Study finds all 4D Lie groups with harmonic curvature.
problem Finding Lie groups with harmonic curvature in 4D.
method Analyzing pp-wave Lie groups and determining harmonic curvature conditions.
result Description of 4D pp-wave Lie groups obtained.
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48 results for “Lie curvatures”
Paper unifies curvature concepts for Lie groupoids and algebroids.
problem Separate theories for Lie groups, bundles, and submersions.
method Using Lie groupoids with source-fibre metrics and Riemannian Lie algebroids.
result Derives sectional curvature formula for Lie groupoids and Lie algebroid curvature formulas.
New metrics with special curvature properties are shown to be parallel in certain Lie groups.
problem Characterizing metrics with harmonic curvature in Lie groups.
method Analyzing left invariant metrics on solvable and low-dimensional Lie groups.
result Left invariant metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6.
Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.
problem Characterizing Lie algebras with constant curvature and their geometric implications.
method Constructing statistical manifolds and Sasakian structures to relate Lie algebras to locally conformally Kähler structures.
result Statistical Lie algebras of constant curvature correspond to locally conformally Kähler Lie algebras.
Investigates solving curvature equations on special Lie groups.
problem Solving curvature equations on non-compact simple Lie groups.
method Analyzes left-invariant naturally reductive metrics and conditions for solvability.
result Obtains conditions for the solvability of curvature equations.
Study on prescribing Ricci curvature on compact Lie groups.
problem Prescribing Ricci curvature in naturally reductive metrics on compact Lie groups.
method Derive necessary and sufficient conditions for solvability.
result Provide a series of examples.
Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
problem Characterize left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
method Analyze left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups, classify Lie algebras and groups.
result New results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator.
Study on subgroups' evolution in 3D Lie groups using mean curvature flow.
problem Existence of solutions to Mean Curvature Flow for 2D Lie subgroups in 3D Lie groups.
method Investigation of Lie groups with fixed left-invariant metrics, focusing on non-unimodular cases.
result Evolution of Lie subgroups is self-similar for abelian subgroups, but not for others.
Study on curvatures of surfaces in specific Lie groups.
problem Analyzing curvatures of surfaces in 3D contact sub-Riemannian Lie groups.
method Riemannian approximation scheme to derive formulas for curvatures.
result Classification of surfaces with constant horizontal curvatures.
Compact Dupin hypersurfaces without constant Lie curvatures found.
problem Finding compact Dupin hypersurfaces with non-constant Lie curvatures.
method Two constructions of compact proper Dupin hypersurfaces in .
result Examples of compact proper Dupin hypersurfaces without constant Lie curvatures.
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.
This paper describes cyclic Riemannian Lie groups and their curvatures.
problem Understanding cyclic Riemannian Lie groups and their properties.
method Analyzing left-invariant vector fields and Riemannian metrics.
result Complete description and detailed analysis of cyclic Riemannian Lie groups and their curvatures.
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.
Formula for sectional curvatures on matrix groups.
problem Calculating curvatures on matrix groups.
method Simple formula derivation for sectional curvatures.
result Valid formula for general linear and reductive Lie groups.
Left invariant Lorentzian metrics and curvatures on non-unimodular Lie groups of dimension threemath.DG
Study left invariant Lorentzian metrics on 3D non-unimodular Lie groups.
problem Classify and analyze left invariant Lorentzian metrics on 3D non-unimodular Lie groups.
method Classify metrics up to automorphism, study curvature functions.
result Obtain Ricci operator, scalar curvature, and sectional curvatures as functions of metrics.
A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In th…
Study curvature of complex Finsler metrics on Lie groups.
problem Explicitly calculate and characterize curvature of complex Finsler metrics.
method Analyzes left-invariant complex Finsler metrics on Lie groups using their Lie algebra.
result Characterizes conditions for a Finsler metric to be Kähler or weakly Kähler.
Lie minimal surfaces are characterized by differential equations of principal curvatures.
problem Characterizing Lie minimal surfaces in Riemannian space forms.
method Using Euler-Lagrange equations and differential equations of principal curvatures.
result Rotational surfaces are found for certain relationships between principal curvatures.
