Journal ArticleDOI
Bundle of nilpotent Lie algebras over a nonholonomic manifold (nilpotentization)
TLDR
On the tangent bundle of a nonholonomic manifold one defines a canonical structure of bundle of homogeneous nilpotent Lie algebras, and on the local ring of a nonsmooth manifold one determines a quasijet structure with the help of these constructions one uniformizes and simplifies a number of results as mentioned in this paper.Abstract:
On the tangent bundle of a nonholonomic manifold one defines a canonical structure of bundle of homogeneous nilpotent Lie algebras. On the local ring of a nonholonomic manifold one defines a quasijet structure. With the help of these constructions one uniformizes and simplifies a number of results of nonholonomic geometry.read more
Citations
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MonographDOI
A Tour of Subriemannian Geometries, Their Geodesics and Applications
TL;DR: Geodesics in subriemannian manifolds: Dido meets Heisenberg Chow's theorem: Getting from A to B A remarkable horizontal curve Curvature and nilpotentization Singular curves and geodesics A zoo of distributions Cartan's approach The tangent cone and Carnot groups Discrete groups tending to Carnot geometries Open problems Mechanics and geometry of bundles: Metrics on bundles Classical particles in Yang-Mills fields Quantum phases Falling, swimming, and orbiting Appendices: Geometric mechanics Bundles and the Hopf fibration The S
Journal ArticleDOI
Nonholonomic tangent spaces: intrinsic construction and rigid dimensions
TL;DR: In this paper, the intrinsic construction of these infinitesimal objects and all rigid (i.e. not deformable) cases are given. But the intrinsic analysis of these objects is limited.
Journal ArticleDOI
The Riemann tensor for nonholonomic manifolds
TL;DR: For every nonholonomic manifold, i.e., manifold with nonintegrable distribution the analog of the Riemann tensor is introduced in this paper, which is interpreted as modiÞcations of the Spencer cohomology.
Journal ArticleDOI
Small-Time Reachable Sets of Linear Systems with Integral Control Constraints: Birth of the Shape of a Reachable Set
TL;DR: The main result is the existence of a limit shape of the reachable sets as time tends to zero, and a precise estimate for the rate of convergence is given.
Journal ArticleDOI
Tangent Maps and Tangent Groupoid for Carnot Manifolds
Woocheol Choi,Raphael Ponge +1 more
TL;DR: In this article, the authors introduce a notion of differential, called Carnot differential, for maps that are compatible with the Carnot manifold structure, which is obtained as a group map between the corresponding tangent groups.
References
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Journal ArticleDOI
Nonholonomic manifolds and nilpotent analysis
V. Gershkovich,Anatoly Vershik +1 more
TL;DR: In this paper, the theory of nonholonomic manifolds is studied and a systematic exposition of the nilpotent analysis apparatus can be found from the unique point of view of the manifolds.
Book
Lecture notes on nil-theta functions
TL;DR: In this paper, the finite Fourier transform Abelian varieties, nil-theta and theta functions have been studied in the context of the Jacobi theta function and the Abelian variety.