This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results determine Lie groups admitting flat bi-invariant affine or projective structures. Th…
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48 results for “flat Lie group”
Flat symplectic Lie algebrasmath.DG
The study of flat symplectic Lie algebras and groups.
problem Characterizing and understanding flat symplectic Lie algebras and groups.
method Analyzing the derived ideal, curvature, and double extension process.
result Every flat symplectic Lie algebra is obtained by a sequence of double extensions starting from the trivial algebra.
The paper identifies all flat CR Lie groups and their structures.
problem Identifying flat CR Lie groups and their structures.
method Analyzing Lie algebras and structures of Lie groups.
result Only specific Lie algebras are Cartan flat non-degenerate CR Lie algebras.
Complete description of flat Lorentzian Lie groups solved.
problem Long-standing open problem in pseudo-Riemannian Lie groups.
method Refined analysis of double extension process and explicit classification of Lie algebras.
result All flat Lorentzian Lie algebras arise from flat Euclidean ones.
A Lorentzian flat Lie group is a Lie group with a flat left invariant metric with signature . The Lie algebra of endowed with is called flat Lorentzian Lie algebra. It is known that the metric of a flat Lorentzian Lie group is geodesical…
Flat Hermitian Lie algebras are always Kähler.
problem Classifying Lie groups with Hermitian structures that are flat.
method Analysis on the Hermitian geometry of 2-step solvable Lie groups.
result Flat Hermitian Lie algebras are Kähler.
We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We characterize flat affine symplectic Lie groups using symplectic étale affine representations and as a consequence of this, we show that a …
In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita connection is flat. We show that no connected semisimple Lie group admits a left invarian…
We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. M…
Study flat connections on Courant algebroids using Lie groups.
problem Flatness conditions on Courant algebroids.
method Metric generalized connections, flatness condition analysis.
result Existence of compact simple Lie groups as building blocks for flat transitive Courant algebroids.
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 Riemannia…
We call the Lie algebra of a Lie group with a left invariant pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new proof of a classical result of Milnor on Riemannian flat Lie algebras. We reduce the study of Lorentzian flat Lie algebras to those with trivial center or those with degenerate ce…
New methods find Ricci-flat metrics on specific Lie groups.
problem Finding Ricci-flat metrics on Lie groups.
method Two constructions based on gradings and filtrations.
result Every nilpotent Lie algebra of dimensions up to 9 admits an indefinite Ricci-flat metric.
Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.
problem Computing Dolbeault cohomology for Levi-flat CR structures on compact Lie groups.
method Algebraic classification of left-invariant CR structures combined with Pittie's result on compact Lie groups.
result Generalization of Dolbeault cohomology computation to Levi-flat CR structures.
We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension , every nice nilpotent Lie group of dimension and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct inf…
The paper constructs a new Ricci-flat metric on almost abelian Lie groups.
problem Finding Lorentzian homogeneous Ricci-flat metrics on almost abelian Lie groups.
method Constructing left-invariant metrics on almost abelian Lie groups, focusing on dimensions four or higher.
result A new Ricci-flat metric that generalizes the Petrov solution to higher dimensions.
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.
Flat surfaces in Lie groups with constant curvature are flat.
problem Characterizing surfaces in Lie groups with constant Gaussian curvature.
method Analyzing surfaces as products of curves and using bi-invariant metrics.
result All surfaces of constant curvature in 3D Lie groups are flat.
We prove that any real Lie group of dimension \leq 5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension \leq 5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.
Study exact Lie bialgebras from flat Lie groups, classifying them.
problem Classifying Lie bialgebras from flat Lie groups.
method Analyzing noncommutative deformations of Poisson-Lie groups with a metric, solving the classical Yang-Baxter equation.
result Complete classification of exact metaflat Lie bialgebras.
This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection. Then we specialize the characterization to the case of a left invariant connecti…
Semisimple Lie groups act transitively on pseudo-Riemannian manifolds, making them flat.
problem Understanding actions of semisimple Lie groups on pseudo-Riemannian manifolds.
method Analyzing the pseudo-Riemannian Lichnerowicz conjecture in homogeneous settings.
result Compact pseudo-Riemannian manifolds on which a semisimple group acts conformally, essentially and transitively, are conformally flat.
Complete classification of para-Kähler structures on Lie groups.
problem Classifying para-Kähler structures on Lie groups.
method Complete classification via automorphism consideration.
result Classification of para-Kähler structures on four-dimensional Lie groups.
In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical Yang-Baxter equation given by Boucetta and Medina. On the other hand, using Koszul's method, we prove the existence of an immersion of L…
Characterizes hypercomplex Lie groups and their solvmanifolds.
problem Understanding hypercomplex structures on Lie groups and their solvmanifolds.
method Characterization of almost abelian Lie groups with hypercomplex structures, analysis of Obata and Bismut connections, classification of hypercomplex Lie groups, and construction of solvmanifolds.
result Classification of hypercomplex almost abelian Lie groups in dimension 8 and properties of their solvmanifolds.
A Lie group has a unique metric when viewed as a flat absolute parallelism.
problem Understanding different metrics on Lie groups and their curvature properties.
method Shifted perspective of viewing Lie groups as flat absolute parallelisms.
result Lie groups have a canonical metric when viewed in this new way.
