Dmitry A. Berdinsky, Iskander A. Taimanov
We derive the Weierstrass (or spinor) representation for surfaces in three-dimensional Lie groups Nil, \tilde{SL}_2, and Sol with Thurston's geometries and establish the generating equations for minimal surfaces in these groups. By using the spectral properties of the corresponding Dirac operators we find analogs of the Willmore functional for surfaces in these geometries.
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In the present paper we study the Lie sphere geometry of Legendre surfaces by the method of moving frame and we prove an existence theorem for real-analytic Lie-minimal Legendre surfaces.
We propose a twistor construction of surfaces in Lie sphere geometry based on the linear system which copies equations of Wilczynski's projective frame. In the particular case of Lie-applicable surfaces this linear system describes joint eigenfunctions of a pair of commuting Schrödinger operators with magnetic fields.
A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a \textit{$t$-symplectic Lie groupoid}; the "$t$" is motivated by the fact that each target fiber of a $t$-symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid $\mathcal{G}\rightrightarrows M$, we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid structures on $A\mathcal{G}$ (the associated …
We show that any left invariant metric with harmonic curvature on a solvable Lie group is Ricci-parallel. We show the same result for any Lie group of dimension $\leq$ 6.
We investigate curved flats in Lie sphere geometry. We show that in this setting curved flats are in one-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces related by Darboux transformation.
The conditions for a cuspidal edge, swallowtail and other fundamental singularities are given in the context of Lie sphere geometry. We then use these conditions to study the Lie sphere transformations of a surface.
We discuss channel surfaces in the context of Lie sphere geometry and characterise them as certain $Ω_{0}$-surfaces. Since $Ω_{0}$-surfaces possess a rich transformation theory, we study the behaviour of channel surfaces under these transformations. Furthermore, by using certain Dupin cyclide congruences, we characterise Ribaucour pairs of channel surfaces.
We give a detailed account of the gauge-theoretic approach to Lie applicable surfaces and the resulting transformation theory. In particular, we show that this approach coincides with the classical notion of $Ω$- and $Ω_{0}$-surfaces of Demoulin.
Using the gauge theoretic approach for Lie applicable surfaces, we characterise certain subclasses of surfaces in terms of polynomial conserved quantities. These include isothermic and Guichard surfaces of conformal geometry and $L$-isothermic surfaces of Laguerre geometry. In this setting one can see that the well known transformations available for these surfaces are induced by the transformations of the underlying Lie applicable surfaces. We also consider linear Weingarten surfaces in this setting and develop a new Bäcklund-type transformation for these surfaces.
Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie algebras and compact Lie algebras we recover by simple methods interesting co-homological known results. The Lie algebra of prederivations encompasses the one of derivations as a subalgebra. We find out that Lie algebras of cotangent Lie groups (which are not semi-simple) of sem…
Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we call current Lie groupoids. We then study basic differential geometry and Lie theory for these Lie g…
We discuss a Moser type argument to show when a deformation of a Lie group homomorphism and of a Lie subgroup is trivial. For compact groups we obtain stability results.
In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this Hexp map. We also describe a Hom-Lie group action on a smooth manifold. Subsequently, we give the notion of an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie algebra $(\mathfrak{gl}(V),[\cdot,\cdot],\mathsf{Ad})$, and the …
In this paper we will show the existence and uniqueness of the solution of the Björling problem for minimal surfaces in a 3-dimensional Lorentzian Lie group.
We discuss the basic properties of Lie groupoids, Lie algebroids and Lie pseudo-groups in view of applying these techniques to the analysis of Jordan-Hölder resolutions and, subsequently, to the integration of partial differential equations. The present introduction is an extension to Lie groupoids, as far as possible, of the so well known properties and techniques much useful in Lie groups theory. We mention as far as possible since, in the case of Lie groupoids, just the first Lie Theorem holds. As for the prolongation algorithm, it is extremely useful when dealing with groupoids whereas rat…
This is a survey of results on surfaces in noncommutative three-dimensional Lie groups obtained by using the Weierstrass (spinor) representation of surfaces. It is based on the talk given at the conference "Geometry related to the theory of integrable systems" (RIMS, Kyoto, September 2007).
We generalize the Uhlenbeck-Segal theory for harmonic maps into compact semi-simple Lie groups to general Lie groups equipped with torsion free bi-invariant connection.
