Affine Connection Spaces Research Articles

О размерности алгебр Ли инфинитезимальных аффинных преобразований прямых произведений более двух пространств аффинной связности первого типа

The theory of motions in generalized spaces is one of the directions in modern differential geometry. Such scientists as E. Cartan, P. K. Rashev­sky, P. A. Shirokov, I. P. Egorov, A.Ya. Sultanov and other scientists were engaged in the study of movements in various spaces of affine connec­tions. The question of movements in direct products of two spaces of af­fine connection was considered in M. V. Morgun’s work. In the case of a direct product of more than two spaces of affine con­nection, the question of the dimension of Lie algebras of infinitesimal affine transformations of a given space remained open. In this article, an estimate of the upper bound of the dimension of the Lie algebra of infinitesimal affine transformations of affine connection spaces, representing a direct reproduction of at least three non-projective Euclidean spaces of a certain type, is obtained. To solve this problem, a system of linear homogeneous equations is obtained, which is satisfied by the components of an arbitrary infinitesi­mal affine transformation. This system is found using the properties of the Lie derivative applied to the tensor field of curvature of the spaces under consideration. The evaluation of the rank of this system allows us to ob­tain an estimate from below of the rank of the matrix of the system under consideration.

Space of Affine Connections of an Almost Hermitian Manifold

Space of Affine Connections of an Almost Hermitian Manifold

On Equivalence between Kinetic Equations and Geodesic Equations in Spaces with Affine Connection

Discrete kinetic equations describing binary processes of agglomeration and fragmentation are considered using formal equivalence between the kinetic equations and the geodesic equations of some affinely connected space A associated with the kinetic equation and called the kinetic space of affine connection. The geometric properties of equations are treated locally in some coordinate chart (x;U). The peculiarity of the space A is that in the coordinates (x) of some selected local chart, the Christoffel symbols defining the affine connection of the space A are constant. Examples of the Smoluchowski equation for agglomeration processes without fragmentation and the exchange-driven growth equation are considered for small dimensions in terms of geodesic equations. When fragmentation is taken into account, the kinetic equations can be written as equations of quasigeodesics. Particular cases of spaces with symmetries are discussed.

Invariants for equitorsion geometric mappings

New method of obtaining invariants for torsion-preserving mappings is the main subject of this research. This research is consisted of introduction and main part of research. After introduction, where we remind to some of obtained invariants, we will develop general formula (Vesic, 2020) for obtaining invariants for mappings. As special cases, we will remind to some invariants for symmetric affine connection spaces which are not obtained yet.

Two Invariants for Geometric Mappings

Two invariants for mappings of affine connection spaces with a special form of deformation tensors are obtained in this paper. We used the methodology of Vesić to obtain the form of these invariants. At the end of this paper, we used these forms to obtain two invariants for third-type almost-geodesic mappings of symmetric affine connection.

Beyond Lorentzian symmetry

This thesis presents a framework in which to explore kinematical symmetries beyond the standard Lorentzian case. This framework consists of an algebraic classification, a geometric classification, and a derivation of the geometric properties required to define physical theories on the classified spacetime geometries. The work completed in substantiating this framework for kinematical, super-kinematical, and super-Bargmann symmetries constitutes the body of this thesis. To this end, the classification of kinematical Lie algebras in spatial dimension $D = 3$, as presented in [3, 4], is reviewed; as is the classification of spatially-isotropic homogeneous spacetimes of [5]. The derivation of geometric properties such as the non-compactness of boosts, soldering forms and vielbeins, and the space of invariant affine connections is then presented. We move on to classify the $\mathcal{N}=1$ kinematical Lie superalgebras in three spatial dimensions, finding 43 isomorphism classes of Lie superalgebras. Once these algebras are determined, we classify the corresponding simply-connected homogeneous (4|4)-dimensional superspaces and show how the resulting 27 homogeneous superspaces may be related to one another via geometric limits. Finally, we turn our attention to generalised Bargmann superalgebras. In the present work, these will be the $\mathcal{N}=1$ and $\mathcal{N}=2$ super-extensions of the Bargmann and Newton-Hooke algebras, as well as the centrally-extended static kinematical Lie algebra, of which the former three all arise as deformations. Focussing solely on three spatial dimensions, we find $9$ isomorphism classes in the $\mathcal{N}=1$ case, and we identify $22$ branches of superalgebras in the $\mathcal{N}=2$ case.

