Dimensional Lie Groups Research Articles

On flat manifold bundles and the connectivity of Haefliger’s classifying spaces

We investigate a conjecture due to Haefliger and Thurston in the context of foliated manifold bundles. In this context, Haefliger-Thurston’s conjecture predicts that every M M -bundle over a manifold B B where dim ⁡ ( B ) ≤ dim ⁡ ( M ) \operatorname {dim}(B)\leq \operatorname {dim}(M) is cobordant to a flat M M -bundle. In particular, we study the bordism class of flat M M -bundles over low dimensional manifolds, comparing a finite dimensional Lie group G G with D i f f 0 ( G ) \mathrm {Diff}_0(G) .

Systems of first order ordinary differential equations allowing a given 3-dimensional Lie group as a subgroup of their symmetry group

AbstractWe determine systems of the first order ordinary differential equations such that their group of symmetries contains a three-dimensional Lie subgroup G. We represent the basis vectors of the Lie algebra $$\mathfrak {g}$$ g of G by vector fields in the three-dimensional real space. Two cases are distinguished according to whether the infinitesimal generators of $$\mathfrak {g}$$ g do not contain any component or contain component with respect to the independent variable of the system.

Geometry of infinite dimensional Cartan developments

The Cartan development takes a Lie algebra valued 1-form satisfying the Maurer–Cartan equation on a simply connected manifold [Formula: see text] to a smooth mapping from [Formula: see text] into the Lie group. In this paper, this is generalized to infinite dimensional [Formula: see text] for infinite dimensional regular Lie groups. The Cartan development is viewed as a generalization of the evolution map of a regular Lie group. The tangent mapping of a Cartan development is identified as another Cartan development.

Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

Let M = ( M , ω ) M=(M,\omega ) be either S 2 × S 2 S^2 \times S^2 or C P 2 # C P 2 ¯ \mathbb {C}P^2\# \overline {\mathbb {C}P^2} endowed with any symplectic form ω \omega . Suppose a finite cyclic group Z n \mathbb {Z}_n is acting effectively on ( M , ω ) (M,\omega ) through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism Z n ↪ H a m ( M , ω ) \mathbb {Z}_n\hookrightarrow Ham(M,\omega ) . In this paper, we investigate the homotopy type of the group S y m p Z n ( M , ω ) Symp^{\mathbb {Z}_n}(M,\omega ) of equivariant symplectomorphisms. We prove that for some infinite families of Z n \mathbb {Z}_n actions satisfying certain inequalities involving the order n n and the symplectic cohomology class [ ω ] [\omega ] , the actions extend to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on J J -holomorphic techniques, on Delzant’s classification of toric actions, on Karshon’s classification of Hamiltonian circle actions on 4 4 -manifolds, and on the Chen-Wilczyński classification of smooth Z n \mathbb {Z}_n -actions on Hirzebruch surfaces.

Classification of contact seaweeds

A celebrated result of Gromov ensures the existence of a contact structure on any connected, non-compact, odd-dimensional Lie group. In general, such structures are not invariant under left translation. The problem of finding which Lie groups admit a left-invariant contact structure resolves to the question of determining when a Lie algebra g is contact; that is, admits a one-form φ∈g⁎ such that φ∧(dφ)k≠0.In full generality, this remains an open question; however we settle it for the important category of the evocatively named seaweed algebras by showing that an index-one seaweed is contact precisely when it is quasi-reductive. Seaweeds were introduced by Dergachev and Kirillov who initiated the development of their index theory – since completed by Joseph, Panyushev, Yakimova, and Coll, among others. Recall that a contact Lie algebra has index one – but not characteristically so. Leveraging recent work of Panyushev, Baur, Moreau, Duflo, Khalgui, Torasso, Yakimova, and Ammari, who collectively classified quasi-reductive seaweeds, our equivalence yields a full classification of contact seaweeds. We remark that since type-A and type-C seaweeds are de facto quasi-reductive (by a result of Panyushev), in these types index one alone suffices to ensure the existence of a contact form.

Homogeneous Geodesics of $4$-dimensional Solvable Lie Groups

We study homogeneous geodesics in $4$-dimensional solvable Lie groups $\mathrm{Sol}_0^4$, $\mathrm{Sol}_1^4$, $\mathrm{Sol}_{m,n}$ and $\mathrm{Nil}_4$.

Comments on Parallel Curves in 3-Dimensional Lie Group G

In this study, firstly, basic concepts in 3-dimensional Euclidean space and basic information about curves are given and some special curves are examined. Then, basic information about Lie algebra and Lie group basic concepts and curves are given and special curves such as helix, involute-evolute, Bertrand, Mannheim, Smarandache, are defined. Secondly, inspired by these special curves examined in the Lie group, the definitions of the parallel curve in the vector direction, the parallel curve in the direction of the vector B and the parallel curve in the direction of the linear combination of the vectors B and N of a curve according to the Frenet frame are given and characterized. Some theorems and results are obtained by finding the Frenet apparatus of these characterized curves. Finally, the findings are examined in more specific circumstances and new results are found.