The jet bundle $J^kG$ of $k$-jets of curves in a Lie group $G$ has a natural Lie group structure. We present an explicit formula for the group multiplication in the right trivialization and for the group 2-cocycle describing the abelian Lie group extension $\mathfrak{g}\to J^{k}G\to J^{k-1}G$.
Groupoids are mathematical structures able to describe symmetry properties more general than those described by groups. They were introduced (and named) by H. Brandt in 1926. Around 1950, Charles Ehresmann used groupoids with additional structures (topological and differentiable) as essential tools in topology and differential geometry. In recent years, Mickael Karasev, Alan Weinstein and Stanisław Zakrzewski independently discovered that symplectic groupoids can be used for the construction of noncommutative deformations of the algebra of smooth functions on a manifold, with potential applica…
Isometry groups of simply connected nonunimodular Lie groups of dimension four
本研究针对四维单连通非幺模李群上的所有左不变黎曼度量，确定了各自对应的完整等距群，推进了李群几何领域的相关研究。
Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisingly far the geometry of principal bundles: parallel transport exists and flat connections integrate to horizontal foliations as in finite dimensions. As consequences we obtain that Lie algebra homomorphisms intergrate to Lie group homomorphisms, if the source group is simply con…
Three explicit examples of Lie 2-groupoids
该研究遵循Mehta与Tang的结论，借助Artin和Mazur的棒构造，从三类双李群胚构造出三个显式李2群胚，丰富了李2群胚的具体实例。
In this paper we study contact structure on 2-step nilpotent, Heisenberg type Lie groups. We decompose this Lie groups to center and orthogonal complement, then investigate properties of both orthogonal Lie subgroups. Finally, we provide a connection between matchings in groups and field extensions and 2-step nilpotent Lie groups.
Tanaka-Webster curvatures of surfaces in 3D Lie groups with a CR structure
研究具有CR结构的3D李群中嵌入曲面的Tanaka-Webster几何，定义Gauss曲率和平均曲率并给出具体例子。
We give the expression of the metric derived from Lie groups. For the metric derived from classical Lie groups such as the unitary group, the orthogonal group and the symplectic group, we conjecture that the metric becomes the Einstein metric.
In this paper, we completely classify three-dimensional Lorentzian $Ein(2)$ Lie groups.
We introduce and study some mixed product Poisson structures on product manifolds associated to Poisson Lie groups and Lie bialgebras. For quasitriangular Lie bialgebras, our construction is equivalent to that of fusion products of quasi-Poisson G-manifolds introduced by Alekseev, Kosmann- Schwarzbach, and Meinrenken. Our primary examples include four series of holomorphic Poisson structures on products of flag varieties and related spaces of complex semi-simple Lie groups.
Using vertical and complete lifts, any left invariant Riemannian metric on a Lie group induces a left invariant Riemannian metric on the tangent Lie group. In the present article we study the Riemannian geometry of tangent bundle of two families of Lie groups. The first one is the family of special Lie groups considered by J. Milnor and the second one is the class of Lie groups with one-dimensional commutator groups. The Levi-Civita connection, sectional and Ricci curvatures have been investigated.
We describe the full group of isometries of absolutely simple, compact, connected real Lie groups, of SO(4) and of U(n), endowed with suitable bi-invariant Riemannian metrics.
We study Wick-rotations of left-invariant metrics on Lie groups, using results from real GIT (\cite{1}, \cite{2}, \cite{3}). An invariant for Wick-rotation of Lie groups is given, and we describe when a pseudo-Riemannian Lie group can be Wick-rotated to a Riemannian Lie group. We also prove a general version (for general Lie algebras) of $\acute{E}$. Cartan's result, namely the existence and conjugacy of Cartan involutions.
Isometry groups of simply connected unimodular 4-dimensional Lie groups
本文刻画四维单连通幺模幂零或可解 $(R)$ 型李群上任意左不变黎曼度量的完整等距群。
A Lie group is called orthogonal if it carries a bi-invariant pseudo Riemannian metric. Oscillator Lie groups constitutes a subclass of the class of orthogonal Lie groups. In this paper, we determine the Lie bialgebra structures and the solutions of the classical Yang-Baxter equation on a generic class of oscillator Lie groups. On the other hand, we show that any solution of the classical Yang-Baxter equation on an orthogonal Lie group induces a metric in the dual Lie groups associated to this solution. This metric is geodesically complete if and only if the dual are unimodular. More generally…
We define and study the invariant linear and nonlinear horizontal double complexes of a local Lie group.
