Sub-Riemannian geometry and nonholonomic mechanics

We investigate certain natural connections between sub-Riemannian geometry and hyperbolic dynamical systems. In particular, we study dynamically defined horizontal distributions which split into two integrable ones and ask: how is the energy of a sub-Riemannian geodesic shared between its projections onto the integrable summands? We show that if the horizontal distribution is the sum of the strong stable and strong unstable distributions of a special type of a contact Anosov flow in three dimensions, then for any short enough sub-Riemannian geodesic connecting points on the same orbit of the Anosov flow, the energy of the geodesic is shared equally between its projections onto the stable and unstable bundles. The proof relies on a connection between the geodesic equations and the harmonic oscillator equation, and its explicit solution by the Jacobi elliptic functions. Using a different idea, we prove an analogous result in higher dimensions for the geodesic flow of a closed Riemannian manifold of constant negative curvature.

Chapter VI Finslerian manifolds of constant sectional curvature [4

In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection and adapted frame fields, we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affine constraints. In the second part of the paper, we show that the Kaluza–Klein theory is best presented and explained by using the framework of nonholonomic mechanical systems. We show that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projections on the spacetime are the geodesics from general relativity. Any other geodesic of the Kaluza–Klein space that is not tangent to the electromagnetic distribution is also a solution of Lagrange–d’Alembert equations, but for affine constraints. In particular, some of these geodesics project exactly on the solutions of the Lorentz force equations of the spacetime.

In this paper, variational techniques are used to analyze the dynamics of nonholonomic mechanical systems with impacts. Implicit nonholonomic smooth Lagrangian and Hamiltonian systems are extended to a nonsmooth context appropriate for collisions. In particular, we provide a variational formulation for implicit nonholonomic mechanical systems with collisions, for those collisions that preserve energy and momentum at the impact. Lastly, the theoretical results are illustrated by examining the example of a rolling disk hitting a wall.

In nonholonomic mechanical systems with constraints that are affine (linear nonhomogeneous) functions of the velocities, the energy is typically not a first integral. It was shown in Fassò and Sansonetto (2016 J. Nonlinear Sci. 26 519–44) that, nevertheless, there exist modifications of the energy, called there moving energies, which under suitable conditions are first integrals. The first goal of this paper is to study the properties of these functions and the conditions that lead to their conservation. In particular, we enlarge the class of moving energies considered in Fassò and Sansonetto (2016 J. Nonlinear Sci. 26 519–44). The second goal of the paper is to demonstrate the relevance of moving energies in nonholonomic mechanics. We show that certain first integrals of some well known systems (the affine Veselova and LR systems), which had been detected on a case-by-case way, are instances of moving energies. Moreover, we determine conserved moving energies for a class of affine systems on Lie groups that include the LR systems, for a heavy convex rigid body that rolls without slipping on a uniformly rotating plane, and for an n-dimensional generalization of the Chaplygin sphere problem to a uniformly rotating hyperplane.

We obtain a non-Kahler almost Hermitian manifold of constant holomorphic sectional curvature by changing the almost complex structure in a Kahler manifold of constant holomorphic sectional curvature.

The simplest chaotic Hamiltonian systems arise in geometry: a point particle moving freely on a 2 or 3 dimensional manifold of constant negative curvature. There is tremendous variety representing both bound state and scattering situations, quite in contrast to the non-chaotic motion on surfaces of constant positive curvature. The classical mechanics has been studied for almost a century and reduces to an enumeration and description of geodesic lines which in turn are closely connected with the topology of the manifold. Physics enters only in a trivial manner since everything can be scaled with the (kinetic) energy of the particle. In the quantum mechanics of these motions, however, the energy is the essential parameter, and one can ask how the two mechanics are related to each other. Quite surprisingly, Feynman's path integral gives the same answer as its classical approximation, just as in the harmonic oscillator. In 3 dimensions, this result holds for the propagator itself, while in 2 dimensions it is true only for the trace and yields Selberg's trace formula. A few striking examples of constant negative curvature spaces will be shown. Thus, quantum chaos can be understood in terms of classical chaos in spite of their qualitatively different features: fractal for the latter, smooth for the former.