Three dimensional Lie groups of scalar Randers type

Three dimensional Lie groups of scalar Randers type

Extended finite similitude and dimensional analysis for scaling

The theory of scaling called finite similitude does not involve dimensional analysis and is founded on a transport-equation approach that is applicable to all of classical physics. It features a countable infinite number of similitude rules and has recently been extended to other types of governing equations (e.g., differential, variational) by the introduction of a scaling space Omega _{beta }, within which all physical quantities are deemed dependent on a single dimensional parameter beta . The theory is presently limited to physical applications but the focus of this paper is its extension to other quantitative-based theories such as finance. This is achieved by connecting it to an extended form of dimensional analysis, where changes in any quantity can be associated with curves projected onto a dimensional Lie group. It is shown in the paper how differential similitude identities arising out of the finite similitude theory are universal in the sense they can be formed and applied to any quantitative-based theory. In order to illustrate its applicability outside physics the Black-Scholes equation for option valuation in finance is considered since this equation is recognised to be similar in form to an equation from thermal physics. It is demonstrated that the theory of finite similitude can be applied to the Black-Scholes equation and more widely can be used to assess observed size effects in portfolio performance.

On the dimension of the algebras of local infinitesimal isometries of 3-dimensional special sub-Riemannian manifolds

Suppose that we are given a contact sub-Riemannian manifold (M, H, g) of dimension 3 such that the Reeb vector field is an infinitesimal isometry (such manifolds will be referred to as special). For a point qin M denote by mathfrak {i}^*(q) the Lie algebra of germs at q of infinitesimal isometries of (M, H, g). It is proved that for a generic point qin M, dim mathfrak {i}^*(q) can only assume the values 1, 2, 4. Moreover, dim mathfrak {i}^*(q) = 4 if and only if the curvature function determined by the canonical sub-Riemannian connection is constant in a neighborhood of q. The latter case is possible if (M, H, g) is locally isometric, in a neighborhood of q, to a left-invariant sub-Riemannian structure on a 3-dimensional Lie group.

Cohomogeneity one central Kähler metrics in dimension four

Abstract A Kähler metric is called central if the determinant of its Ricci endomorphism is constant; see [12]. For the case in which this constant is zero, we study on 4-manifolds the existence of complete metrics of this type which have cohomogeneity one for three unimodular 3-dimensional Lie groups: SU(2), the group E(2) of Euclidean plane motions, and a quotient by a discrete subgroup of the Heisenberg group nil3. We obtain a complete classification for SU(2), and some existence results for the other two groups, in terms of specific solutions of an associated ODE system.

B_{n}-Generalized Pseudo-Kähler Structures

We define the notions of B_{n}-generalized pseudo-Hermitian and B_{n}-generalized pseudo-Kähler structure on an odd exact Courant algebroid E. When E is in the standard form (or of type B_{n}) we express these notions in terms of classical tensor fields on the base of E. This is analogous to the bi-Hermitian viewpoint on generalized Kähler structures on exact Courant algebroids. We describe left-invariant B_{n}-generalized pseudo-Kähler structures on Courant algebroids of type B_{n} over Lie groups of dimension two, three and four.

Sweeping Surfaces Due to Conjugate Bishop Frame in 3-Dimensional Lie Group

In this work, we present a new Bishop frame for the conjugate curve of a curve in the 3-dimensional Lie group G3. With the help of this frame, we derive a parametric representation for a sweeping surface and show that the parametric curves on this surface are curvature lines. We then examine the local singularities and convexity of this sweeping surface and establish the sufficient and necessary conditions for it to be a developable ruled surface. Additionally, we provide detailed explanations and examples of its applications.

Magnetic unit vector fields

We show that a unit vector field on an oriented Riemannian manifold is a critical point of the Landau Hall functional if and only if it is a critical point of the Dirichlet energy functional. Therefore, we provide a characterization for a unit vector field to be a magnetic map into its unit tangent sphere bundle. Then, we classify all magnetic left invariant unit vector fields on $3$-dimensional Lie groups.

Szabos algorithm and applications

In this paper, Szabos algorithm is used as the main tool to find locally symmetric left invariant Riemannian metrics on some $4$-dimensional Lie groups. Locally symmetric left invariant Riemannian Lie groups constitute an important subclass of Riemannian Lie groups with zero-divergence Weyl-tensor the so-called C-manifolds. Some properties of the curvature operator of these 4-dimensional C-manifolds are studied.

