Egorov’s Theorem for Transversally Elliptic Operators on Foliated Manifolds and Noncommutative Geodesic Flow

Abstract

Abstract. 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.

Теория Индекса И Некоммутативная Геометрия На Многообразиях Со Слоением

“…Поэтому определения различных геометрических и аналитических объ-ектов, данные в унитальном случае, необходимо модифицировать, учитывая поведение этих объектов на "бесконечности". В работе [139] была построена ал-гебра Ψ * 0 (A ), ассоциированная с произвольной гладкой спектральной тройкой (A , H , D). Эту алгебру можно рассматривать как аналог алгебры псевдодиф-ференциальных операторов на некомпактном многообразии, символы которых обращаются в нуль на бесконечности вместе с производными любого порядка.…”

Noncommutative geometry of foliations

“…In [116] a version of the Egorov theorem for transversally elliptic operators on compact foliated manifolds is proved. Suppose that (M, F) is a compact foliated manifold.…”

“…In [50,43], the definition of the algebra Ψ * (A) of pseudodifferential operators is given for a unital algebra A. In this Section, following the paper [116], we introduce the algebra Ψ * 0 (A), which can be considered as an analogue of the algebra of pseudodifferential operators on a noncompact manifold, whose symbols vanish at infinity along with all the derivatives of an arbitrary order. First, we introduce some auxiliary notions.…”

Noncommutative Spectral Geometry of Riemannian Foliations: Some Results and Open Problems

“…In this section, we describe the corresponding noncommutative object. To do this, we will follow the construction of the cotangent bundle T * B to the base B from the cotangent bundle T * M to the total space M for a fibration M → B (as explained in [39], this construction can be considered as a particular case of the foliation reduction in symplectic geometry) and, when it will be necessary, switch to noncommutative algebras.…”

“…[18,39] The noncommutative geodesic flow is the oneparameter group α t of automorphisms of the algebra S * A defined by (17).…”