Prescribed scalar curvature on a [formula omitted] compact Riemannian manifold of dimension two

ABSTRACTSegway is a self-balancing motorized two-wheeled vehicle which is able to carry the human body. In the presented paper, a problem of nonholonomic constrained mechanical systems is treated. New methods in nonholonomic mechanics are applied to a problem of a two-wheeled self-balancing robots motion ‘SEGWAY’. This method of the geometrical theory of general nonholonomic constrained systems on fibred manifolds and their jet prolongations, based on so-called Chetaev-type constraint forces, was proposed and developed in the last decade by Krupková and others. The equations of motion of a two-wheeled self-balancing robot are highly nonlinear and rolling without slipping condition can only be expressed by nonholonomic constraint equations. In this paper, the geometrical theory is applied to the above mentioned mechanical problem using the above mentioned Krupková approach. Additionally, the results of numerical solutions of constrained equations of motion derived within the theory are in good agreement with results of (1) [Maddahi, A., Shamekhi, A. H., & Ghaffari, A. (2015). A Lyapunov controller for self-balancing two-wheeled vehicles. Robotica, 33(1), 225–239]. using Lyapunov's feedback control design technique. The existence, continuity, and uniqueness of the solution for the proposed control system are proved utilizing the Filippov's solution (2). And with fuzzy controller proposed by [Qian, Q., Wu, J., & Wang, Z. (2017). A novel configuration of two-wheeled self-balancing robot/Nova konfiguracija samobalansirajuceg robota na dva kotaca (Original scientific paper/Izvorni znastveni clanak). Tehnicki Vjesnik-Technical Gazette, 24(2), 459–465].

We find some curvature properties of 3-quasi-Sasakian manifolds which extend certain well-known identities holding in the Sasakian case. As an application, we prove that any 3-quasi-Sasakian manifold of constant horizontal sectional curvature is necessarily either 3-α-Sasakian or 3-cosymplectic.

Let $\pi \colon X \to B$ be a holomorphic submersion between compact K\ahler manifolds of any dimensions, whose fibres and base have no non-zero holomorphic vector fields and whose fibres admit constant scalar curvature K\ahler metrics. This article gives a sufficient topological condition for the existence of a constant scalar curvature K\ahler metric on $X$. The condition involves the $\CM$-line bundle---a certain natural line bundle on $B$---which is proved to be nef. Knowing this, the condition is then implied by $c_1(B) <0$. This provides infinitely many K\ahler manifolds of constant scalar curvature in every dimension, each with K\ahler class arbitrarily far from the canonical class.

Warped products provide an elegant and versatile framework for exploring and understanding a wide range of geometric structures. Their ability to combine two distinct manifolds through a warping function introduces a rich and diverse set of geometries, thus making them a powerful tool in various mathematical, physical, and computational applications. This article addresses the central query related to warped product submanifolds in the context of statistics. It focuses on deriving two new and distinct inequalities for a statistical warped product submanifold in a statistical manifold of a constant (quasi-constant) curvature. This article then finishes with some concluding remarks.

Nonholonomic systems provide an important class of mechanical control systems. One reason for this importance is that nonintegrability is essential to both the mechanics and the control: Nonintegrable constraint distributions are the essence of nonholonomic systems, while a nonintegrable distribution of control vector fields is the key to controllability of nonlinear systems. We will learn how these two different types of nonintegrability work together when we study control of nonholonomic mechanical systems.

In this paper, we show that a generalized Sasakian space form of dimension greater than three is either of constant sectional curvature; or a canal hypersurface in Euclidean or Minkowski spaces; or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold; or locally a wapred product of a real line and a generalized complex space form; or an $\alpha$-Sasakian space form; or it is of five dimension and admits an $\alpha$-Sasakian Einstein structure. In particular, a local classification for generalized Sasakian space forms of dimension greater than five is obtained. A local classification of Riemannian manifolds of quasi constant sectional curvature of dimension greater than three is also given in this paper.

Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and $\varepsilon$-spaces exhaust the class of $n$-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least ${1/2} n(n-1)+1$, for almost all values of $n$ [Patrangenaru V., Geom. Dedicata 102 (2003), 25-33]. We shall prove that the curvature tensor of these spaces satisfy several interesting algebraic properties. In particular, we will show that Egorov spaces are Ivanov-Petrova manifolds, curvature-Ricci commuting (indeed, semi-symmetric) and $\mathcal P$-spaces, and that $\varepsilon$-spaces are Ivanov-Petrova and curvature-curvature commuting manifolds.