Affine Connection Research Articles

Colesanti type inequalities for affine connections

Colesanti type inequalities for affine connections

Аффинные преобразования касательного расслоения общего пространства путей

In this paper, we study infinitesimal transformations of the tangent bundle of a common path space. The general path space is a genera­liza­tion space of the affine connectivity. By affine connectivity of the com­mon path space, we construct an affine connection on the tangent bundle. For the infinitesimal transformation of the tangent bundle, a system of in­va­riance equations for the constructed affine connectivity is compiled. This system is a system of second-order differential equations with res­pect to the components of the infinitesimal transformation. The main re­sults of the article are obtained by analyzing this system taking into account the properties of homogeneous functions. It is shown that the comp­lete lift of an infinitesimal transformation of base is an infinitesimal af­fine motion of a tangent bundle if and only if the infinitesimal transfor­ma­tion of base is an affine motion in the general path space. Necessary and sufficient conditions are found that the infinitesimal transformation of a tangent bundle generated by a vertical vector field leaves the affine connec­ti­vity of the tangent bundle invariant. Conditions are given that are neces­sa­ry and sufficient so that the infinitesimal transformation of a tangent bundle with affine connectivity that preserves layers is an affine motion.

Canonical F-planar mappings of spaces with affine connection onto 3-symmetric spaces

We consider canonical F-planar mappings of spaces with affine connection onto 3-symmetric spaces. The main equations for the mappings have been obtained as a closed system of PDEs of Cauchy type in covariant derivatives. We have found the maximum numbers of essential parameters which the general solution of the system depends on.

Линейные и проективные связности над гладким многообразием

The principal bundles of the first order coframes and the second order coframes, as well as factor bundle of centroprojective (coaffine) coframes are considered. In the bundle of linear coframes a connection is given with the help of the field of connection object. The torsion and curvature tensors of this linear connection are determined. Special connections are singled out: torsion-free, curvature-free. The space of a linear connection devoid of torsion and curvature is an affine group, that served as the basis for classical name «affine connection». Under the specializations of a manifold, strong and weak projectivity conditions was introduced, which make it possible to single out the cof­ra­me bundles. The connections in these principal bundles are called strong and weak projective connections. In the case of symmetric linear connection, when the torsion is absent, the object of classic projective connection is considered. Connec­tion forms are introduced and their structure equations are found. Hence it follows that classic projective connection is neither fundamental-group nor linear differential-geometric. It is proved, that the curvature object of this connection forms a quasitensor together with the connection object only. It is shown, that classic projective connection degenerates into diffe­rent from the original linear connection on the image of a section of some homogeneous bundle.

A study of different class of almost Para-Sasakian Riemannian manifold

In this work, the author thought about and looked into the Nijenhuis tensor, the Affine Connection, and the conformal curve tensor on Nearly Para Sasakian different class-Riemannian differential Manifold. We developed and proved the different results for above mentioned class. We also established new condition of Geodesic for different class-Riemannian differential Manifold.

Canonical almost geodesic mappings π_2(e), e=±1, of spaces with affine connection onto m-symmetric spaces

Canonical almost geodesic mappings π_2(e), e=±1, of spaces with affine connection onto m-symmetric spaces

Hybrid optimization algorithm based on the combination of swarm intelligence models: a swarm of chromosomes and bees

A hybrid algorithm for solving the problem of placing VLSI elements has been developed, which combines the procedures for the behavior of a group of bees and a swarm of chromosomes. The optimization criterion is based on estimates of the number of nets crossing the given lines of the routing area. A modified chromosome swarm paradigm is described, which, in contrast to the canonical method, provides the possibility of searching for solutions in an affine position space with integer parameter values. As the weight of the affine connection between chromosomes on the affine-relaxation model of a swarm of chromosomes, estimates of the degree of «difference» between them are used. A modified structure of the bee algorithm is proposed. For each base position, a probabilistic choice of a set of positions located in its vicinity is implemented. The time complexity of the algorithm is О(n2) – О(n3).

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

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.

