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In this chapter we give the basic concept of the “topological numbers” in the theory of quasiperiodic functions. Attention is especially paid to appearance of such quantities in transport phenomena, including galvanomagnetic phenomena in normal metals (Sect. 2.1) and the modulations of 2D electron gas (Sect. 2.3). We give a detailed introduction to both these areas and explain in a simple way the appearance of the “integral characteristics” in both these problems. Though this chapter cannot be considered a detailed survey in the area, it explains the main basic features of the corresponding phenomena.

It is well known that the projective space ℂP n carries a canonical symplectic structure (see, for instance, Arnol’d [1974] or Arnol’d and Givental’ [1985]). We consider a representation ρ: G →GL(V) of a Lie group G on a finite-dimensional complex vector space V. There is an induced action of G on the projective space P(V) associated with V. We denote by G(z) the orbit of this action passing through a point z ϵ P(V). It is now natural to ask when the symplectic structure of P(V) induces a symplectic structure on the orbits of G in P(V). Let us assume that G is compact, let T be a maximal torus in G, let g and t be the Lie algebras of G and T, respectively, and let L ℂ denote the complexification of a vector space L. We then have the root space decomposition \( {g^C} = {t^C} + \sum\limits_{\alpha \ne 0} {\mathbb{C}{E_\alpha }{H_\alpha }} = \left[ {{E_\alpha }{E_{ - \alpha }}} \right] \) (see, for instance, Helgason [1962], Jacobson [1962] or the Appendix). Let n: V\0 → P(V) be the natural projection which maps a vector υ ∊ V\0 into the straight line passing through υ and the origin.

In this chapter we consider the simplest nonholonomic variational problems. We study three-dimensional nonholonomic Lie groups, i.e. groups with a left-invariant nonholonomic distribution. Our main subject is the study of the nonholonomic geodesic flow (NG-flow), more precisely, of the nonholonomic sphere, of the wave front (Section 1), and of the general dynamical properties of the flow (Section 2). The mixed bundle for Lie groups is the direct product G × (V ⊕ V⊥). In Section 1.1 we show that the NG-flow on the mixed bundle is the semidirect product with base V ⊕ V⊥ and fiber G. In Section 1.2 we describe left-invariant metric tensors on Lie algebras; in Section 1.3 the normal forms for the equations of nonholonomic geodesics are obtained. In Section 1.4 we study the reduced flow on V ⊕ V ⊥. In the subsequent Sections (1.5–1.7) we describe local properties of the flow on the fiber; in Section 1.5 we describe the e-wave front of the NG-flow and the e-sphere of the nonholonomic metrics which appear to be manifolds with singularities, the same for all three-dimensional nonholonomic Lie groups. In Section 1.6 we describe their topology and in Section 1.7 their metric structure.

In the sequel without further notice all objects, such as manifolds, functions, mappings, distributions, vector fields, forms, etc., are supposed to be infinitely differentiable.

A pair (M, ω) consisting of a 2n-dimensional manifold M together with a closed 2-form ω is called a symplectic manifold if the form ω is nondegenerate, i.e. if ω n = ω ∧ · ... · ω ≢ 0.

In this section we consider the nonholonomic variational problem on the minimum of length on a Riemannian manifold with constraints given by a nonholonomic distribution. The solutions to this problem, the nonholonomic geodesics, satisfy the Euler-Lagrange equations of a conditional problem. They generate a nonholonomic geodesic flow defined on the mixed bundle which is the direct sum of the distribution and its annihilator in the cotangent bundle (see Section 1.3). This flow allows to extend to the nonholonomic case the notion of exponential mapping.

I am going to discuss here some results in the soliton theory of a Moscow group during the last years. The group of people who worked here include B.A.Dubrovin, I.M. Krichever, S.P. Tsarev (and the present author). More details may be found in the survey article [1]. Modern needs in the large new classes of hydrodynamic type systems appear in connection with very interesting asymptotic method - so called “nonlinear analog of WKB-method”, method of the slow modulations of parameters or “Whitham method” in the theory of solitons (see for example the book [2], chapter 4). This method is based on the large family of exact solutions φ0(x,t,u)periodic or quasiperiodic in x and t, of the form which we call “soliton lattice ”: $${{\varphi }_{0}}(x,t;u) = F({{\eta }_{0}} + Ux + Vt;{{u}^{1}}, \ldots ,{{u}^{N}}).$$ (0.1) Here \(F({{\eta }_{1}}, \ldots ,{{\eta }_{m}};{{u}^{1}}, \ldots ,{{u}^{N}})\) is the function, which is 2π-periodic in each variable η j, and depends on the N parameters up. All quantities (aj, bk,up,) are constants and φ0(x,t,u) satisfies some nonlinear P.D. equation $${{\varphi }_{t}}(x,t) = K(\varphi ,{{\varphi }_{x}}, \ldots ,{{\varphi }^{{(s)}}}),$$ (0.2) describing the propagation of solitons or some other nonlinear waves.

We consider a translation invariant evolution equation of the form $$ \dot u = K[u],({u^1}; = \frac{{\partial u}}{{\partial x}},\dot u = \frac{{\partial u}}{{\partial x}})$$ (10.1) where u is a vector-function of a continuous variable x or of a discrete variable n; the equation (10.1) is invariant with respect to translations in x (or n).

This paper deals mainly with problems connected with topological and homotopic invariance of Pontrjagin classes and some closely related problems of algebraic topology and stable algebra. These questions have arisen from the authors’ papers on topological and homotopic invariance of rational Pontrjagin classes, based on the discovery of deep connections between characteristic classes and the fundamental group. There are a number of new stable algebraic problems connected with the diffeomorphism problem and Pontrjagin classes of nonsimply connected manifold (especially when π1= Z × • • • × Z).