Nonholonomic manifolds and nilpotent analysis (Q908517)

This paper contains a systematic exposition of the theory of nonholonomic manifolds and of the nilpotent analysis apparatus from a unifying point of view. In the first part the fundamental notions of the analysis on nonholonomic manifolds are introduced: the geometry of distributions, construction of the bundle of nonholonomic Lie algebras, homogeneous nilpotent Lie algebras and their classification. Then, the graded Lie algebras of vector field jets on a nonholonomic manifold are introduced and the notion of a nonholonomic Riemannian metric is defined. This makes it possible to introduce the first and second differentials of a function defined on a nonholonomic manifold and the critical point and nonholonomic Hessian as well. The second part of the paper is devoted to nonholonomic Riemannian geometry and to dynamical systems generated by a nonholonomic geodesic flow. One presents also the questions connected with a nonholonomic Laplacian: definition, hypoharmonic function, nonholonomic Green's formula, theorems about asymptotics of spectral functions. The connection between the asymptotic behaviour of eigenvalues' growth and nonholonomic diffusion is also given.