Lie Grubunda Bir Eğri Boyunca Sabit Gauss Eğrilikli Yüzeyler Üzeine Notlar

In this study, sufficient conditions are derived and examples are created to derive surfaces with constant Gauss curvature along a given curve in terms of the linear combination of its Frenet frame in the 3-dimensional Lie group.

Ground state representations of topological groups

Let α:R→Aut(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha : {{\\mathbb {R}}}\\rightarrow \\mathop {\ extrm{Aut}}\ olimits (G)$$\\end{document} define a continuous R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}$$\\end{document}-action on the topological group G. A unitary representation (π♭,H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\pi ^\\flat ,\\mathcal {H})$$\\end{document} of the extended group G♭:=G⋊αR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G^\\flat := G \\rtimes _\\alpha {{\\mathbb {R}}}$$\\end{document} is called a ground state representation if the unitary one-parameter group π♭(e,t)=eitH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi ^\\flat (e,t) = e^{itH}$$\\end{document} has a non-negative generator H≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H \\ge 0$$\\end{document} and the subspace H0:=kerH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^0 := \\ker H$$\\end{document} of ground states generates H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document} under G. In this paper, we introduce the class of strict ground state representations, where (π♭,H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\pi ^\\flat ,\\mathcal {H})$$\\end{document} and the representation of the subgroup G0:=Fix(α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G^0 := \\mathop {\ extrm{Fix}}\ olimits (\\alpha )$$\\end{document} on H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^0$$\\end{document} have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of G0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G^0$$\\end{document}. This is particularly effective if the occurring representations of G0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G^0$$\\end{document} can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations.

Real semisimple Lie groups and balanced metrics

Given any non-compact real simple Lie group \mathrm{G}_o of inner type and even dimension, we prove the existence of an invariant complex structure \mathrm{J} and a Hermitian balanced metric on \mathrm{G}_o and on any compact quotient \mathrm{M}= \Gamma\backslash\mathrm{G}_o , with \Gamma a cocompact lattice. We also prove that (\mathrm{M},\mathrm{J}) does not carry any pluriclosed metric, in contrast to the case of even dimensional compact Lie groups, which admit pluriclosed but not balanced metrics.

On the Symmetric Einstein Equation for Three-Dimensional Lie Groups with Left-Invariant Riemannian Metric and Semi-Symmetric Connection

Riemannian manifolds with a Levi-Civita connection and constant Ricci curvature, or Einstein manifolds, were studied in the works of many mathematicians. This question has been most studied in the homogeneous Riemannian case. In this direction, the most famous ones are the results of works by D.V. Alekseevsky, M. Wang, V. Ziller, G. Jensen, H.Laure, Y.G. Nikonorov, E.D. Rodionov and other mathematicians. At the same time, the question of studying Einstein manifolds has been little studied for the case of an arbitrary metric connection. This is primarily due to the fact that the Ricci tensor of metric connection is not, generally speaking, symmetric.
 In this paper, we consider semisymmetric connections on 3-dimensional Lie groups with a left-invariant Riemannian metric. For the symmetric part of the Ricci tensor, the Einstein equation is studied. As a result of the research carried out, a classification of 3-dimensional metric Lie groups and the corresponding semisymmetric connections in the case of the Einstein symmetric equation has been obtained.
 Earlier in the works of P.N. Klepikov, E.D. Rodionov and O.P. Khromova, the classical Einstein equation was studied, and it was proved that if the classical.
 The Einstein equation holds for a 3-dimensional Lie group with left-invariant (pseudo) Riemannian metric and semi-symmetric connection. Then, either the connection is a Levi-Civita connection, or the curvature tensor of the connection is equal to zero.

Semigroups in 3-graded Lie groups and endomorphisms of standard subspaces

Let V be a standard subspace in the complex Hilbert space H and U : G \to U(H) be a unitary representation of a finite dimensional Lie group. We assume the existence of an element h in the Lie algebra of G such that U(exp th) is the modular group of V and that the modular involution J_V normalizes U(G). We want to determine the semigroup $S_V = \{ g\in G : U(g)V \subseteq V\}.$ In previous work we have seen that its infinitesimal generators span a Lie algebra on which ad h defines a 3-grading, and here we completely determine the semigroup S_V under the assumption that ad h defines a 3-grading. Concretely, we show that the ad h-eigenspaces for the eigenvalue $\pm 1$ contain closed convex cones $C_\pm$, such that $S_V = exp(C_+) G_V exp(C_-)$, where $G_V$ is the stabilizer of V in G. To obtain this result we compare several subsemigroups of G specified by the grading and the positive cone $C_U$ of U. In particular, we show that the orbit U(G)V, endowed with the inclusion order, is an ordered symmetric space covering the adjoint orbit $Ad(G)h$, endowed with the partial order defined by~$C_U$.