Noncommutative geometry of foliations
We review basic notions and methods of noncommutative geometry and their applications to analysis and geometry on foliated manifolds.
math.DG↗
arXiv subjects
Yuri A. Kordyukov
Publications and source records attributed to Yuri A. Kordyukov.
Egorov's theorem for transversally elliptic operators on foliated manifolds and noncommutative geodesic flow
The main result of the paper is Egorov's theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.
math.DG↗
Spectral gaps for periodic Schrödinger operators with strong magnetic fields
We consider Schrödinger operators $H^h = (ih d+{\bf A})^* (ih d+{\bf A})$ with the periodic magnetic field ${\bf B}=d{\bf A}$ on covering spaces of compact manifolds. Under some assumptions on $\bf B$, we prove that there are arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of strong magnetic field $h\to 0$.
math.SP↗
The trace formula for transversally elliptic operators on Riemannian foliations
We prove a Duistermaat-Guillemin trace formula for transversally elliptic operators on a compact foliated manifold.
math.DG↗
The transversal wave equation and the noncommutative geodesic flow in Riemannian foliations
We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and the corresponding transversal bicharacteristic flow and give a description of the noncommutative geodesic flow in transversal geometry of Riemannian foliations.
dg-ga↗
Noncommutative spectral geometry of Riemannian foliations
We construct spectral triples in a sense of noncommutative differential geometry, associated with a Riemannian foliation on a compact manifold, and describe its dimension spectrum.
dg-ga↗
Adiabatic Limits and Spectral Geometry of Foliations
We study spectral asymptotics for the Laplace operator on differential forms on a Riemannian foliated manifold equipped with a bundle-like metric in the case when the metric is blown up in directions normal to the leaves of the foliation. The asymptotical formula for the eigenvalue distribution function is obtained. The relationships with the spectral theory of leafwise Laplacian and with the noncommutative spectral geometry of foliations are discussed.
dg-ga↗