Locally Homogeneous Manifolds Defined by Lie Algebra of Infinitesimal Affine Transformations

This article deals with Lie algebra G of all infinitesimal affine transformations of the manifold M with an affine connection, its stationary subalgebra ℌ⊂G, the Lie group G corresponding to the algebra G, and its subgroup H⊂G corresponding to the subalgebra ℌ⊂G. We consider the center ℨ⊂G and the commutant [G,G] of algebra G. The following condition for the closedness of the subgroup H in the group G is proved. If ℌ∩ℨ+G;G=ℌ∩[G;G], then H is closed in G. To prove it, an arbitrary group G is considered as a group of transformations of the set of left cosets G/H, where H is an arbitrary subgroup that does not contain normal subgroups of the group G. Among these transformations, we consider right multiplications. The group of right multiplications coincides with the center of the group G. However, it can contain the right multiplication by element 𝒽¯, belonging to normalizator of subgroup H and not belonging to the center of a group G. In the case when G is in the Lie group, corresponding to the algebra G of all infinitesimal affine transformations of the affine space M and its subgroup H corresponding to its stationary subalgebra ℌ⊂G, we prove that such element 𝒽¯ exists if subgroup H is not closed in G. Moreover 𝒽¯ belongs to the closures H¯ of subgroup H in G and does not belong to commutant G,G of group G. It is also proved that H is closed in G if P+ℨ∩ℌ=P∩ℌ for any semisimple algebra P∈G for which P+ℜ=G.

A comment on metric vs metric-affine gravity

We consider the sum of the Einstein-Hilbert action and a Pontryagin density (PD) in arbitrary even dimension D≥4. All curvatures are functions of independent affine (torsionless) connections only. In arbitrary even dimension, not only in D=4n, these first order PD terms are shown to be covariant divergences of “Chern-Simons” currents. The field equation for the connection leads to it being Levi-Civita, and to the metric and affine field equations being equivalent to the second order metric theory. This result is a counterexample to the theorem stating that purely metric and metric-affine models can only be equivalent for Lovelock theories.

Hommage to Chentsov’s theorem

Chentsov’s theorem, which characterises Markov invariant Riemannian metric and affine connections of manifolds of probability distributions on finite sample spaces, is undoubtedly a cornerstone of information geometry. This article aims at providing a comprehensible survey of Chentsov’s theorem as well as its modest extensions to generic tensor fields and to parametric models comprising continuous probability densities on Rk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb R}^k$$\\end{document}.

Bi-Invariant Dissimilarity Measures for Sample Distributions in Lie Groups

Data sets sampled in Lie groups are widespread, and as with multivariate data, it is important for many applications to assess the differences between the sets in terms of their distributions. Indices for this task are usually derived by considering the Lie group as a Riemannian manifold. Then, however, compatibility with the group operation is guaranteed only if a bi-invariant metric exists, which is not the case for most noncompact and noncommutative groups. We show here that if one considers an affine connection structure instead, one obtains bi-invariant generalizations of well-known dissimilarity measures: Hotelling statistic, Bhattacharyya distance, and Hellinger distance. Each of the dissimilarity measures matches its multivariate counterpart for Euclidean data and is translation invariant, so that biases, e.g., through an arbitrary choice of reference, are avoided. We further derive nonparametric two-sample tests that are bi-invariant and consistent. We demonstrate the potential of these dissimilarity measures by performing group tests on data of knee configurations and epidemiological shape data. Significant differences are revealed in both cases.

Inflating and reheating the Universe with an independent affine connection

It is shown that a component of the dynamical affine connection, which is independent of the metric, can drive inflation in agreement with observations. This provides a geometrical origin for the inflaton. It is also found that the decays of this field, which has spin 0 and odd parity, into Higgs bosons can reheat the universe up to a sufficiently high temperature.

On quantum‐induced revisiting Einstein tensor in the relativistic regime

AbstractFor a rigorous approach to reconcile principles of general relativity (GR) and quantum mechanics (QM), we suggest applying the generalized relativistic noncommutative Heisenberg algebra on spacetime coordinates and momenta of a test particle on curved Riemann geometry manifold, which in turn is extended to an eight‐dimensional manifold. The natural generalization of the four‐dimensional Riemann geometry is the Finsler geometry, in which the quadratic restriction on the length measure is relaxed. With the minimum measurable length derived from the relativistic generalized four‐dimensional uncertainty principle, the quantum‐induced corrections to the fundamental tensor could be determined in the relativistic regime. Accordingly, the affine connections could be revisited. With the quantum‐induced revisiting Riemann curvature tensor and its contractions, the Ricci curvature tensor, and then the Ricci scalar, we have been able to construct a quantum‐induced revision of the Einstein tensor, in which besides quantum‐induced corrections, additional geometric structures emerge. On the surface of the 2‐sphere, we compared the quantum‐induced revisiting and non‐revisiting Einstein tensor and concluded that the difference between both versions strongly depends on the minimum measurable length and the local geodesics of the test particle through the additional curvature.

