Wiener distribution on holonomy groups

grou~, we examin~ t~e group of pa.!~~i' holonomx groups and nonintegrable phase factors: (i) Conslderng the seml·dlrect product EPo=[Path]®P o, where [Path] stands for the group of paths in a four· dimensional Minkowski space, we construct a linear representation of EPo by defining the action:of this group on the fiber bundle P. The parallel transports in P, the holonomy group and the nonintegrable phase factor are all described by this representation. (ii) Correpsonding discussions are given for the affine frame bundle over the spacetime. Remarkable is the fact that the motions of a point and of a vector are both described by the nonintegrable phase factor. Discussions can be easily extended to the case in which Poincare gauge and Yang· Mills gauge fields coexist. Nonintegrable phase factors are equally important in Poincare gauge theory of gravity as well as in the Yang·Mills gauge theory.

In \overlinePoincar é gauge theory of gravity, which is based on the principal fiber bundle P over the spacetime having the covering group P0 of the proper orthochronous Poincaré group as the structure group, we examine the group of paths, holonomy groups and nonintegrable phase factors: (i) Considering the semi-direct product EP0def =[Path]⊗ P0 , where [Path] stands for the group of paths in a four-dimensional Minkowski space, we construct a linear representation of EP0 by defining the action of this group on the fiber bundle P . The parallel transports in P , the holonomy group and the nonintegrable phase factor are all described by this representation. (ii) Corresponding discussions are given for the affine frame bundle over the spacetime. Remarkable is the fact that the motions of a point and of a vector are both described by the nonintegrable phase factor. Discussions can be easily extended to the case in which \overlinePoincar é gauge and Yang-Mills gauge fields coexist. Nonintegrable phase factors are equally important in \overlinePoincar é gauge theory of gravity as well as in the Yang-Mills gauge theory.

An introduction to the theory of generalized conics and their applications

It has been proposed to abandon the requirement that parallel transporters in gauge theories are unitary (or pseudoorthogonal). This leads to a geometric interpretation of Vierbein fields as parts of gauge fields, and nonunitary parallel transport in extra directions yields Higgs fields. In such theories, the holonomy group H is larger than the gauge group G. Here we study a 1-dimensional model with fermions which retains only the extra dimension, and which is soluble in the sense that its renormalization group flow may be exactly computed, with G=SU(2) and noncompact subgroup H of GL(2,C), or G=U(2), H=GL(2,C). In all cases the asymptotic behavior of the Higgs potential is computed, and with one possible exception for G=SU(2), H=GL(2,C), there is a flow of the action from an UV-fix point which describes a SU(2)-gauge theory with unitary parallel transporters, to a IR-fixpoint. We explain how a splitting of the masses of fermions of different flavor can arise through spontaneous symmetry breaking.

A connection between the Yang–Mills gauge fields on 4-dimensional orientable compact Riemannian manifolds and modified Lévy Laplacians is studied. A modified Lévy Laplacian is obtained from the Lévy Laplacian by the action of an infinite dimensional rotation. Under the assumption that the 4-manifold has a nontrivial restricted holonomy group of the bundle of self-dual 2-forms, the following is proved. There is a modified Lévy Laplacian such that a parallel transport in some vector bundle over the 4-manifold is a solution of the Laplace equation for this modified Lévy Laplacian if and only if the connection corresponding to the parallel transport satisfies the Yang–Mills anti-self-duality equations. An analogous connection between the Laplace equation for the Lévy Laplacian and the Yang–Mills equations was previously known.

In this paper we prove a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that (M, H, g) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field xi is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point qin M such that the holonomy group Psi (q) acts reducibly on H(q) yielding a decomposition H(q) = H_1(q)oplus cdots oplus H_m(q) into Psi (q)-irreducible factors. Using parallel transport we obtain the decomposition H = H_1oplus cdots oplus H_m of H into sub-distributions H_i. Unlike the Riemannian case, the distributions H_i are not integrable, however they induce integrable distributions Delta _i on M/xi , which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that T(U/xi )=Delta _1oplus cdots oplus Delta _m, and the latter decomposition of T(U/xi ) induces the decomposition of U/xi into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. We also give a version of the theorem for indefinite metrics.

We introduce horizontal holonomy groups, which are groups defined using parallel transport only along curves tangent to a given subbundle D of the tangent bundle. We provide explicit means of computing these holonomy groups by deriving analogues of Ambrose–Singer’s and Ozeki’s theorems. We then give necessary and sufficient conditions in terms of the horizontal holonomy groups for existence of solutions of two problems on foliated manifolds: determining when a foliation can be either (a) totally geodesic or (b) endowed with a principal bundle structure. The subbundle D plays the role of an orthogonal complement to the leaves of the foliation in case (a) and of a principal connection in case (b).

In the context of two-particle interferometry, we construct a parallel transport condition that is based on the maximization of coincidence intensity with respect to local unitary operations on one of the subsystems. The dependence on correlation is investigated and it is found that the holonomy group is generally non-Abelian, but Abelian for uncorrelated systems. It is found that our framework contains the Lévay geometric phase (2004 J. Phys. A: Math. Gen. 37 1821) in the case of two-qubit systems undergoing local SU(2) evolutions.

Berger's Theorem classifies the linear holonomy groups of irreducible, simply connected Riemannian manifolds. For physical applications, however, it is at least as important to have a classification of the possible spin holonomy groups (defined by the parallel transport of spinors) of non-simply-connected manifolds. We establish a complete classification of the spin holonomy groups of all compact, locally irreducible, Einstein Riemannian spin manifolds of non-negative scalar curvature.