There are studied Lie groups considered as almost hypercomplex Hermitian-Norden manifolds, which are integrable and have the lowest dimension four. It is established a correspondence of the derived Lie algebras of types of invariant hypercomplex structures and the explicit matrix representation of their Lie groups. There are constructed examples of the considered structure of different types on some known Lie groups.
In this paper, we study the Ricci-Bourguignon flow on higher dimensional classical Heisenberg nilpotent Lie groups and construct a solution of this flow on Heisenberg and quaternion nilpotent Lie groups. In the end, we investigate the deformation of spectrum and length spectrum on compact nilmanifolds obtained of Heisenberg and quaternion nilpotent Lie groups.
Lie minimal Weingarten surfaces
本文研究黎曼空间形式中 Lie 球面不变能量的临界点 Lie 极小曲面，通过 Euler-Lagrange 方程刻画其主曲率，并证明常平均曲率曲面及平坦外围空间中满足仿射关系的曲面和椭圆线性 Weingarten 曲面均具有旋转性。
We show that for any coboundary Poisson Lie group G, the Poisson structure on G^* is linearizable at the group unit. This strengthens a result of Enriquez-Etingof-Marshall, who had established formal linearizability of G^* for quasi-triangular Poisson Lie groups G. We also prove linearizability properties for the group multiplication in G^* and for Poisson Lie group morphisms, with similar assumptions.
The notions of \emph{Poisson Lie group} and \emph{Poisson homogeneous space} are extended to the Dirac category. The theorem of Drinfel$'$d (\cite{Drinfeld93}) on the one-to-one correspondence between Poisson homogeneous spaces of a Poisson Lie group and a special class of Lagrangian subalgebras of the Lie bialgebra associated to the Poisson Lie group is proved to hold in this more general setting.
This thesis treats two main topics: calibrated symplectic foliations, and local Lie groupoids. Calibrated symplectic foliations are one possible generalization of taut foliations of 3-manifolds to higher dimensions. Their study has been popular in recent years, and we collect several interesting results. We then show how de Rham's theory of currents, and Sullivan's theory of structure currents, can be applied in trying to understand the calibratability of symplectic foliations. Our study of local Lie groupoids begins with their definition and an exploration of some of their basic properties. N…
Four-dimensional pp-wave Lie groups and harmonic curvature
本文研究了四维pp波李群与调和曲率的关系，确定了具有调和曲率的四维李群结构。
In this note, we completely classify left-invariant Riemann solitons on three-dimensional Lorentzian Lie groups.
In this paper, we denone the generalized bicomplex numbers and give some algebraic properties of them. Also, we show that some hyperquadrics in R4 and R42 are Lie groups by using generalized bicomplex number product and obtain Lie algebras of these Lie groups. Morever, by using tensor product surfaces, we determine some special Lie subgroups of these hyperquadrics.
We study relation of the Ricci Flow on 3-dimensional Lie groups and 4-dimensional Ricci-flat manifolds. In particular, we construct Ricci-flat cohomogeneity one metrics with respect to 3-dimensional Lie groups.
To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannian metric. We give a useful necessary and sufficient condition that guaranties the flatness of a left invariant semi Riemannian metric defined on a quadratic Lie group. All these semi Riemannian metrics are complete. We show that there are no Riemannian or Lorentzian flat left inv…
We construct harmonic morphisms on the compact simple Lie group G2. The construction uses eigenfamilies in a representation theoretic scheme.
In this paper we classify those three-dimensional Riemannian Lie groups which admit harmonic morphisms to surfaces.
In this note we construct an infinite-dimensional Lie group structure on the group of vertical bisections of a regular Lie groupoid. We then identify the Lie algebra of this group and discuss regularity properties (in the sense of Milnor) for these Lie groups. If the groupoid is locally trivial, i.e. a gauge groupoid, the vertical bisections coincide with the gauge group of the underlying bundle. Hence the construction recovers the well known Lie group structure of the gauge groups. To establish the Lie theoretic properties of the vertical bisections of a Lie groupoid over a non-compact base, …
In this article we endow the group of bisections of a Lie groupoid with compact base with a natural locally convex Lie group structure. Moreover, we develop thoroughly the connection to the algebra of sections of the associated Lie algebroid and show for a large class of Lie groupoids that their groups of bisections are regular in the sense of Milnor.