Projective and amplified symmetries in metric–affine theories

In this paper we extend the projective symmetry of the full metric–affine Einstein–Hilbert theory to a new symmetry transformation in the space of affine connections called the amplified symmetry. We prove that the Lagrangian of the standard model of particle physics is invariant under this new symmetry. We also show that the gravitational Lagrangian can be modified so that the amplified symmetry extends to the gravitational sector and hence to the whole action. The new theory so constructed is shown to be dynamically equivalent to Einstein–Cartan’s though genuinely metric–affine.

Geometric Regularity Results on B k α , β -Manifolds, I: Affine Connections

In this paper we consider existence and multiplicity results concerning affine connections on $C^{k}$-manifolds $M$ whose coefficients are as regular as one needs, following the regularity theory introduced in [10]. We show that if $M$ admits a $B_{\alpha,\beta}^{k}$-structure, then the existence of such regular connections can be established in terms of properties of the structural presheaf $B$. In other words, we propose a solution to the existence problem in this setting. With regard to the multiplicity problem, we show that the space of regular affine connections is an affine space of the space of regular $\operatorname{End}(TM)$-valued 1-forms, and that if two regular connections are locally additively different, then they are actually locally different. The existence of a topology in which the space of regular connections is a nonempty open dense subset of the space of all regular $\operatorname{End}(TM)$-valued 1-forms is suggested. <br><br> У цій роботі розглядаються результати існування та кратності для афінних зв’язностей на $C^{k}$-многовидах $M$, коефіцієнти яких настільки регулярні, наскільки це необхідно, відповідно до теорії регулярності, введеної в [10]. Показано, що якщо $M$ допускає $B_{\alpha,\beta}^{k}$-структуру, то існування таких регулярних зв’язностей можна встановити з точки зору властивостей структурного передпучка $B$. Іншими словами, ми пропонуємо розв'язання проблеми існування в цій постановці. Стосовно проблеми кратності, ми показуємо, що простір регулярних зв’язностей є афінним простором простору регулярних 1-форм зі значеннями в $\operatorname{End}(TM)$, і що якщо дві регулярні зв’язності локально адитивно різні, то вони насправді є локально різні. Запропоновано топологію, в якій простір регулярних зв’язностей є непорожньою відкритою щільною підмножиною простору всіх регулярних $\operatorname{End}(TM)$-значних 1-форм.

A concurrence theorem for alpha-connections on the space of t-distributions and its application

The space of Student's t-distributions with $\nu$ degrees of freedom is the upper half-plane $\mathbb H$ with the location-scale coordinates. A normal distribution is a t-distribution with $\nu=\infty$. The $\alpha$-connections for t-distributions form a line ${\mathbb L}^\nu$ in the space of affine connections on $\mathbb H$. We show that the family $\{{\mathbb L}^\nu\}_{\nu\in(1,\infty]}$ has the concurrent point which presents the e-connection for normal distributions. As an application, generalizing the previous result of the author, we construct a contact Hamiltonian flow which visualizes certain Bayesian learnings.

Modeling Vector Fields in Space of Affine Connection

The authors have been constructed the splitting of the basic geometric images vector field (points, straights, hyperplanes and hyperguadrics) in transition from n-dimensional affine space to the space of affine connection. All invectigations have been fulfilled in the moving coordinate system of zero order.

Geometry and BMS Lie algebras of spatially isotropic homogeneous spacetimes

Simply-connected homogeneous spacetimes for kinematical and aristotelian Lie algebras (with space isotropy) have recently been classified in all dimensions. In this paper, we continue the study of these “maximally symmetric” spacetimes by investigating their local geometry. For each such spacetime and relative to exponential coordinates, we calculate the (infinitesimal) action of the kinematical symmetries, paying particular attention to the action of the boosts, showing in almost all cases that they act with generic non-compact orbits. We also calculate the soldering form, the associated vielbein and any invariant aristotelian, galilean or carrollian structures. The (conformal) symmetries of the galilean and carrollian structures we determine are typically infinite-dimensional and reminiscent of BMS Lie algebras. We also determine the space of invariant affine connections on each homogeneous spacetime and work out their torsion and curvature.

