Index theory and non-commutative geometry on foliated manifolds

Kordyukov, Yuri A.

doi:10.1070/rm2009v064n02abeh004616

Cited by 13 publications

(10 citation statements)

References 172 publications

(287 reference statements)

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“…In the case D is a foliation, ∆ D is a longitudinally elliptic operator. We refer the reader to [15] for a survey of longitudinally elliptic operators on regular foliations and to [1,4] for the case of singular foliations.…”

Section: The Horizontal Laplacian Of a Distributionmentioning

confidence: 99%

“…Let us recall some necessary information on noncommutative geometry of regular foliations (for more information and details, see [14,15] and references therein).…”

Section: The Horizontal Laplacian As a Multipliermentioning

confidence: 99%

“…×T the corresponding coordinate chart on G [9] (see also [14,15] In the charts φ and φ ′ , the density µ is written as µ = µ(x, y)|dx||dy| and µ = µ ′ (x ′ , y ′ )|dx ′ ||dy ′ |, respectively, and the density α as α = α(x, y)|dx| and α = α ′ (x ′ , y ′ )|dx ′ |, respectively. Then δ ∈ C ∞ (U × U × T ) is given by (see [9,Proposition VIII.12])…”

Section: The Horizontal Laplacian As a Multipliermentioning

confidence: 99%

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Riemannian metrics and Laplacians for generalized smooth distributions

Androulidakis

Kordyukov

2019

J. Topol. Anal.

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In this paper, we discuss spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold. First, we give a survey of results on generalized smooth distributions on manifolds, Riemannian structures and associated Laplacians. Then, under the assumption that the singular foliation generated by the distribution is regular, we prove that the Laplacian associated with the distribution defines an unbounded multiplier on the foliation C * -algebra. To this end, we give the construction of a parametrix.1991 Mathematics Subject Classification. 58J60, 53C17, 46L08, 58B34.

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Section: The Horizontal Laplacian Of a Distributionmentioning

confidence: 99%

“…Let us recall some necessary information on noncommutative geometry of regular foliations (for more information and details, see [14,15] and references therein).…”

Section: The Horizontal Laplacian As a Multipliermentioning

confidence: 99%

Section: The Horizontal Laplacian As a Multipliermentioning

confidence: 99%

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Riemannian metrics and Laplacians for generalized smooth distributions

Androulidakis

Kordyukov

2019

J. Topol. Anal.

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“…We demonstrate this fact inside this section. is not clear, but it is an obstruction to some transverse curvature and other geometric conditions (see [26], [30], [41], [28]). Suppose F has codimension q.…”

Section: Basic Clifford Bundlesmentioning

confidence: 99%

Index theory for basic Dirac operators on Riemannian foliations

Brüning¹,

Kamber²,

Richardson³

2011

Noncommutative Geometry and Global Analysis

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In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of integrals over blowups of the strata of the foliation and also involves eta invariants of associated elliptic operators. As a special case, a Gauss-Bonnet formula for the basic Euler characteristic is obtained using two independent proofs.

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“…This notion was introduced by Atiyah and Singer[8,61] and actively studied since then (see especially[31][32][33] and the references cited there).…”

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confidence: 99%

Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds

Savin

Sternin

2013

Pseudo-Differential Operators, Generalized Functions and Asymptotics

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Abstract. In this paper we give a survey of elliptic theory for operators associated with diffeomorphisms of smooth manifolds. Such operators appear naturally in analysis, geometry and mathematical physics. We survey classical results as well as results obtained recently. The paper consists of an introduction and three sections. In the introduction we give a general overview of the area of research. For the reader's convenience here we tried to keep special terminology to a minimum. In the remaining sections we give detailed formulations of the most important results mentioned in the introduction.

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Index theory and non-commutative geometry on foliated manifolds

Cited by 13 publications

References 172 publications

Riemannian metrics and Laplacians for generalized smooth distributions

Riemannian metrics and Laplacians for generalized smooth distributions

Index theory for basic Dirac operators on Riemannian foliations

Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds

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