SYMPLECTIC STRUCTURE ON A GRASSMANNIAN FIBRATION

This chapter is divided into three parts. The first and most substantial of these is devoted to symplectic vector spaces. Using methods of exterior algebra which are due to Elie Cartan, we study the rank of an exterior 2-form on a real vector space as well as that of the 2-form induced on a vector subspace. By means of associated canonical bases we show that there is only one model of a symplectic form and that vector subspaces may be classified by the rank of the induced form. In mechanics one uses mainly isotropic and coisotropic subspaces, particularly Lagrangian subspaces which will be discussed here in some detail. In particular we study the reduction of a symplectic vector space.

In this chapter, we introduce an important generalization of tangent bundles: if M is a smooth manifold, a vector bundle over M is a collection of vector spaces, one for each point in M, glued together to form a manifold that looks locally like the Cartesian product of M with ℝn, but globally may be “twisted.” We then go on to discuss local and global sections of vector bundles (which correspond to vector fields in the case of the tangent bundle). At the end of the chapter, we discuss the natural maps between bundles, called bundle homomorphisms, and subsets of vector bundles that are themselves vector bundles, called subbundles.KeywordsVector BundleFiber BundleTangent BundleSmooth ManifoldGlobal SectionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The object of this work is to define, by analogy with algebra, the Witt group and the graded Brauer group of a topological space X.A homomorphism is defined between them analogous to the generalized Hasse invariant.Upon evaluation, the Witt group is seen to be KO(X), the graded Brauer group 1+H1(X;ZZ) + H2(X; Z2) with truncated cup product multiplication, while the homomorphism is given by Stiefel-Whitney classes: I + m^ + h^.Introduction.The Hasse invariant of a quadratic form on a vector space over a field k is an element of the Brauer group of k.It can be generalized to a homomorphism from the Witt group of classes of quadratic forms to the graded Brauer group of classes of graded ^-algebras [3], [16].Curiously enough, in this guise it is quite analogous to the homomorphism given by the first two Stiefel-Whitney classes which maps real vector bundles over a space X to the group 1 + H\X; Z2) + H2(X;Z2), where multiplication is by truncated cup product.The purpose of this paper is to display this analogy.Delzant has defined Stiefel-Whitney classes of quadratic forms [4], [13] using the discriminant and Hasse invariant.It should add insight into his definition to see that the first two topological Stiefel-Whitney classes can also be defined in an algebraic way.A vector bundle L is known to be orientable for real L-theory if its first two Stiefel-Whitney classes vanish.In this case it can be read off from the diagram of Theorem 3.1 that the Clifford bundle of L is a bundle of graded endomorphism algebras of a graded module bundle, which is then used to construct the Thorn class of L. Donovan and Karoubi in studying this orientability question have done independently much of this same work, applying it further to define L-theory with local coefficients [6], [7].The connection between algebra and topology comes from considering bundles over X whose fibers are vector spaces with quadratic forms or graded algebras and subjecting them to the same analysis the fiber alone is accorded in algebra.This is described in some detail in 1.11 is devoted to the algebraic computations and 111 to a study of the topological problems.

Geometry of slow–fast Hamiltonian systems and Painlevé equations

We present a topological method for analysing dynamical systems, starting with a smooth (or, alternatively, real analytic) manifold M representing configuration space and its cotangent bundle T*M with given symplectic structure [1]. We assume k independent first integrals in involution, F=(f1,…,fk):T*M→Rk to be given, one of which, f, represents the Hamiltonian. For each c∈Rk, the set Ic≔F−1(c) is an invariant manifold for the vector field Xf [2]. Information about the dynamics can be obtained by studying the changes in topological type of these manifolds as c varies. We illustrate this with two examples, the compound pendulum and the plane Kepler problem; we see that a bifurcation in the nature of the dynamics occurs at points where the topological degree of the invariant manifold jumps.

A Cartan manifold is a smooth manifold M whose slit cotangent bundle 0 * M T is endowed with a regular Hamiltonian K which is positively homogeneous of degree 2 in momenta. The Hamiltonian K defines a (pseudo)-Riemannian metric ij g in the vertical bundle over 0 * M T and using it, a Sasaki type metric on 0 * M T is constructed. A natural almost complex structure is also defined by K on 0 * M T in such a way that pairing it with the Sasaki type metric an almost Kahler structure is obtained. In this paper we deform ij g to a pseudo-Riemannian metric ij G and we define a corresponding almost complex Kahler structure. We determine the Levi-Civita connection of G and compute all the components of its curvature. Then we prove that if the structure

Quantum kinematics and geometric quantization

Let V be a vector space of dimension 2n, n even, over a field F, equipped with a nonsingular symplectic form. We define a new algebraic/combinatorial structure, a spread of nonsingular pairs, or nsp-spread, on V and show that nsp-spreads exist in considerable generality. We further examine in detail some particular cases.

