Invariant Connections Research Articles

Covariant and manifestly projective invariant formulation of Thomas-Whitehead gravity

Thomas-Whitehead (TW) gravity is a recently formulated projectively invariant extension of Einstein-Hilbert gravity. Projective geometry was used long ago by Thomas to succinctly package equivalent paths encoded by the geodesic equation. Projective invariance in gravity has further origins in string theory through a geometric action constructed from the method of coadjoint orbits using the Virasoro algebra. A projectively invariant connection arises from this construction, a part of which is known as the diffeomorphism field. TW gravity exploits projective Gauss-Bonnet terms in the action functional to endow the diffeomorphism field with dynamics, while allowing the theory to collapse to general relativity in the limit that the diffeomorphism field vanishes and the connection becomes Levi-Civita. In the original formulation of TW gravity, the diffeomorphism field is projectively invariant but not tensorial and the connection is projectively invariant but not affine. In this paper we reformulate TW gravity in terms of projectively invariant tensor fields and a projectively invariant covariant derivative, derive field equations respecting these symmetries, and show that the field equations obtained are classically equivalent across formulations. This provides a “Rosetta Stone” between this newly constructed covariant and projective invariant formulation of TW gravity and the original formulation that was manifestly projective invariant, but not covariant. Published by the American Physical Society 2024

A note on Weyl gauge symmetry in gravity

Abstract A scale invariant theory of gravity, containing at most two derivatives, requires, in addition to the Riemannian metric, a scalar field and (or) a gauge field. The gauge field is usually used to construct the affine connection of Weyl geometry. In this note, we incorporate both the gauge field and the scalar field to build a generalised scale invariant Weyl affine connection. The Ricci tensor and the Ricci scalar made out of this generalised Weyl affine connection contain, naturally, kinetic terms for the scalar field and the gauge field. This provides a geometric interpretation for these terms. It is also shown that scale invariance in the presence of a cosmological constant and mass terms is not completely lost. It becomes a duality transformation relating various fields.

A computational methodology for invariant manifold connections between quasi-periodic libration point orbits in non-autonomous problems

Semi-analytical and numerical techniques to systematically analyze and compute natural connections between quasi-periodic orbits associated to non-autonomous systems are considered. Focusing in the non-autonomous Sun–Earth+Moon coherent QBCP model, the center-unstable and center-stable invariant manifolds of Lyapunov quasi-periodic orbits are parameterized and a methodology to detect heteroclinic connections between manifolds is introduced. The methodology aims to decrease the dimensionality of the problem and to address the issue of searching good initial conditions inside the high dimensional state space. Then, connections are identified by searching for approximate patch points in a state-time Poincaré section. These points are subsequently refined to obtain smooth paths that are further continued inside their families.

Invariant connections on non-irreducible symmetric spaces with simple Lie group

Consider a symmetric space G / H G/H with simple Lie group G G . We demonstrate that when G / H G/H is not irreducible, it is necessarily even dimensional and noncompact. Furthermore, the subgroup H H is also both noncompact and non-semisimple. Additionally, we establish that the only G G -invariant connection on G / H G/H is the canonical connection. On the other hand, we show that if G / H G/H has an odd dimension, it must be irreducible, and the subgroup H H must be semisimple. Finally, we present an explicit example, and we show that there exists no other torsion-free G G -invariant connection on a symmetric space G / H G/H with semisimple Lie group G G which has the same curvature as the canonical one.

Metrizability of SO(3)-invariant connections: Riemann versus Finsler

Abstract For a torsion-free affine connection on a given manifold, which does not necessarily arise as the Levi-Civita connection of any pseudo-Riemannian metric, it is still possible that it corresponds in a canonical way to a Finsler structure; this property is known as Finsler (or Berwald-Finsler) metrizability.
In the present paper, we clarify, for 4-dimensional SO(3)-invariant, Berwald-Finsler metrizable connections, the issue of the existence of an affinely equivalent pseudo-Riemannian structure. In particular, we find all classes of SO (3)-invariant connections which are not Levi-Civita connections for any pseudo-Riemannian metric - hence, are non-metric in a conventional way - but can still be metrized by SO(3)-invariant Finsler functions. The implications for physics, together with some examples are briefly discussed.

Covariant Derivatives on Homogeneous Spaces: Horizontal Lifts and Parallel Transport

We consider invariant covariant derivatives on reductive homogeneous spaces corresponding to the well-known invariant affine connections. These invariant covariant derivatives are expressed in terms of horizontally lifted vector fields on the Lie group. This point of view allows for a characterization of parallel vector fields along curves. Moreover, metric invariant covariant derivatives on a reductive homogeneous space equipped with an invariant pseudo-Riemannian metric are characterized. As a by-product, a new proof for the existence of invariant covariant derivatives on reductive homogeneous spaces and their the one-to-one correspondence to certain bilinear maps is obtained.