Study of curvature flow on complex Lie groups, leading to soliton convergence.
problem Characterizing long-time behavior of curvature flow on complex 2-step nilpotent Lie groups.
method Analyzing left-invariant metrics and using Cheeger-Gromov topology.
result Normalized solutions converge to a non-flat algebraic soliton.
Let G be a Lie group. On the trivial principal G-bundle over the Lie algebra of G there is a natural connection whose curvature is the Lie bracket. The exponential map is given by parallel transport of this connection. If G is the diffeomorphism group of a manifold, the curvature of the natural connection is the Lie br…
On Ricci negative Lie groupsmath.DG
Study on Lie groups with negative Ricci curvature, including open questions and a new cone.
problem Understanding Lie groups with negative Ricci curvature metrics.
method Overview and introduction of a new cone C(n) for solvable Lie algebras.
result Introduction of a new open cone C(n) that parametrizes solvable Lie algebras with negative Ricci curvature metrics.
The paper classifies specific types of metrics on 5D Lie groups.
problem Classifying -metrics on five dimensional nilpotent Lie groups.
method Left-invariant metrics and vector fields classification.
result Geometric properties of classified metrics.
Study on surface geometry in Lie groups with CR structures.
problem Understanding surface curvature in Lie groups with CR structures.
method Defined Gauss and mean curvature in Tanaka-Webster geometry.
result Gave specific examples of surface curvature calculations.
The paper finds solutions for specific curvature conditions on 5D Lie groups.
problem Finding metrics with prescribed Ricci curvature on 5D nilpotent Lie groups.
method Applied Milnor-type theorem technique to prove global existence of (g, c).
result Global existence of (g, c) for prescribed Ricci curvature on 5D nilpotent Lie groups.
Derives spacetime regularity under specific curvature conditions.
problem Ensuring smoothness in spacetime models with given curvature constraints.
method General regularity estimate for 4-d spacetimes, using Ricci curvature and Lie derivatives.
result Establishes conditions for smoothness in spacetime models.
Considering prolongation of a Lie algebroid equipped with a spray, defining some classical tensors, we show that a Lie symmetry of a spray is a curvature collineation for these tensors.
Lie sphere geometry helps classify Dupin hypersurfaces.
problem Classifying Dupin hypersurfaces in Lie sphere geometry.
method Generalizing Dupin hypersurfaces to Lie sphere geometry and using Lie sphere transformations.
result Many classifications of proper Dupin hypersurfaces have been obtained.
Introduces Lie group actions in smoothing processes for currents and spaces with curvature.
problem Regularization of currents and metrics on manifolds and spaces with curvature.
method Actions of compact Lie groups in De Rham approximation and smoothing of Riemannian metrics.
result Effective smoothing processes for currents and metrics on manifolds and spaces with curvature.
Study of curvature flow on specific Lie groups, leading to soliton solutions.
problem Curvature flow on 2-step nilpotent Lie groups with complex structures.
method Left-invariant metrics and complex structures on Lie groups, convergence analysis.
result Existence and convergence of flow to soliton solutions.
The paper studies automorphisms of generalized Kähler manifolds and their Lie algebras.
problem Understanding the existence or non-existence of generalized Kähler structures with constant scalar curvature.
method Analyzing the Lie algebra of automorphisms of generalized complex manifolds under specific conditions.
result The Lie algebra of automorphisms is reductive if a generalized Kähler structure of symplectic type with constant scalar curvature exists.
Extends Cheeger's method to Lie groupoid actions on manifolds.
problem Smooth Lie group actions on manifolds with singularities.
method Extension of Cheeger's deformation techniques to Lie groupoid actions.
result Explicit sectional curvature description of the deformation.
The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.
problem Understanding surfaces with spherical curvature lines and their generation mechanisms.
method The approach involves Lie sphere transformations, Legendre curves, and polynomial conserved quantities of connections.
result Lie applicable surfaces with exactly one family of spherical curvature lines are generated by the lift of constrained elastic curves.
