Characteristic number

Abstract: In this note we define a global, R-valued invariant of a compact, strictly pseudoconvex 3-dimensional CR-manifold M whose holomorphic tangent bundle is trivial. The invariant arises as the evaluation of a deRham cohomology class on the fundamental class of the manifold. To construct the relevant form, we start with the CR structure bundle Y over M (see [Ch-Mo], whose notation we follow). The form is a secondary characteristic form of this structure. By fixing a contact form and coframe, i.e., a section of Y, we obtain a form on M. Surprisingly, this form is well-defined up to an exact term, and thus its cohomology class is well-defined in H 3 (M, R). Our motivation for studying this invariant was its analogy with the R/Z secondary characteristic number associated by Chern and Simons to the conforreal class of a Riemannian 3-manifold N, which provides an obstruction to the conformal immersion of N in R 4. Though several formal analogies to the conformal case are valid for our invariant, this one does not hold up: specifically, in w below, we calculate examples which show that the CR invariant can take on any positive real value for hypersurfaces embedded in C 2. It is also clear that the invariant is neither a homotopy nor concordance invariant, but depends in an elusive way on the CR structure. Our inspiration came from the seminal papers of Chern and Moser and Chern and Simons. The idea of looking at secondary characteristic forms of higher order geometric structures in general appears in [Ko-Oc], though with a different intention. In w 2 we will quickly review the definition of a CR structure, the construction of y and its reduction to a pseudo-hermitian structure ~ ld Webster [-We]. In w 3 we define the invariant and prove that it is, in fact, R-valued, and not R/Z-valued as in the Riemannian case. We also prove that if the invariant is stationary as a function of the CR structure, then M is locally CR equivalent to the standard three sphere in C 2, paralleling a result of Chern and Simons. As noted already, w 4 is devoted to the calculation of several examples.

Abstract: I. A well-known theorem of J. Milnor [8] states that on an oriented surface of genus h any flat Sl(2,~)-bundle has Euler number of numerical value at most h-]. Here "flat" means that there exists a system of local trivializations for the bundle such that all the transition functions are constant. D. Sullivan [9] has generalized this result by finding bounds for the Euler number of a flat Sl(2n, ~)-bundle on a 2n-dimensional manifold. In this note we shall generalize Milnor's theorem in a different direction by finding bounds for the 2-dimensional real characteristic numbers of flat G-bundles where G is any connected semi-simple Lie group with finite center. By a real characteristic number we simply mean the evaluation of a real characteristic class (i.e. the pull-back under the classifying map of a class in H2(BG,~)) on a given homology class in the base and we want to estimate this number independently of the flat bundle. Actually it suffices to consider the characteristic numbers of flat bundles over surfaces (see Remark 2 following Proposition 2.2 below) and in this case our results are given by Proposition 2.2 and Theorem 4.1 below. The results depend on the particular simple description due to Guichardet and Wigner [4] which one has for 2-dimensional continuous cochains on Lie groups and I am indebted to Professor A. Guichardet for informing me about his work.

Abstract: Wedescribe a newmethod of distinguishing between ordered and chaotic orbits, which is much faster than the methods used up to now, namely (1) the distribution of the Poincare consequents, (2) the Lyapunov characteristic number and (3) the distribution of the rotation angles This method is based on the distribution of the helicity angles (the angles of small deviations ξ from a given orbit with a fixed direction), or of the twist angles (the differences of successive helicity angles), and the stretching numbers (the logarithms of the ratios of successive deviations | ξ |, also called ’short time Lyapunov characteristic numbers’) We apply this method to 2-D mappings and 4-D mappings, representing Hamiltonian systems of 2 and 3 degrees of freedom respectively