Motivated by a number of recent investigations, we define and investigate the various properties of the ruled surfaces depend on three dimensional Lie groups with a bi-variant metric. We give useful results involving the characterizations of these ruled surfaces. Some special ruled surfaces such as normal surface, binormal surface, tangent developable surface, rectifying developable surface and Darboux developable surface are worked. From those applications, we make use of such a work to interpret the Gaussian, mean curvatures of these surfaces and geodesic, normal curvature and geodesic torsi…
We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left canonical action of the group on itself is Hamiltonian. The principal tool used for our description is a canonical affine structure associated with the symplectic form. We also characterize the Hamiltonian symplectic Lie groups among the connected symplectic Lie groups. We special…
The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π_2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integra…
We extend the Nambu bracket to 1-forms. Following the Poisson-Lie case, we define Nambu-Lie groups as Lie groups endowed with a multiplicative Nambu structure. A Lie group G with a Nambu structure P is a Nambu-Lie group iff P=0 at the unit and the Nambu bracket of left (right) invariant forms is left (right) invariant. We define a corresponding notion of a Nambu-Lie algebra. We give several examples of Nambu-Lie groups and algebras.
We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.
In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and Lie groups in that their second groups classify extensions. We use this fact together with an adapted van Est map to prove the integrability of Lie 2-algebras anew.
We introduce two basic invariant forms which define generic surface in 3-space uniquely up to Lie sphere equivalence. Two particularly interesting classes of surfaces associated with these invariants are considered, namely, the Lie-minimal surfaces and the diagonally-cyclidic surfaces. For diagonally-cyclidic surfaces we derive the stationary modified Veselov-Novikov equation, whose role in the theory of these surfaces is similar to that of Calapso's equation in the theory of isothermic surfaces. Since Calapso's equation itself turns out to be related to the stationary Davey-Stewartson equatio…
We define stacky Lie groups to be group objects in the 2-category of differentiable stacks. We show that every connected and etale stacky Lie group is equivalent to a crossed module of the form (H,G) where H is the fundamental group of the given stacky Lie group and G is the connected and simply connected Lie group integrating the Lie algebra of the stacky group. Our result is closely related to a strictification result of Baez and Lauda.
In the present paper, we describe two geometric notions, holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups, and prove that on Hom-Lie groups in the left invariant setting, these structures are related to each other. We study Kähler-Norden structures with abelian complex structures and give the curvature properties of holomorphic Norden structures on Hom-Lie groups. Finally, we show that any left-invariant holomorphic Hom-Lie group is a flat (holomorphic Norden Hom-Lie algebra carries a Hom-Left-symmetric algebra) if its left-invariant complex structure (complex struc…
In this article we define the twisted product of groups as the generalization of the semidirect product of groups. We will find the necessary and sufficient condition in order that the twisted product of groups to be a group. In particular, for two copies of the same group, the twisted product of group by itself through the action of inner automorphisms is a group if and only if the initial group is a metabelian group. Further we will construct Lie algebra for Lie group of a twisted product of Lie groups. In the case of twisted product of Lie group by itself by means of the action of inner aut…
In this survey, we discuss a series of linearization problems--for Poisson structures, Lie algebroids, and Lie groupoids. The last problem involves a conjecture on the structure of proper groupoids. Attempting to prove this by the method of averaging leads to problems concerning almost actions of compact groups and almost invariant submanifolds for compact group actions. The paper ends with a discussion of possible extensions of the convexity theorems for momentum maps of hamiltonian actions of compact groups.
We develop the structure theory of symplectic Lie groups based on the study of their isotropic normal subgroups. The article consists of three main parts. In the first part we show that every symplectic Lie group admits a sequence of subsequent symplectic reductions to a unique irreducible symplectic Lie group. The second part concerns the symplectic geometry of cotangent symplectic Lie groups and the theory of Lagrangian extensions of flat Lie groups. In the third part of the article we analyze the existence problem for Lagrangian normal subgroups in nilpotent symplectic Lie groups.
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