Bundle Of Linear Frames Research Articles

On the structure of an affine connection object and the torsion tensor in the bundle of linear frames

On the structure of an affine connection object and the torsion tensor in the bundle of linear frames

Most Probable Paths for Anisotropic Brownian Motions on Manifolds

Brownian motion on manifolds with non-trivial diffusion coefficient can be constructed by stochastic development of Euclidean Brownian motions using the fiber bundle of linear frames. We provide a comprehensive study of paths for such processes that are most probable in the sense of Onsager–Machlup, however with path probability measured on the driving Euclidean processes. We obtain both a full characterization of the resulting family of most probable paths, reduced equation systems for the path dynamics where the effect of curvature is directly identifiable, and explicit equations in special cases, including constant curvature surfaces where the coupling between curvature and covariance can be explicitly identified in the dynamics. We show how the resulting systems can be integrated numerically and use this to provide examples of most probable paths on different geometries and new algorithms for estimation of mean and infinitesimal covariance.

О расширении касательного пространства 2-го порядка гладкого многообразия

This paper relates to differential geometry, and the research technique is based on G. F. Laptev’s method of extensions and envelopments, which generalizes E. Cartan’s method of moving frame and exterior forms. We consider a smooth m-dimensional manifold, its tangent and cotangent spaces, as well as the second-order frames and coframes on this manifold. Using the perturbation of the exterior derivative and ordinary diffe­ren­tial, mappings are introduced that enable us to construct non-sym­met­rical second-order frames and coframes on a smooth manifold. It is shown that the extension of the second order tangent space to a smooth m-dimen­sional manifold is carried out by adding the vertical vectors to the linear frame bundle over the manifold to the second order tangent vectors to this manifold. A deformed external differential is widely used, which is a differen­tial, i. e., its reapplication vanishes. We introduce a deformed external dif­ferential being a differential along the curves on the manifold, i. e., its re­peated application along the curves on the manifold gives zero.

О тензоре кручения аффинной связности на двумерном и трехмерном многообразиях

The basis for this study of affine connections in linear frame bundle over a smooth manifold is the structure equations of the bundle. An affine connection is given in this bundle by the Laptev — Lumiste method. The differential equations are written for components of the deformation ten­sor from an affine connection to the symmetrical canonical one. The ex­pressions for the components of the torsion tensor for two-dimensional and three-dimensional manifolds were found. For a two-dimensional manifold, the affine torsion is a fraction, in the numerator there is a linear combination of two fiber coordinates which coefficients are two functions depending on the base coordinates (the co­ordinates on the base), and in the denominator there is the determinant composed of the fiber coordinates (the coordinates in a fiber). For a three-dimensional manifold, the arbitrariness of the numerator is determined by nine functions depending on the base coordinates.

О максимальной размерности алгебр Ли инфинитезимальных аффинных преобразований касательных расслоений с синектической связностью А.П. Широкова

The theory of tangent bundles over a differentiable manifold M be­longs to the geometry and topology of manifolds and is an intensively developing area of the theory of fiber spaces. The foundations of the theo­ry of fibered spaces were laid in the works of S. Eresman, A. Weil, A. Mo­ri­moto, S. Sasaki, K. Yano, S. Ishihara. Among Russian scientists, tangent bund­les were investigated by A. P. Shirokov, V. V. Vishnevsky, V. V. Shu­rygin, B. N. Shapukov and their students. In the study of automorphisms of generalized spaces, the question of infinitesimal transformations of connections in these spaces is of great importance. K. Yano, G. Vrancianu, P. A. Shirokov, I. P. Egorov, A. Z. Pet­rov, A. V. Aminova and others have studied movements in different spac­es. The works of K. Sato and S. Tanno are devoted to the motions and au­tomorphisms of tangent bundles. Infinitesimal affine collineations in tan­gent bundles with a synectic connection were considered by H. Shadyev. At present, the question of the motions of fibered spaces is considered in the works of A. Ya. Sultanov, in which infinitesimal transformations of a bundle of linear frames with a complete lift connection, the Lie algebra of holomorphic affine vector fields in arbitrary Weyl bundles are investi­gated. In this paper we obtain exact upper bounds for the dimensions of Lie algebras of infinitesimal affine transformations in tangent bundles with a synectic connection A. P. Shyrokov.

Flat affine manifolds and their transformations

We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental form of the bundle of linear frames of the manifold. We show that the group of affine transformations of a real flat affine $n$-dimensional manifold, acts on $\mathbb{R}^n$ leaving an open orbit when its dimension is greater than $n$. Moreover, when the dimension of the group of affine transformations is $n$, this orbit has discrete isotropy. For any given Lie subgroup $H$ of affine transformations of the manifold, we show the existence of an associative envelope of the Lie algebra of $H$, relative to the connection. The case when $M$ is a Lie group and $H$ acts on $G$ by left translations is particularly interesting. We also exhibit some results about flat affine manifolds whose group of affine transformations admits a flat affine bi-invariant structure. The paper is illustrated with several examples.