КОНЦЕПЦІЯ ГЕНЕТИЧНОГО ЯДРА В СТРУКТУРНІЙ ОРГАНІЗАЦІЇ І ЕВОЛЮЦІЇ СКЛАДНИХ ЕЛЕКТРОМЕХАНІЧНИХ СИСТЕМ

The interdisciplinary aspect in the study of the core concept in complex genetically organized systems of natural and anthropogenic origin is analyzed. The purpose of the article is scientific substantiation of the nature and essence of the genetic core of the electromechanical system. Based on the provisions of the theory of genetic evolution of electromechanical systems, the information essence of the genetic core as a carrier of genetic information of the primary source of the electromagnetic field is revealed. The analysis of the invariant relationships of the genetic nucleus with the structure of groups, subgroupsand small periods, the generative system was carried out. An invariant connection between the core concept and the processes of evolutionary speciation and the main taxonomic categories of electromechanical systems has been established. Based on the results of the system-genetic analysis, a definition of the concept of the genetic core is proposed. The relationship between genetic and energy cores in the hierarchy of levels of complexity of electromechanical systems is studied. The principles of genetic structuring of complex systems of themultinuclear type are revealed. According to the results of the system-genetic analysis of the conception of the genetic core of electro-mechanical systems, its main properties are summarized. The im-portance of the obtained research results for the spreadof genetic prediction technology and interdisciplinary synthesis to complex electromechanical complexes with nuclei of different physical nature is emphasized

Stretched non-positive Weyl connections on solvable Lie groups

We determine the structure of solvable Lie groups endowed with invariant stretched non-positive Weyl connections and find classes of solvable Lie groups admitting and not admitting such connections. In dimension 4 we fully classify solvable Lie groups which admit invariant SNP Weyl connections.

GENERAL STRATEGIC CITY PLANNING DESIGNING AND PROBLEMS OF PLANNING MANAGEMENT FOR SETTLING (the second part : problems, hindrances concerning their solution, actual targets and technological demands to the content of city planning documentations)

GENERAL STRATEGIC CITY PLANNING DESIGNING AND PROBLEMS OF PLANNING MANAGEMENT FOR SETTLING (the second part : problems, hindrances concerning their solution, actual targets and technological demands to the content of city planning documentations)

Homogeneous spaces with invariant Koszul-Vinberg structures

Let M be a manifold endowed with a flat torsionless connection ∇. A symmetric bivector field h on (M,∇) is said to be a Koszul-Vinberg (K-V for short) bivector field if it satisfies: (∇α#h)(β,γ)=(∇β#h)(α,γ), where α#:=h#(α) and h#:T⁎M→TM is the contraction of h. The pair (∇,h) is called a K-V structure and the triple (M,∇,h) is called a K-V manifold. When h is non-degenerate, (M,∇,h−1) is a pseudo-Hessian manifold, otherwise, we have a singular foliation defined by Imh# called the associated affine foliation whose leaves are pseudo-Hessian manifolds. In this paper, we study invariant K-V structures on homogeneous spaces. More precisely, we give an algebraic characterization of these structures in the same spirit of Nomizu's theorem on invariant connections on homogeneous spaces, and we give many classes of examples mainly for reductive or symmetric pairs (G,H). We establish many properties of pseudo-Hessian homogeneous manifolds, in particular, we show that when G is a semi-simple Lie group then G/H does not admit any non trivial G-invariant pseudo-Hessian structure. Finally, we show that the leaves of the affine foliation associated to an invariant K-V structure are homogeneous pseudo-Hessian manifolds.

Flat symplectic Lie algebras

Let be a symplectic Lie group, i.e., a Lie group endowed with a left invariant symplectic form. If is the Lie algebra of G then we call a symplectic Lie algebra. The product on defined by extends to a left invariant connection on G which is torsion free and symplectic (. When has vanishing curvature, we call a flat symplectic Lie group and a flat symplectic Lie algebra. In this paper, we study flat symplectic Lie groups. We start by showing that the derived ideal of a flat symplectic Lie algebra is degenerate with respect to ω. We show that a flat symplectic Lie group must be nilpotent with degenerate center. This implies that the connection of a flat symplectic Lie group is always complete. We prove that the flat symplectic double extension process can be applied to characterize all flat symplectic Lie algebras. More precisely, we show that every flat symplectic Lie algebra is obtained by a sequence of flat symplectic double extension of flat symplectic Lie algebras starting from {0}. As examples in low dimensions, we classify all flat symplectic Lie algebras of dimension .

Topological invariant and domain connectivity in moiré materials

Recently, a moir\'e material has been proposed in which multiple domains of different topological phases appear in the moir\'e unit cell due to a large moir\'e modulation. Topological properties of such moir\'e materials may differ from those of the original untwisted layered material. In this paper, we study how the topological properties are determined in moir\'e materials with multiple topological domains. We show a correspondence between the topological invariant of moir\'e materials at the Fermi level and the topology of the domain structure in real space. We also find a bulk-edge correspondence that is compatible with a continuous change in the truncation condition, which is specific to moir\'e materials. We demonstrate these correspondences in the twisted Bernevig-Hughes-Zhang model that describes general ${\mathbb{Z}}_{2}$-indicated topological insulator phases, by tuning its domain structure in the moir\'e unit cell. The obtained result can only be used to determine the topology at the Fermi level in the charge-neutral condition, but it can be combined with the traditional Wilson loop analysis to determine topologies of other gaps around. These results give a feasible method to evaluate a topological invariant for all occupied bands of a moir\'e material and contribute to the design of topological moir\'e materials and devices.