Affine connections on 3-Sasakian and manifolds

The space of invariant affine connections on every 3-Sasakian homogeneous manifold of dimension at least seven is described. In particular, the subspace of invariant affine metric connections and the subclass with skew torsion are also determined. To this aim, an explicit construction of all 3-Sasakian homogeneous manifolds is exhibited. It is shown that the 3-Sasakian homogeneous manifolds which admit nontrivial Einstein with skew torsion invariant affine connections are those of dimension seven, that is, $${\mathbb {S}}^7$$, $${\mathbb {R}}P^7$$ and the Aloff–Wallach space $${\mathfrak {W}}^{7}_{1,1}$$. On $${\mathbb {S}}^7$$ and $${\mathbb {R}}P^7$$, the set of such connections is bijective to two copies of the conformal linear transformation group of the Euclidean space, while it is strictly bigger on $${\mathfrak {W}}^{7}_{1,1}$$. The set of invariant connections with skew torsion whose Ricci tensor satisfies that its eigenspaces are the canonical vertical and horizontal distributions, is fully described on 3-Sasakian homogeneous manifolds. An affine connection satisfying these conditions is distinguished, by parallelizing all the Reeb vector fields associated with the 3-Sasakian structure, which is also Einstein with skew torsion on the 7-dimensional examples. The invariant metric affine connections on 3-Sasakian homogeneous manifolds with parallel skew torsion have been found. Finally, some results have been adapted to the non-homogeneous setting.

On one problem of connections in the space of non-symmetric affine connection and its subspace

Let XM be a submanifold of a differentiable manifold XN (XM ? XN). If on XN a non-symmetric affine connection L is defined by coefficients Lijk ? Likj and on XM a non-symmetric basical tensor g(g?? ? g??) is given, in the present paper we investigate the problem: Find a relation between induced connection L from LN into XM end the connection ??, defined by the tensor 1 in XM. The solutions is given in the Theorem 3.1., that is by the equation (3.9). Some examples are constructed.

Dynamical projective curvature in gravitation

By using a projective connection over the space of two-dimensional affine connections, we are able to show that the metric interaction of Polyakov two-dimensional gravity with a coadjoint element arises naturally through the projective Ricci tensor. Through the curvature invariants of Thomas and Whitehead, we are able to define an action that could describe dynamics to the projective connection. We discuss implications of the projective connection in higher dimensions as related to gravitation.

Projective invariants for equitorsion geodesic mappings of semi-symmetric affine connection spaces

In the present paper we study invariants of equitorsion geodesic mappings of non-symmetric affine connection spaces. We generalize invariants of this mapping herein. As the main result, we examine the forms of these invariants in the case of equitorsion geodesic mappings of semi-symmetric spaces.

Homogeneous affine surfaces: Moduli spaces

We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type A surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and examine the structure of the associated moduli space. For Type B surfaces which are not Type A we show the corresponding moduli space is a simply connected real analytic 4-dimensional manifold with second Betti number equal to 1.

Some Invariants of Equitorsion Third Type Almost Geodesic Mappings

In this paper, we consider equitorsion third type almost geodesic mappings of non-symmetric affine connection spaces. Applying different computational methods, we find invariants of these mappings obtained from curvature tensors of a corresponding non-symmetric affine connection space. Weyl projective tensor of a symmetric affine connection space is generalized in this way for the case of a third type almost geodesic mapping.

Basic equations of G-almost geodesic mappings of the second type, which have the property of reciprocity

We study G-almost geodesic mappings of the second type $$\mathop {{\pi _2}}\limits_\theta (e),\theta = 1,2$$ , θ = 1,2 between non-symmetric affine connection spaces. These mappings are a generalization of the second type almost geodesic mappings defined by N. S. Sinyukov (1979). We investigate a special type of these mappings in this paper. We also consider e-structures that generate mappings of type $$\mathop {{\pi _2}}\limits_\theta (e),\theta = 1,2$$ , θ = 1,2. For a mapping $$\mathop {{\pi _2}}\limits_\theta (e,F),\theta = 1,2$$ , θ = 1,2, we determine the basic equations which generate them.

On special first-type almost geodesic mappings of affine connection spaces preserving a certain tensor

On special first-type almost geodesic mappings of affine connection spaces preserving a certain tensor

Special Almost Geodesic Mappings of the First Type of Non-symmetric Affine Connection Spaces

We introduce a special kind of almost geodesic mappings of the first type \(\pi _1^*\) of spaces with non-symmetric affine connections. Also, we investigate a special class of equitorsion almost geodesic mappings of type \(\pi _1^*\) and find some invariant geometric objects of these mappings.