Продолжения аффинной связности и горизон­тальных векторов

The linear frame bundle over a smooth manifold is considered. The mapping dе defined by the differentials of the first-order frame e is a lift to the normal N, i. e., a space complementing the first-order tangent space to the second-order tangent space to this bundle. In particular, the map­ping defined by the differentials of the vertical vector of this frame is a vertical lift into normal N. The lift dе allows us to construct a prolongation both of the tangent space and its vertical subspace into the second-order tangent space, more precisely into the normal N. The normal lift dе defines the normal prolon­gation of the tangent space (i. e. the normal N) and its vertical subspace. The vertical lift defines the vertical prolongation of the tangent space and its vertical subspace. The differential of an arbitrary vector field on the linear frame bundle is a complete lift from the first-order tangent space to the second-order tangent space to this bundle. It is known that the action of vector fields as differential operators on functions coincides with action of the differentials of these functions as 1-forms on these vector fields. Horizontal vectors played a dual role in the fibre bundle. On the one hand, the basic horizontal vectors serve as opera­tors for the covariant differentiation of geometric objects in the bundle. On the other hand, the differentials of these geometric objects can be con­sidered as forms (including tangential-valued ones) and their values on basic horizontal vectors give covariant derivatives of these geometric ob­jects. For objects which covariant derivatives require the second-order con­nection, the covariant derivatives are equal to the values of the differen­tials of these objects on horizontal vectors in prolonged affine connectivi­ty. Prolongations of the basic horizontal vectors, i. e., the second-order horizontal vectors for prolonged connection, were constructed. The sec­ond-order tangent space is represented as a straight sum of the first-order tangent space, vertical prolongations of the vertical and horizontal sub­spaces, and horizontal prolongation of the horizontal subspace.

Prolongations generated by horizontal vectors

The linear frame bundle over a smooth manifold is considered. It is shown that covariant derivatives of geometrical objects given on the linear frame bundle are the images of the horizontal frame vectors under maps defined by the differentials of the geometrical objects. In a (first-order) affine connection that is true for tensors, vertical and horizontal frame vectors, and arbitrary vectors from the tangent space to the linear frame bundle. For the very first-order affine connection object and non-vertical frame vectors this property holds in a prolongation of the affine connection. Prolongations of the horizontal frame vectors are constructed.

Geometry of stochastic delay differential equations with jumps in manifolds

In this article we propose a model for stochastic delay differential equation with jumps (SDDEJ) in a differentiable manifold $M$ endowed with a connection $\nabla $. In our model, the continuous part is driven by vector fields with a fixed delay and the jumps are assumed to come from a distinct source of (càdlàg) noise, without delay. The jumps occur along adopted differentiable curves with some dynamical relevance (with fictitious time) which allow to take parallel transport along them. In the last section, using a geometrical approach, we show that the horizontal lift of the solution of an SDDEJ is again a solution of an SDDEJ in the linear frame bundle $BM$ with respect to a horizontal connection $\nabla ^H$ in $BM$.

CANONICAL FORM IN SECOND-ORDER FRAME BUNDLES IN A NATURAL WAY

Using horizontal n-bases of the tangent bundle of the linear frame bundle [Formula: see text] of an n-dimensional manifold M, the canonical form in the non-holonomic second-order frame bundle of M is introduced as a restriction of the canonical form of the bundle [Formula: see text]. This construction generalizes the ones in the corresponding semi-holonomic and holonomic second-order frame bundles. We prove that the natural projection of the set of all non-holonomic second-order frames of M into [Formula: see text] defines a principal bundle structure.

Why time flows

The question used as the title of this article has been asked frequently since the introduction of space–time in which both space and time function as coordinates in a four-dimensional manifold. After a brief review of space–time geometry using the affine bundle it is shown how to construct it from a bundle of linear frames of a five-dimensional manifold. The flow of time is a consequence of the construction.

Curvatures of the diagonal lift from an affine manifold to the linear frame bundle

Abstract We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle LM. This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on LM investigated by L.A. Cordero and M. de León in 1986. The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold. We receive analogous and, surprisingly, even stronger results in our affine setting.

Almost symplectic structures on the linear frame bundle from linear connection

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Diffeomorphism-invariant covariant hamiltonians of a linear frame and a linear connection

Let π : FN → N (resp. p : CN → N) be the bundle of linear frames (resp. the bundle of linear connections) of a C∞ manifold N. The Ehresmann connections on FN ×N CN for which the covariant Hamiltonian of every DiffN-invariant Lagrangian density is also DiffN-invariant, are determined.