In this note it is shown that the Maslov index for pairs of Lagrangian paths as introduced by Leray and later canonized by Cappell, Lee and Miller appears by parallel transporting elements of (a certain complex line-subbundle of) the symplectic spinor bundle over Euclidean space, when pulled back to an (embedded) Lagrangian submanifold \(L\), along closed or non-closed paths therein. In especially, the CLM-Index mod \(4\) determines the holonomy group of this line bundle w.r.t. the Levi-Civita-connection on \(L\), hence its vanishing mod 4 is equivalent to the existence of a trivializing parallel section. Moreover, it is shown that the CLM-Index determines parallel transport in that line-bundle along arbitrary paths when compared to the parallel transport w.r.t. to the canonical flat connection of Euclidean space, if the Lagrangian tangent planes at the endpoints either coincide or are orthogonal. This is derived from a result on parallel transport of certain elements of the dual spinor bundle along closed or endpoint-transversal paths.

The local geometrical structure of general relativity is analyzed in detail from the standpoint of a formulation of gravity as a gauge theory of the de Sitter group SO(3,2). In order to reproduce the structure of the Einstein-Cartan theory, it is essential that the SO(3,2) gauge symmetry be spontaneously broken down to the Lorentz group. In the geometrical analysis of this spontaneously broken theory, the Goldstone field of the symmetry-breaking mechanism plays a central role, representing the coordinates of a point in an internal anti-de Sitter space where the motions induced by parallel transport across space-time take place. In order to establish the connection between the SO(3,2) gauge theory and the Einstein-Cartan theory, the gravitational vierbein and spin connection are derived from the original SO(3,2) gauge fields by passing over to a set of nonlinearly-transforming fields through a redefinition involving the Goldstone field. The original SO(3,2) gauge fields have a different but equally important role: they generate pseudotranslations and rotations in the internal anti-de Sitter space under a kind of parallel transport across space-time that is called "development." Development maps curves in space-time into image curves in the internal space, and vector fields along the curves in space-time into image vector fields along the image curves. Considering development along infinitesimal closed curves in space-time leads to the proper interpretation of the effects of torsion and of curvature in terms of the nonclosure of image curves and of the rotation of image vectors with respect to their original values.

Given a Finsler space (M, F ) on a manifold M , the averaging method associates to Finslerian geometric objects affine geometric objects living on M . In particular, a Riemannian metric is associated with the fundamental tensor g and an affine, torsion free connection is associated with the Chern-Rund connection. As an illustration of the applications of theory, a generalization of the Gauss-Bonnet theorem to Berwald surfaces using the average metric is presented. The parallel transport and curvature endomorphisms of the average connection are obtained. The holonomy group for a Berwald space is discussed. Finally, isometries and symmetric spaces are considered and the heritage of the property of symmetric space from the Finsler space to the average Riemannian metric is proved.

An affine manifold is a differentiable manifold together with an atlas of coordinate charts whose coordinate changes extend to affine automorphisms of Euclidean space. These charts are called atline coordinates. A map between affine manifolds is called affine it its expression in affine coordinates is the restriction of an aftine map between vector spaces. Thus we form the category of affine manifolds and affine m_nps. Let M be a connected affine manifold of dimension n-> 1, locally isomorphic to the vector space E. Its universal covering/~/ inheri ts a unique affine structure for which the covering projection/~/--~ M is an aifine immersion. The group ~r of deck transformations acts on /V/by afline automorphisms. It is well known that there is an affine immersion D :/~5/--~ E, called the developing map. This follows, for example, from Chevalley's Monodromy Theorem; a proof is outlined in Section 2. Such an immersion is unique up to composition with an atiine automorphism of E. Thus for every g ~ n there is a unique affine automorphism a(g) of E such that D o g =ct(g)oD. The resulting homomorphism a : ~r --~ Aft (E) from 7r into the group of affine automorphisms of E is called the affine holonomy representation. It is unique up to inner automorphisms of Aft (E). The composition A : 7r ~ G L (E) is called the linear holonomy representation. The affine structure on M is completely determined by the pair (D, a) . M is called complete when D is a homeomorphism. This is equivalent to geodesic completeness of the connection on M (in which parallel transport is locally defined by affine charts as ordinary parallel transport in E). It is notorious that compactness does not imply completeness. The main results of this paper are about aftine manifolds whose affine holonomy groups a(Tr) are nilpotent. An important class of such manifolds are the affine nilmanifolds 7r\G. Here 7r is a discrete subgroup of a simply connected nilpoint Lie group G. It is assumed that G has a left-invariant afline structure; the space of right cosets of r then inherits an affine structure.

A Berger-type theorem for metric connections with skew-symmetric torsion

The paper exposits Wiener distribution theory for I(1) time series as an overview to a special issue on testing integration and cointegration. The behavior of an I(1) series is related to a Wiener process to derive the limiting distribution of its sample mean. Other Wiener processes are related to functions of the normal distribution. The analysis is applied to an autoregressive process, a bivariate regression, and the similarity and power properties of two single-equation tests for cointegration. Systems analyses of cointegration based on the Johansen approach are derived by successive concentration of the likelihood function. An empirical model for Norwegian consumption expenditure is examined. Copyright 1992 by Blackwell Publishing Ltd