Study of Riemannian geometry on tangent Lie groups of two families.
problem Investigate Riemannian geometry on tangent Lie groups.
method Use vertical and complete lifts to induce metrics, analyze Levi-Civita connection, sectional, and Ricci curvatures.
result Investigated Riemannian geometry on tangent bundles of two families of Lie groups.
New dynamics derived from Lie groups using 2nd order tangent groups.
problem Deriving dynamics on complex Lie groups.
method Using double cross product groups and 2nd order Euler-Lagrange equations.
result 2nd order Lagrangian dynamics on double cross product groups derived.
New complex structures found on tangent bundles of Lie groups.
problem Finding integrable complex structures on tangent bundles of Lie groups.
method Inspired by Samelson's construction, a left-invariant integrable almost complex structure is defined on the tangent bundle of any compact Lie group.
result Tangent bundles of compact Lie groups admit left-invariant integrable almost complex structures.
Paper defines tangent Lie algebra for non-smooth subgroups of diffeomorphism groups.
problem Understanding tangent spaces of non-smooth subgroups of diffeomorphism groups.
method Introduced tangent Lie algebra TG for non-smooth subgroups of Diff(M) of a compact manifold M.
result Tangent Lie algebra TG is a Lie subalgebra of smooth vector fields on M.
Tangent Lie groups have a special Riemannian metric structure.
problem Characterizing the Riemannian structure of tangent Lie groups.
method Proving the existence of a left-invariant naturally reductive metric and connection on tangent Lie groups.
result Tangent Lie groups admit a special Riemannian metric structure.
New structure found on Lie group tangent bundle.
problem Finding new structures on Lie group tangent bundles.
method Analyzing left invariant structures on Lie groups.
result Tangent bundle of Lie group admits a left-invariant nearly pseudo-Kähler structure.
In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…
Study on Randers metrics on tangent Lie groups and their geometric properties.
problem Characterizing Randers metrics of Berwald type on tangent Lie groups.
method Analyzing the relations between flag curvature and sectional curvature.
result Identifying all 3-dimensional Lie groups with tangent bundles admitting Berwald type Randers metrics.
The paper studies geometric properties of tangent Poisson-Lie groups.
problem Geometric properties of tangent Poisson-Lie groups.
method Expressed Levi-Civita connection, curvature, and metacurvature of tangent Poisson-Lie groups in terms of the base group.
result Proved that the space of differential forms on a Poisson-Lie group is a differential graded Poisson algebra if and only if the space on its tangent Poisson-Lie group is.
Study geodesic Lie groups' convergence to limits with quantitative estimates.
problem Quantifying convergence rates of geodesic Lie groups to their limits.
method Estimates on the difference between original metrics and asymptotic/tangent metrics.
result Sharpens existing bounds on convergence rates.
The paper studies special metrics on tangent Lie groups.
problem Investigating lifted (α,β)-metrics of Douglas type on tangent Lie groups. method Constructing and analyzing vertical and complete lifted (α,β)-metrics on tangent Lie groups. result Necessary and sufficient conditions for these metrics to be of Douglas type are derived.
Study Riemann-Finsler geometry on tangent bundles of Lie groups with 2D commutator subgroup.
problem Characterize Riemannian and Finslerian properties of tangent bundles of Lie groups with specific commutator subgroups.
method Investigate sectional curvatures, define Randers metrics, and compute flag curvatures on tangent bundles.
result Explicit formulas for Riemannian curvature tensor on tangent bundles of Lie groups with 2D commutator subgroup.
Survey on finite dimensional Lie groups over real numbers.
problem Lack of rigorous proofs for Lie groups using tangent space formalism.
method Formalism of tangent space via chart and vector methods, curve and derivation methods.
result Rigorous proofs of Lie groups facts with this formalism.