THE LIE DERIVATIVE OF SPINOR FIELDS: THEORY AND APPLICATIONS

Starting from the general concept of a Lie derivative of an arbitrary differentiable map, we develop a systematic theory of Lie differentiation in the framework of reductive G-structures P on a principal bundle Q. It is shown that these structures admit a canonical decomposition of the pull-back vector bundle [Formula: see text] over P. For classical G-structures, i.e. reductive G-subbundles of the linear frame bundle, such a decomposition defines an infinitesimal canonical lift. This lift extends to a prolongation Γ-structure on P. In this general geometric framework the concept of a Lie derivative of spinor fields is reviewed. On specializing to the case of the Kosmann lift, we recover Kosmann's original definition. We also show that in the case of a reductive G-structure one can introduce a "reductive Lie derivative" with respect to a certain class of generalized infinitesimal automorphisms, and, as an interesting by-product, prove a result due to Bourguignon and Gauduchon in a more general manner. Next, we give a new characterization as well as a generalization of the Killing equation, and propose a geometric reinterpretation of Penrose's Lie derivative of "spinor fields". Finally, we present an important application of the theory of the Lie derivative of spinor fields to the calculus of variations.

Integrability of the field equations of invariant variational problems on linear frame bundles

The integrability of the Euler–Lagrange equations and the Jacobi fields of the natural basis of Lagrangian densities on the bundle of linear frames of a manifold which are invariant under diffeomorphisms, is stated. Applications to the reducibility of the pre-symplectic structure attached to such variational problems as defined in [Symp. Math. 14 (1974) 219], are also given.

Linear frames in manifolds, riemannian structures and description of internal degrees of freedom

Discussed is affine model of internal degrees of freedom, i.e. infinitesimal Affnely rigid body in a manifold. The treatment is nonrelativistic and classical. The special stress is laid on geodetic models and invariance classification of a priori possible kinetic energies. Geometrically this is equivalent to deriving natural Riemannian structures in the bundle of linear frames from some geometry on the base manifold (affine connection, metric). It turns out that the general structure of dynamical balance laws resembles that derived in general relativity for the pole-dipole particle. Linear momentum and (affine) spin are coupled to the torsion and curvature tensors.

Reductive G-structures and Lie derivatives

Reductive G-structures on a principal bundle Q are considered. It is shown that these structures, i.e. reductive G-subbundles P of Q, admit a canonical decomposition of the pull-back vector bundle i P ∗(TQ)≡P× QTQ over P. For classical G-structures, i.e. reductive G-subbundles of the linear frame bundle, such a decomposition defines an infinitesimal canonical lift. This lift extends to a prolongation Γ-structure on P. In this general geometric framework the theory of Lie derivatives is considered. Particular emphasis is given to the morphisms which must be taken in order to state what kind of Lie derivative has to be chosen. On specializing the general theory of gauge-natural Lie derivatives of spinor fields to the case of the Kosmann lift, we recover the result originally found by Kosmann. We also show that in the case of a reductive G-structure one can introduce a “reductive Lie derivative” with respect to a certain class of generalized infinitesimal automorphisms. This differs, in general, from the gauge-natural one, and we conclude by showing that the “metric Lie derivative” introduced by Bourguignon and Gauduchon is in fact a particular kind of reductive rather than gauge-natural Lie derivative.

Invariant variational problems on linear frame bundles

The Hamiltonian structure of variational problems defined by the natural basis jki of diff M-invariant Lagrangians on the 1-jet bundle of linear frames of a m-dimensional manifold M is described. Diffeomorphism invariance on J1(FM) and its infinitesimal counterpart, i.e., invariance under the natural representation of vector fields of M, are analysed. The Lagrangians jki are proved to be the basic tools required to factor diff M×G-invariance into diff M-invariance and G-invariance. The densities Ωjki = jkiθ1∧∧θm, where the θi are the components of the canonical form, are shown to define two types of variational problem according to whether i{j,k} or i {j,k}. The field equations for their extremals are deduced. These equations are examples of underdetermined non-linear systems of partial differential equations. Extremals defining a Lie algebra structure are characterized. The functions in the real linear space spanned by jki are the only Lagrangians on FM admitting a Hamiltonian formalism of order zero. Infinitesimal symmetries and Noether invariants of the densities Ωjki are studied in detail. In particular, it is proved that the Noether invariant of every vertical symmetry vanishes. Hence only the horizontal symmetries appear in the Hamiltonian structure. The equations of the Jacobi fields along an extremal are explicitly obtained. The pre-symplectic structure attached to Ωjki is defined to be an alternate bilinear map (ω2)s on the space of Jacobi fields along an extremal s with values in closed (m-1)-forms on M and its kernel is related to vertical infinitesimal symmetries of the Lagrangian. For m = 3,4, the equations of the extremals are integrated explicitly; we thus obtain normal forms, which, when transformed by an arbitrary diffeomorphism, yield the general solution to the field equations.

A frame bundle generalization of multisymplectic geometries

We present a generalized symplectic geometry on a principal bundle over the configuration space of a classical field. This bundle, the vertically adapted linear frame bundle, is obtained by breaking the symmetry of the full linear frame bundle of the field configuration space, and it inherits an “ n-symplectic” structure from the full frame bundle. This geometric structure admits vector-valued field observables and produces Hamiltonian vector fields, from which we can define a Poisson bracket on the field observables. We show that both the linear and the affine models of multisymplectic geometry are obtained by mapping the vertically adapted frame bundle to associated fiber bundles. In addition, the new geometry resolves a fundamental problem that arises in each model.