Heat kernel expansions in vector bundles

A vector bundle flow $({\Phi ^t},{\phi ^t})$ on the vector bundle $E$ over a compact metric space $M$ induces a one-parameter group $\{ \Phi _t^\# \}$ of bounded operators acting on the continuous sections of $E$, with infinitesimal generator $L$. An example is given by the tangent flow $(T{\phi ^t},{\phi ^t})$, if ${\phi ^t}$ is a flow on a smooth manifold. In this article, the spectrum of the generator $L$ is used to study the exponential growth rates of bundle trajectories in the neighborhood of a fixed invariant subbundle, e.g. the tangent bundle of a submanifold of $M$. Auxiliary normal and tangential spectra are introduced, and their relationship and fine structure are explored.

Existence of natural and projectively equivariant quantizations

A vector bundle flow ( Φ t , ϕ t ) ({\Phi ^t},{\phi ^t}) on the vector bundle E E over a compact metric space M M induces a one-parameter group { Φ t # } \{ \Phi _t^\# \} of bounded operators acting on the continuous sections of E E , with infinitesimal generator L L . An example is given by the tangent flow ( T ϕ t , ϕ t ) (T{\phi ^t},{\phi ^t}) , if ϕ t {\phi ^t} is a flow on a smooth manifold. In this article, the spectrum of the generator L L is used to study the exponential growth rates of bundle trajectories in the neighborhood of a fixed invariant subbundle, e.g. the tangent bundle of a submanifold of M M . Auxiliary normal and tangential spectra are introduced, and their relationship and fine structure are explored.

A generalization of the Perelomov procedure for the construction of coherent states is proposed. The new procedure is used to construct systems of coherent states in the carrier spaces of unitary irreducible representations of groups G=S∅V, where V is a vector space and S⊂GL(V). The coherent states are shown to be labeled by the points in cotangent bundles T*𝒪* of orbits 𝒪* of S in V*, the dual of V; it is proven that T*𝒪* is a symplectic homogeneous space of G. The generalized procedure for the construction of coherent states presented in this paper is shown to encompass as special cases the constructions known in the literature for the coherent states of the Weyl–Heisenberg, the ‘‘ax+b,’’ and the Galilei and Poincaré groups. Moreover, completely new sets of coherent states are constructed for the Euclidean group E(n), where the Perelomov construction fails.

A non-Hermitian operator may serve as the Hamiltonian for a unitary quantum system, if we can modify the Hilbert space of state vectors of the system so that it turns into a Hermitian operator. If this operator is time-dependent, the modified Hilbert space is generally time-dependent. This in turn leads to a generic conflict between the condition that the Hamiltonian is an observable of the system and that it generates a unitary time-evolution via the standard Schr\"odinger equation. We propose a geometric framework for addressing this problem. In particular we show that the Hamiltonian operator consists of a geometric part, which is determined by a metric-compatible connection on an underlying Hermitian vector bundle, and a non-geometric part which we identify with the energy observable. The same quantum system can be locally described using a time-dependent Hamiltonian that acts in a time-independent state space and is the sum of a geometric part and the energy operator. The full global description of the system is achieved within the framework of a moderate geometric extension of quantum mechanics where the role of the Hilbert space of state vectors is played by a Hermitian vector bundle $\mathcal{E}$ endowed with a metric compatible connection, and observables are given by global sections of a real vector bundle that is determined by $\mathcal{E}$. We examine the utility of our proposal to describe a class of two-level systems where $\mathcal{E}$ is a Hermitian vector bundle over a two-dimensional sphere.

A one-parameter linear group in a vector space L is a continuous map \( g:t\mapsto {g^t}=g(t) \) on the space of parameters ℝ, taking values in a space of the linear operators on L and having the following properties: (1) g 0 = E and (2) the sum of parameter values corresponds to the composition \( {g^{s+t }}={g^s}\circ {g^t} \). Thus, one-parameter linear groups in ℝ are the continuous solutions of the functional equations g(0) = 1, g(t + s) = g(t)g(s). As is well known, exponential functions satisfy these equations. After solving the first problems in this chapter, readers will find that the continuous solutions are all exponential and, thus, will extract the usual properties of exponential functions from these functional equations (which corresponds to a real historical development).The second part of the chapter is devoted to proper generalizations for one-parameter matrix groups. Readers will find that all of them are given by the matrix exponential series, as in the one-dimensional case. However, a multidimensional picture looks much more complicated. Working on these problems will familiarize readers with matrix analysis. In addition, readers may need familiarity with some tools used by number theory and the theory of differential equations, such as Euclid’s algorithm, Chinese remainder theorem, and Poincaré’s recurrence theorem; we introduce and discuss these tools.In the third part of the chapter readers will find some elementary classic applications of one-parameter matrix groups in differential equation theory (Liouville formula), in complex analysis (e.g., complex exponential functions, Euler’s formula), for finite-dimensional functional spaces (spaces of quasipolynomials), and others. Readers will see how to deal with these problems using elementary tools of analysis and linear algebra.Further, part of this chapter contains problems for readers possessing more advanced preliminary experience (although we introduce and discuss the necessary tools). While working on these problems, readers will come across many interesting things regarding analysis and linear algebra and become acquainted with important concepts (symplectic forms and others).Readers will encounter far-reaching advances and applications of the subjects considered in the present chapters in powerful mathematical theories: of differential equations, Lie groups, and group representations (a guide to the literature is provided).KeywordsProblem GroupNumerical SeriesGreat Common DivisorExponential SeriesLess Common MultipleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.