On Gromov's theory of rigid transformation groups: A dual approach

There is an error in the proof of Theorem 1.1 that invalidates proofs of other theorems. Theorem 1.5 is unaffected.

It is shown that a locally geometrical structure of arbitrarily curved Riemannian space is defined by a deformed group of its diffeomorphisms

The paper presents the geometry of Lie algebroids and its applications to optimal control. The first part deals with the theory of Lie algebroids, connections on Lie algebroids and dynamical systems defined on Lie algebroids (mainly Lagrangian and Hamiltonian systems). In the second part we use the framework of Lie algebroids in the study of distributional systems (drift less control affine systems) with holonomic or nonholonomic distributions.

A comparative analysis of two different versions of the Legendre transformation is presented. We provide an almost complete although somewhat superficial review of the geometric background for analytical mechanics. Complete coordinate characterizations of all structures are provided. Intrinsic constructions of most of the objects are given. Examples of applications to a number of physical systems is given

This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late 19th century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or infinitesimal groups, are a recurring theme in 20th-century research on Lie algebras. I will focus on so-called transitive or even primitive Lie algebras, and explain their theory due to Lie, Morozov, Dynkin, Guillemin, Sternberg, Blattner, and others....

This paper analyses the parabolic geometries generated by a free 3-distribution in the tangent space of a manifold. It shows the existence of normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to SO(4,2) and SO(3,3), respectively. There is also a fascinating construction of a `dual' distribution when the holonomy reduces to $G_2'$. The paper concludes with some holonomy constructions for free $n$-distributions for $n>...

This is a survey paper on several aspects of differential geometry for the last 30 years, especially in those areas related to non-linear analysis. It grew from a talk I gave on the occasion of seventieth anniversary of Chinese Mathematical Society. I dedicate the lecture to the memory of my teacher S.S. Chern who had passed away in December 2004.

This book contains the material of my research on stereotype duality theories in geometry. It was intended as a continuation of my recently published monograph in De Gruyter on stereotype spaces and algebras.

DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improvements in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in grav...

Given a real-analytic Riemannian manifold M there exists a canonical complex structure on part of its tangent bundle which turns leaves of the Riemannian foliation on TM into holomorphic curves. A Grauert tube over M of radius r, denoted as T^rM, is the collection of tangent vectors of M of length less than r equipped with this canonical complex structure. We say the Grauert tube T^rM is rigid if Aut(T^rM) is coming from Isom (M). In this article, we prove the rigidity fo...