We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which is tangent o…
The paper integrates Lie-Leibniz triples into Lie group-rack triples.
problem Integrating Lie-Leibniz triples into Lie group structures.
method Defining Lie group-rack triples and integrating finite-dimensional Lie-Leibniz triples.
result Any finite-dimensional Lie-Leibniz triple can be integrated to a local Lie group-rack triple.
Study of Lagrangian dynamics on matched Lie groups.
problem Understanding dynamics on matched Lie groups.
method Isomorphic tangent bundle and Euler-Lagrange/Poincaré equations.
result Covering semi-direct product theory and explicit equations.
The paper studies Riemannian metrics on tangent Lie groups using two left-invariant metrics.
problem Exploring Riemannian structures on tangent Lie groups.
method Defining a new left-invariant Riemannian metric on the tangent Lie group using two left-invariant metrics and symplectic forms.
result Explicit formulas for the Levi-Civita connection, tensor curvature, and sectional curvature of the new metric in terms of the original metrics.
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
The paper extends Cartan development to infinite dimensional Lie groups.
problem Generalizing Cartan development to infinite dimensional Lie groups.
method Generalization of Cartan development to infinite dimensional manifolds and Lie groups.
result The tangent mapping of a Cartan development is another Cartan development.
New integrators for mechanical systems on Lie groups simplify based on group properties.
problem Designing numerical integrators for mechanical systems on Lie groups.
method Leverage retraction maps and Lie group properties to design structure-preserving integrators.
result Simplified design of integrators for Euler-Poincare and Lie-Poisson equations.
Proposes an auto-encoder for Gaussian distributions using Lie group theory.
problem Generative models for Gaussian distributions.
method Lie group auto-encoder with UTDATs, incorporating geometric properties.
result Eliminates matrix exponential operator and derives intrinsic loss.
Lie algebras of quotient groups defined under specific conditions.
problem Conditions for Lie differentiation of quotient groups.
method Diffeological group theory, tangent structure, Lie functor instantiation.
result Lie algebra structure on quotient groups derived from Lie algebras of parent groups.
Lie Calculus connects differential and Lie theory using groupoids.
problem Understanding the relationship between differential and Lie theories.
method Using groupoids to link differential and Lie theories.
result Higher order theory involves higher algebra (n-fold groupoids).
Frölicher spaces form a cartesian closed category which contains the category of smooth manifolds as a full subcategory. Therefore, mapping groups such as C^\infty(M,G) or \Diff(M), but also projective limits of Lie groups are in a natural way objects of that category, and group operations are morphisms in the category…
Characterizes magnetic unit vector fields on Lie groups.
problem Classifying magnetic unit vector fields on Lie groups.
method Characterization through critical points of Landau Hall and Dirichlet energy functionals.
result Classification of all magnetic left invariant unit vector fields on 3-dimensional Lie groups.
As a step toward proving an index theorem for hypoelliptic operators Heisenberg manifolds, including those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes' tangent groupoid of a manifold M. As it is well known for a Heisenberg manifold (M,H) the relevant notion of tangent is…
This work defines a categorical notion of principal bundles.
problem Different definitions of principal bundles in various categories.
method Formulated in join-restriction categories, which generalize partial maps.
result Shows the tangent bundle as the product of tangent space and group object.
Our goal in this paper is to make an attempt to find the largest Lie algebra of vector fields on the indicatrix such that all its elements are tangent to the holonomy group of a Finsler manifold. First, we introduce the notion of the curvature algebra, generated by curvature vector fields, then we define the infinitesi…
The study explores Lie superalgebras constructed from Lie algebras using Schouten-like brackets.
problem Investigate how the core Lie algebra controls the Lie superalgebra.
method Construct Lie superalgebras from abstract Lie algebras using Schouten-like brackets and analyze Betti numbers of super homology groups.
result For low dimensional non-abelian Lie algebras, the Betti numbers of super homology groups provide insights into the control of the core Lie algebra.