Infinite families of homogeneous Bismut Ricci flat manifolds

Starting from compact symmetric spaces of inner type, we provide infinite families of compact homogeneous spaces carrying invariant non-flat Bismut connections with vanishing Ricci tensor. These examples turn out to be generalized symmetric spaces of order [Formula: see text] and (up to coverings) they can be realized as minimal submanifolds of the Bismut flat model spaces, namely compact Lie groups. This construction generalizes the standard Cartan embedding of symmetric spaces.

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}.

Bismut Ricci flat manifolds with symmetries

We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way that the generalized Alekseevsky–Kimelfeld theorem does not hold. The classification of compact homogeneous Bismut Ricci flat spaces in dimension$5$is also provided. Moreover, we investigate compact homogeneous spaces with non-trivial third Betti number, and we point out other possible ways to construct Bismut Ricci flat manifolds. Finally, since Bismut Ricci flat connections correspond to fixed points of the generalized Ricci flow, we discuss the stability of some of our examples under the flow.

Invariant connections on a Whitney sum of bundles

Invariant connections on a Whitney sum of bundles

Interaction of Codazzi Pairs with Almost Para Norden Manifolds

In this paper, we research some properties of Codazzi pairs on almost para Norden manifolds. Let $(M_{2n},\ \varphi ,\ g,G)$ be an almost para Norden manifold. Firstly, $g$-conjugate connection, $G$-conjugate connection and $\varphi $-conjugate connection of a linear connection $\mathrm{\nabla }$ on $M_{2n}$ denoted by ${\mathrm{\nabla }}^{*\
 },\ {\mathrm{\nabla }}^{\dagger \ }$ and ${\mathrm{\nabla }}^{\varphi \ }$ are defined and it is demonstrated that on the spaces of linear connections, $\left(id,\ *,\dagger ,\varphi \right)$ acts as the four-element Klein group. We also searched some properties of these three types conjugate
 connections. Then, Codazzi pairs $\left(\mathrm{\nabla },\varphi \right)\ ,\left(\mathrm{\nabla },g\right)$ and $\left(\mathrm{\nabla },G\right)$ are introduced and some properties of them are given. Let $R\ ,\ R^{*\ }$and $R^{\dagger \ }$are $(0,4)$-curvature tensors of conjugate connections
 $\mathrm{\nabla }\mathrm{\ ,\ }{\mathrm{\nabla }}^{*\ }$and ${\mathrm{\nabla }}^{\dagger \ }$, respectively. The relationship among the curvature tensors is investigated. The condition of $N_{\varphi }=0$ is obtained, where $N_{\varphi }$ is Nijenhuis tensor field on $M_{2n}$ and it is known
 that the condition of integrability of almost para complex structure $\varphi $ is $N_{\varphi }=0$. In addition, Tachibana operator is applied to the pure metric $g$ and a necessary and sufficient condition $\left(M,\varphi ,\ g,G\right)$ being a para Kahler Norden manifold is found. Finally, we examine $\varphi $-invariant linear connections and statistical manifolds.

Homogeneous surfaces admitting invariant connections

We compute all the simply connected homogeneous and infinitesimally homogeneous surfaces admitting one or more invariant affine connections. We find exactly six non equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces admitting exactly one invariant connection.

IDENTICAL SEMANTICS OF SENTENCE CONFIGURATION IN RUSSIAN AND GERMAN

Russian as a foreign language. When describing a foreign language for learning purposes, it is necessary to present all language units in their most important functions. When describing a language unit, it is necessary to indicate all its valences. This gives a fairly complete picture of the use of the word. The comparison of a foreign language with a native one is becoming increasingly important. At the initial stage of mastering a foreign language, a student cannot do without relying on his native language. When formulating statements, practically when translating from the native language into Russian, linguistic interference occurs. Comparative linguistics data should help the teacher predict possible interference for the effectiveness of the process of teaching a foreign language. As a result, it should be noted that a review is carried out and the effectiveness of a comprehensive and functional approach to clarifying the valency of the predicate in the sentences of Russian and German is substantiated, and the integrity of the consideration of the sentence here is realized by comparing reality, the individual elements of which are in certain invariant connections and relationships, with the corresponding structure of thinking, displaying these elements invariantly, which are expressed as connected members of the sentence and with the valency structure of the predicate.

Elementary relative tractor calculus for Legendrean contact structures

For a manifold $M$ endowed with a Legendrean (or Lagrangean) contact structure $E\oplus F \subset TM$, we give an elementary construction of an invariant partial connection on the quotient bundle $TM/F$. This permits us to develop a naïve version of relative tractor calculus and to construct a second order invariant differential operator, which turns out to be the first relative BGG operator induced by the partial connection.