Gershkovich, V.; Vershik, A. Nonholonomic manifolds and nilpotent analysis. (English) Zbl 0693.53006 J. Geom. Phys. 5, No. 3, 407-452 (1988). 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. Reviewer: G.Zet Cited in 12 Documents MSC: 53B20 Local Riemannian geometry 58A30 Vector distributions (subbundles of the tangent bundles) Keywords:Lagrange-Hamilton formalism; nonholonomic manifolds; graded Lie algebras; nonholonomic Riemannian geometry; nonholonomic geodesic flow PDFBibTeX XMLCite \textit{V. Gershkovich} and \textit{A. Vershik}, J. Geom. Phys. 5, No. 3, 407--452 (1988; Zbl 0693.53006) Full Text: DOI References: [1] Rothschild, L.; Stein, E. 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