CR embeddings in complex spaces for specific Lie groups.
problem Embedding specific Lie groups in complex spaces.
method Using integrable complex structures on subbundles of tangent bundles.
result CR embeddings possible as the edge of wedges in complex domains.
In our previous paper (arXiv:1306.5449) we have given a sufficient and necessary condition when the coupling between Lie algebra bundle (LAB) and the tangent bundle exists in the sense of Mackenzie (\cite{Mck-2005}, Definition 7.2.2) for the theory of transitive Lie algebroids. Namely we have defined a new topology on …
In this paper, we introduce a study of prolongations of representations of Lie groups. We obtain a faithful (one-to-one) representation of TG where G is a finite-dimensional Lie group and TG is the tangent bundle of G, by using (not necessarily faithful) representations of G. We show that tangent functions of Lie group…
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.
New geometric structures on Lie groups discovered.
problem Understanding geometric properties of Lie groups.
method Investigated 4D Riemannian manifolds with specific endomorphisms.
result Found new Lie groups with circulant structure.
The paper studies complex Finsler metrics on complex Lie groups.
problem Characterizing properties of left invariant complex Finsler metrics on complex Lie groups.
method Using invariant frames, the paper proves properties of the metric and its spray.
result The strongly Kähler, Kähler, and weakly Kähler properties are equivalent for the metric.
We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In…
We prove a reduction theorem for the tangent bundle of a Poisson manifold (M,π) endowed with a pre-Hamiltonian action of a Poisson Lie group (G,πG). In the special case of a Hamiltonian action of a Lie group, we are able to compare our reduction to the classical Marsden-Ratiu reduction of M. If the manifold $M…
The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles...). In particular, we are interested in two cases: singular Lagrangi…
In this paper, we describe a geometric setting for higher-order lagrangian problems on Lie groups. Using left-trivialization of the higher-order tangent bundle of a Lie group and an adaptation of the classical Skinner-Rusk formalism, we deduce an intrinsic framework for this type of dynamical systems. Interesting appli…
The derivation dT on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms on the tangent bundle TM is extended to multivector fields. These tangent lifts are studied with applications to the theory of Poisson structures, their symplectic foliations, canonical vector fields a…
We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic…
A special symplectic Lie group is a triple (G,ω,∇) such that G is a finite-dimensional real Lie group and ω is a left invariant symplectic form on G which is parallel with respect to a left invariant affine structure ∇. In this paper starting from a special symplectic Lie group we show how to ``defo…
Study minimal rational curves on complex manifolds with isotropic VMRT.
problem Understanding minimal rational curves tangent to distributions on complex manifolds.
method Partial equivariant compactification of metabelian groups.
result Any isotropic VMRT can be realized as VMRT of minimal rational curves tangent to a distribution.
Unified product Lie groups and their quotient spaces are analyzed for dynamics.
problem Analyzing dynamics over homogeneous spaces using Lie group theory.
method Reduction and extension of Lie group structures to quotient spaces, formulation of Euler-Lagrange, Hamilton, and Euler-Poincaré equations.
result Unified product Lie groups and their quotient spaces provide a framework for formulating dynamics equations.
New type of spaces with tangent structures for analysis.
problem Defining tangent structures for non-smooth spaces.
method Introducing elastic diffeological spaces and defining tangent structures.
result Elastic spaces have a natural tangent structure with graded commutation relations.
We solve three open problems concerning infinite-dimensional Lie groups posed in a recent survey article by K.-H. Neeb: (1) There exists a subgroup of some infinite-dimensional Lie group G which does not admit an initial Lie subgroup structure; (2) The pathology cannot occur if G is a direct limit of an ascending seque…
The paper introduces Carnot coordinates for Carnot manifolds, simplifying nilpotent approximation.
problem Nilpotent approximation of Carnot manifolds.
method Identification of Carnot coordinates as privileged coordinates with specific properties.
result Carnot coordinates provide a precise and effective nilpotent approximation of Carnot manifolds.
This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (…