positive definite" assumption by the existence of a suitable torsion-free connection, a foliation with a transversely Hessian structure appears naturally. In the paper we develop the study of transversely Hessian foliations in view of applications in information geometry.2011 Mathematics Subject Classification. 53C12

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Foliated and associated geometric structures on foliated manifolds

Cited by 30 publications

References 6 publications

A first approximation for quantization of singular spaces

A first approximation for quantization of singular spaces

A Remark on the Brylinski Conjecture For orbifolds

Transversely Hessian foliations and information geometry

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