Orientable Two-dimensional Manifold Research Articles

Slow divergence integral on a Möbius band

The slow divergence integral has proved to be an important tool in the study of slow-fast cycles defined on an orientable two-dimensional manifold (e.g. R2). The goal of our paper is to study 1-canard cycle and 2-canard cycle bifurcations on a non-orientable two-dimensional manifold (e.g. the Möbius band) by using similar techniques. Our focus is on smooth slow-fast models with a Hopf breaking mechanism. The same results can be proved for a jump breaking mechanism and non-generic turning points. The slow-fast bifurcation problems on the Möbius band require the study of the 2-return map attached to such 1- and 2-canard cycles. We give a simple sufficient condition, expressed in terms of the slow divergence integral, for the existence of a period-doubling bifurcation near the 1-canard cycle. We also prove the finite cyclicity property of “singular” 1- and 2-homoclinic loops (“regular” 1-homoclinic loops of finite codimension have been studied by Guimond).

Constructing reference metrics on multicube representations of arbitrary manifolds

Reference metrics are used to define the differential structure on multicube representations of manifolds, i.e., they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be continuous. This paper introduces a general procedure for constructing reference metrics automatically on multicube representations of manifolds with arbitrary topologies. The method is tested here by constructing reference metrics for compact, orientable two-dimensional manifolds with genera between zero and five. These metrics are shown to satisfy the Gauss–Bonnet identity numerically to the level of truncation error (which converges toward zero as the numerical resolution is increased). These reference metrics can be made smoother and more uniform by evolving them with Ricci flow. This smoothing procedure is tested on the two-dimensional reference metrics constructed here. These smoothing evolutions (using volume-normalized Ricci flow with DeTurck gauge fixing) are all shown to produce reference metrics with constant scalar curvatures (at the level of numerical truncation error).

On Global Hypoellipticity

We consider a first order linear partial differential operator of principal type on a closed connected orientable two-dimensional manifold sending sections of one complex line bundle to sections of another. We prove that the assumption of global hypoellipticity of the operator implies a relation between the degrees of the line bundles and the Euler characteristic of the manifold.

Finite groups and flows on 2-manifolds

We consider flows on compact orientable two-dimensional manifolds all points of which are non-wandering. An inter-relation between so-called Conley-Lyapunov-Peixoto graphs of such flows and Cayley graphs of finite groups is clarified. We describe all finite groups that are `good candidates' to symmetry groups of the flows in question. An application of the above results to the integrability problem of Hamiltonian flows is suggested. We conclude with two questions which might be of interest to group theorists.

Braids and the Wigner-Racah calculus

The Wigner-Racah approach to the system of identical particles is compared with the theory of N anyons for the case when each hard-core particle is moving on an orientable two-dimensional manifold. It results that these two approaches are non-equivalent, and that the main source of this fact is a distinction in the meaning of the notion of indistinguishability, represented by the symmetric and the braid group for the Wigner-Racah and anyon theory, respectively.

THE ABSENCE OF NONCLOSED POISSON-STABLE SEMITRAJECTORIES AND TRAJECTORIES DOUBLY ASYMPTOTIC TO A DOUBLE LIMIT CYCLE FOR DYNAMICAL SYSTEMS OF THE FIRST DEGREE OF STRUCTURAL INSTABILITY ON ORIENTABLE TWO-DIMENSIONAL MANIFOLDS

In the present paper we investigate the absence, for dynamical systems of :he first degree of structural instability on two-dimensional compact orientable manifolds of any genus, of nonclosed Poisson-stable semitrajectories and trajectories that are doubly asymptotic to a double limit cycle. The propositions are some of the basic propositions that must be added to the known conditions for the first degree of structural instability on a plane (or sphere) [1] in order to obtain a description of dynamical systems of the first degree of structural instability on orientable two-dimensional manifolds. Systems of the first degree of structural instability on a torus were considered in [2]. A study of such systems on two-dimensional manifolds of higher genus has not been carried out up to the present time.