Geometric Proca with matter in metric-Palatini gravity

In the present work, we study linear, torsion-free metric-Palatini gravity, extended by the quadratics of the antisymmetric part of the Ricci tensor and extended also by the presence of the affine connection in the matter sector. We show that this extended metric-Palatini gravity reduces dynamically to the general relativity plus a geometrical massive vector field corresponding to non-metricity of the connection. We also show that this geometric Proca field couples to fermions universally. We derive static, spherically symmetric field equations of this Einstein-geometric Proca theory. We study possibility of black hole solutions by taking into account the presence of a dust distribution that couples to the geometric Proca. Our analytical and numerical analyses show that the presence of this dust worsens the possibility of horizon formation. We briefly discuss possible roles of this universally-coupled geometric Proca in the astrophysical and collider processes.

The information geometry of two-field functional integrals

Two-field functional integrals (2FFI) are an important class of solution methods for generating functions of dissipative processes, including discrete-state stochastic processes, dissipative dynamical systems, and decohering quantum densities. The stationary trajectories of these integrals describe a conserved current by Liouville’s theorem, despite the absence of a conserved kinematic phase space current in the underlying stochastic process. We develop the information geometry of generating functions for discrete-state classical stochastic processes in the Doi-Peliti 2FFI form, and exhibit two quantities conserved along stationary trajectories. One is a Wigner function, familiar as a semiclassical density from quantum-mechanical time-dependent density-matrix methods. The second is an overlap function, between directions of variation in an underlying distribution and those in the directions of relative large-deviation probability that can be used to interrogate the distribution, and expressed as an inner product of vector fields in the Fisher information metric. To give an interpretation to the time invertibility implied by current conservation, we use generating functions to represent importance sampling protocols, and show that the conserved Fisher information is the differential of a sample volume under deformations of the nominal distribution and the likelihood ratio. We derive a pair of dual affine connections particular to Doi-Peliti theory for the way they separate the roles of the nominal distribution and likelihood ratio, distinguishing them from the standard dually-flat connection of Nagaoka and Amari defined on the importance distribution, and show that dual flatness in the affine coordinates of the coherent-state basis captures the special role played by coherent states in Doi-Peliti theory.

From Dual Connections to Almost Contact Structures

A dualistic structure on a smooth Riemaniann manifold M is a triple (M,g,∇) with g a Riemaniann metric and ∇ an affine connection generally assumed to be torsionless. From g and ∇, dual connection ∇* can be defined. In this work, we give conditions on the basis of this notion for a manifold to admit an almost contact structure and some related structures: almost contact metric, contact, contact metric, cosymplectic, and co-Kähler in the three-dimensional case.

Affine connections on complex compact surfaces and Riccati distributions

Let $M$ be a complex surface. We show that there is a one-to-one correspondence between torsion-free affine connections on $M$ and Riccati distributions on $\mathbb{P}(TM)$. Furthermore, if $M$ is compact, then this correspondence induces a one-to-one correspondence between affine structures on $M$ and Riccati foliations on $\mathbb{P}(TM)$.

Comparing equivalent gravities: common features and differences

We discuss equivalent representations of gravity in the framework of metric-affine geometries pointing out basic concepts from where these theories stem out. In particular, we take into account tetrads and spin connection to describe the so called Geometric Trinity of Gravity. Specifically, we consider General Relativity, constructed upon the metric tensor and based on the curvature R; Teleparallel Equivalent of General Relativity, formulated in terms of torsion T and relying on tetrads and spin connection; Symmetric Teleparallel Equivalent of General Relativity, built up on non-metricity Q, constructed from metric tensor and affine connection. General Relativity is formulated as a geometric theory of gravity based on metric, whereas teleparallel approaches configure as gauge theories, where gauge choices permit not only to simplify calculations, but also to give deep insight into the basic concepts of gravitational field. In particular, we point out how foundation principles of General Relativity (i.e. the Equivalence Principle and the General Covariance) can be seen from the teleparallel point of view. These theories are dynamically equivalent and this feature can be demonstrated under three different standards: (1) the variational method; (2) the field equations; (3) the solutions. Regarding the second point, we provide a procedure starting from the (generalised) second Bianchi identity and then deriving the field equations. Referring to the third point, we compare spherically symmetric solutions in vacuum recovering the Schwarzschild metric and the Birkhoff theorem in all the approaches. It is worth stressing that, in extending the approaches to f(R), f(T), and f(Q) gravities respectively, the dynamical equivalence is lost opening the discussion on the different number of degrees of freedom intervening in the various representations of gravitational theories.