We determine all Ricci flat left invariant Lorentzian metrics on simply connected 2-step nilpotent Lie groups. We show that the -dimensional Heisenberg Lie group carries a Ricci flat left invariant Lorentzian metric if and only if . We show also that for any , carries a R…
Study the spaces of flat connections for classical Lie groups using Chern-Weil theory.
problem Understanding the weak homotopy type of spaces of flat connections for classical Lie groups.
method Use Chern-Weil theory and relate to the functorial map involving continuous families of representations.
result Relate the spaces of flat connections to the weak homotopy type of the spaces of representations.
We establish the existence of solvable Lie groups of dimension 4 and left-invariant Riemannian metrics with zero Bach tensor which are neither conformally Einstein nor half conformally flat.
Half-flat SU(3)-structures are the natural initial values for Hitchin's evolution equations whose solutions define parallel G_2-structures. Together with the results of arXiv:0912.3486v1, the results of this article completely solve the existence problem of left-invariant half-flat SU(3)-structures on decomposable Lie …
New findings on Chern flat metrics and their criticality.
problem Understanding critical Hermitian metrics on Chern flat manifolds.
method Analyzing Chern flat manifolds as compact quotients of complex Lie groups and studying their criticality.
result Chern flat metrics on semi-simple Lie groups are torsion-critical and vice versa.
Study para-complex structures on specific Lie groups, finding explicit forms and properties.
problem Characterizing para-Kähler structures on six-dimensional nilpotent Lie groups.
method Examined left-invariant para-complex structures on six-dimensional nilpotent Lie groups, obtained explicit expressions and investigated curvature properties.
result Para-complex structures are nilpotent and para-Kähler metrics are Ricci-flat.
We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.
Constructs special Kähler structures on Lie groups.
problem Creating special Kähler structures on Lie groups.
method Introducing twisted cartesian product and double extension process.
result Characterizes left invariant flat special Kähler structures.
We show the correspondence between left invariant flat projective structures on Lie groups and certain prehomogeneous vector spaces. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex Lie groups admitting irreducible left invariant flat complex projective structures. As a r…
New complex manifolds found with flat structure.
problem Finding compact complex manifolds with flat affine structure.
method Using Lie groups with left-invariant complex structure.
result Retrieved Inoue surfaces in 2D.
This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-flat metrics on nilpotent Lie groups o…
We prove that any base space of Riemannian submersion from a compact Lie group (with bi-invariant metric) must have a basic property previously known for normal biquotients; namely, any zero-curvature plane exponentiates to a flat.
Defines and extends flat pseudo-Riemannian F-Lie algebras.
problem Generating weakly flat Lorentzian non-abelian bi-nilpotent F-Lie algebras.
method Constructs double extensions of flat pseudo-Riemannian F-Lie algebras.
result Provides a framework for generating all weakly flat Lorentzian non-abelian bi-nilpotent F-Lie algebras.
Study Lie bialgebra structures on flat metric Lie algebras, leading to explicit Poisson-Lie groups.
problem Understanding Lie bialgebra structures on flat metric Lie algebras.
method Splitting Lie algebras, establishing normal forms, and using invariant Schouten squares.
result Explicit construction of multiplicative Poisson tensors on flat Poisson-Lie groups.
Using canonical 1-parameter family of Hermitian connections on the tangent bundle, we provide invariant solutions to the Strominger system on complex Lie groups. Both flat and non-flat cases are discussed in detail.
Post-Lie algebra structure found on non-flat manifolds with curvature and torsion.
problem Understanding vector fields and endomorphisms on manifolds with curvature and torsion.
method Analyzing the post-Lie algebra structure of vector fields and endomorphisms for non-flat connections.
result A universal Lie algebra is constructed for the post-Lie algebra of vector fields and endomorphisms.
New conformally Einstein metrics on Heisenberg group found.
problem Finding new conformally Einstein metrics on specific Lie groups.
method Described Lorentzian semi-direct extensions of the Heisenberg group.
result Classified Bach-flat left-invariant Lorentzian metrics.
Researchers found a non-Ricci-flat Einstein metric on a 7D nilpotent Lie group.
problem Existence of Einstein metrics on 7D nilpotent Lie algebras.
method Constructed a left-invariant pseudo-Riemannian metric on a 7D nilpotent Lie group.
result Found a non-Ricci-flat Einstein metric on a 7D nilpotent Lie group.
Compact complex manifolds with specific group actions are conformally flat.
problem Compact complex manifolds with invariant conformal holomorphic structures.
method Study of manifolds with transitive and essentially acting complex semi-simple Lie groups.
result If a complex semi-simple Lie group acts transitively and essentially, the manifold is conformally flat.
We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental for…
With a f-left-invariant Riemannian metric on a Lie group , we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor . In this article, we study the geometry of such metrics and give a necessary and sufficient condition for an f-left-invariant Rie…
We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric with specific properties such as being bi-invariant, flat, negatively curved, Einstein, etc. We classify some of such contact Lie groups and derive some obstruction results to the existence of left invariant contact structu…