Distribution (differential geometry)

About: Distribution (differential geometry) is a research topic. Over the lifetime, 911 publications have been published within this topic receiving 10149 citations.

On contact equivalence of systems of ordinary differential equations

Abstract: We consider a problem of equivalence of generic pairs $(X,V)$ on a manifold $M$, where $V$ is a distribution of rank $m$ and $X$ is a distribution of rank one. We construct a canonical bundle with a canonical frame. We prove that two pairs are equivalent if and only if the corresponding frames are diffeomorphic. As a particular case, with $V$ integrable, we provide a new solution to the problem of contact equivalence of systems of $m$ ordinary differential equations: $x^{(k+1)}=F(t,x,x',...,x^{(k)})$, where $k>2$ or $k=2$ and $m>1$.

A Note on Hyperbolic Flows in Sub-Riemannian Structures with Transverse Symmetries

Abstract: The curvature and the reduced curvature are basic differential invariants of the pair (Hamiltonian system, Lagrangian distribution) on a symplectic manifold. We consider the Hamiltonian flows of the curve of least action of natural mechanical systems in sub-Riemannian structures with symmetries. We give sufficient conditions for the reduced flows (after reduction of the first integrals induced from the symmetries) to be hyperbolic in terms of the reduced curvature and show new examples of Anosov flows.

Stability and Hopf Bifurcation Analysis of an Epidemic Model with Time Delay.

Abstract: Epidemic models are normally used to describe the spread of infectious diseases. In this paper, we will discuss an epidemic model with time delay. Firstly, the existence of the positive fixed point is proven; and then, the stability and Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equations. Thirdly, the theory of normal form and manifold is used to drive an explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions. Finally, some simulation results are carried out to validate our theoretic analysis.

On the geometry of nonholonomic mechanical systems with vertical distribution

Abstract: Let (Q,g) be the configuration space of a nonholonomic mechanical system, where g is a Riemannian metric on Q. Suppose the horizontal distribution D on Q admits a vertical distribution D¯, that is D¯ is an integrable complementary (not necessarily orthogonal) distribution to D in TQ. We prove the existence and uniqueness of a linear connection on (Q,g) subject to some conditions on its torsion and on the covariant derivative of g. Then we show that the solutions of the Lagrange-d’Alembert equations are the geodesics of ∇ and vice versa. All the local components of the torsion and curvature tensor fields of ∇ with respect to an adapted frame field are determined. Finally, two examples are given to illustrate the theory we develop in the paper.

Isometries of almost-Riemannian structures on Lie groups

Abstract: A simple Almost-Riemannian Structure on a Lie group G is defined by a linear vector field (that is an infinitesimal automorphism) and dim(G) -- 1 left-invariant ones. It is first proven that two different ARSs are isometric if and only if there exists an isometry between them that fixes the identity. Such an isometry preserves the left-invariant distribution and the linear field. If the Lie group is nilpotent it is an automorphism. These results are used to state a complete classification of the ARSs on the 2D affine and the Heisenberg groups.