In this paper, firstly we study some left invariant Riemannian metrics on para-hypercomplex 4-dimensional Lie groups. In each Lie group, the Levi-Civita connection and sectional curvature have been given explicitly. We also show these spaces have constant negative scalar curvatures. Then by using left invariant Riemann…
Lorentzian left invariant metrics on three dimensional unimodular Lie groups and their curvaturesmath.DG
There are five unimodular simply connected three dimensional unimodular non abelian Lie groups: the nilpotent Lie group , the special unitary group , the universal covering group of the special linear group, the solvable Lie group and…
Born Lie algebrasmath.DG
Born Lie algebras classified up to 6D, with integrable metrics studied.
problem Classifying and understanding Born Lie algebras.
method Bicross product construction from pseudo-Riemannian Lie algebras.
result Classification of Lie algebras up to 6D with integrable Born structures.
Study magnetic curvature on Lie groups, extending Milnor's work.
problem Exploring magnetic curvatures on Lie groups.
method Computing magnetic curvatures and analyzing algebraic properties.
result Extending results from Milnor's classic paper on left-invariant metrics.
Abstract: Characterizes spaces with positive scalar curvature.
problem Spaces with positive scalar curvature and invariant metrics.
method Cohomogeneity one manifolds and homogeneous spaces with compact Lie group actions.
result Characterization of spaces with positive scalar curvature.
Study of curves in Lie sphere geometry using moving frames and variational principles.
problem Characterize curves in Lie sphere geometry using Lie curvatures.
method Moving frames, exterior differential systems, and calculus of variations.
result Critical curves are uniquely determined by Lie curvatures.
Develops scalar curvature in generalized Kahler geometry and shows constant scalar curvature on compact Lie groups.
problem Defines scalar curvature in generalized Kahler geometry.
method Introduces scalar curvature in terms of pure spinors formalism and develops a moment map framework.
result Scalar curvature is given by the moment map, generalizing results from ordinary Kahler geometry.
In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Ran…
Survey of Dupin hypersurfaces in Lie sphere geometry.
problem Classifying Dupin hypersurfaces in Lie sphere geometry.
method Detailed description of Lie sphere geometry concepts and classification results.
result Many classification results relating Dupin hypersurfaces to isoparametric hypersurfaces in spheres.
We consider the question of whether a given solvable Lie group admits a left-invariant metric of strictly negative Ricci curvature. We give necessary and sufficient conditions of the existence of such a metric for the Lie groups the nilradical of whose Lie algebra is either abelian or Heisenberg or standard filiform, a…
The paper studies Riemannian metrics on Lie groups with specific commutator subgroups.
problem Investigating Riemannian metrics on Lie groups with commutator subgroups of dimensions 1 and 2.
method Explicitly provided Levi-Civita connection, sectional curvature, and Ricci curvature; computed necessary and sufficient conditions for Ricci solitons; characterized Ricci solitons on Lie groups with one-dimensional commutator subgroups; examined indecomposable Lie groups with two-dimensional commutator subgroups.
result Characterization of all Ricci solitons on Lie groups with one-dimensional commutator subgroups and examination of indecomposable Lie groups with two-dimensional commutator subgroups.
Study derivations for nilpotent Lie algebras with negative Ricci curvature.
problem Characterize derivations leading to solvable extensions with negative Ricci curvature.
method Investigate the space of diagonalizable derivations for specific Lie algebras.
result Prove conjecture about derivations in dimension 5 and for Heisenberg and standard filiform Lie algebras.
Classifies SNC-algebras in 5D, calculating curvature.
problem Classifying SNC-algebras in higher dimensions.
method Defined SNC-algebras and used Lie group properties.
result Classified SNC-algebras in dimension five.
Study left invariant spray structures on Lie groups, calculating curvature and geodesics.
problem Understanding curvature and geodesics in left invariant spray structures on Lie groups.
method Use invariant frames and canonical bi-invariant Berwald spray structure to analyze left invariant spray structures.
result Established correspondence between geodesics and inverse integral curves of spray vector fields.
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.
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.