Noncommutative geometry of foliations

2019

J. Topol. Anal.

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.

Section: The Horizontal Laplacian As a Multipliermentioning

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%

See 1 more Smart Citation

Riemannian metrics and Laplacians for generalized smooth distributions

Androulidakis

2019

J. Topol. Anal.

“…Before giving the global definition of the principal symbol, we recall several notions (for more details, see e.g. [24] and references therein). Let γ : [0.1] → M be a continuous leafwise path in M with the initial point x = γ (0) and the final point y = γ (1) and T 0 and T 1 arbitrary smooth submanifolds (possibly, with boundary), transversal to the foliation, such that x ∈ T 0 and y ∈ T 1 .…”

Section: The Principal Symbol For Transverse ψ Dosmentioning

confidence: 99%

The Egorov theorem for transverse Dirac-type operators on foliated manifolds

Journal of Geometry and Physics

2007

Self Cite

Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first-order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associated transverse bicharacteristic flow. For a particular case of a transverse Dirac operator, the transverse bicharacteristic flow is shown to be given by the transverse geodesic flow and the parallel transport by the parallel transport determined by the transverse Levi-Civita connection. These results allow us to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.The Egorov theorem is a fundamental fact in microlocal analysis and quantum mechanics. It relates the evolution of pseudodifferential operators on a compact manifold (quantum observables) determined by a first-order elliptic operator with the corresponding evolution of classical observables -the bicharacteristic flow on the space of symbols. More precisely, let M be a compact manifold and let P be a positive, self-adjoint, elliptic, first-order pseudodifferential operator on M with the positive principal symbol p ∈ S 1 (T * M \ 0). Let f t be the bicharacteristic flow of the operator P, that is, the Hamiltonian flow of p on T * M. Egorov's theorem [8] states that, for any pseudodifferential operator A of order 0 with the principal symbol a ∈ S 0 (T * M \ 0), the operator A(t) = e it P Ae −it P is a pseudodifferential operator of order 0. The principal symbol a t ∈ S 0 (T * M \ 0) of this operator is given by the formula a t (x, ξ ) = a( f t (x, ξ )), (x, ξ ) ∈ T * M \ 0.

“…We refer the reader to [20] for basic notions of Poisson geometry and to the survey paper [14] for information and references on noncommutative geometry of foliations.…”

mentioning

confidence: 99%

Noncommutative Hamiltonian dynamics on foliated manifolds

J. Fixed Point Theory Appl.

2010

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First, we review the notion of a Poisson structure on a noncommutative algebra due to Block, Getzler and Xu, and we introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a noncommutative algebra associated with a transversely symplectic foliation and construct a class of Hamiltonian vector fields associated with this Poisson structure.Mathematics Subject Classification (2010). Primary 58B34; Secondary 37J05, 53C12.