The paper studies metrics on Lie groups SO(2) and SO(3) and their integrability.
problem Integrability of gradient systems on Lie groups via Fisher metrics.
method Analysis of Souriau-Fisher metrics and 2-cocycles on Lie groups SO(2) and SO(3).
result Cocycles can locally modify Fisher metrics on Lie group orbits.
Study describes isometry groups of specific Lie groups.
problem Understanding isometry groups of compact Lie groups.
method Analyzing bi-invariant metrics on SO(4) and U(n).
result Full groups of isometries identified for specific Lie groups.
The authors find geodesics, shortest arcs, diameter, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group SO(3), under condition that the metric is right-invariant relative to the Lie subgroup SO(2)⊂SO(3).
The study finds multiple Einstein metrics on SO(n) groups.
problem Finding Einstein metrics on compact Lie groups.
method Proving existence of non-naturally reductive left-invariant Einstein metrics.
result Compact Lie groups SO(n) for n≥10 have at least $2\left([\frac{n-1}{3}]-2
ight)$ such metrics. The author finds geodesics, shortest arcs, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group SO0(2,1) under the condition that the metric is right-invariant relative to the Lie subgroup SO(2)⊂SO0(2,1).
New harmonic self-maps constructed for specific Lie groups.
problem Harmonic self-maps of compact cohomogeneity one manifolds.
method Developed theory of equivariant harmonic self-maps; constructed new maps by solving non-linear boundary value problems.
result Harmonic self-maps of specific Lie groups with novel degrees.
New Einstein metrics found on SO(n) without being naturally reductive.
problem Finding new Einstein metrics on SO(n) that are not naturally reductive. method Imposing symmetry assumptions and solving polynomial equations using symbolic computations with Gröbner bases.
result Invariant Einstein metrics on SO(n) (n≥7) that are not naturally reductive were found. We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is SU(p,q) (resp. SO∗(2n), n odd) and the surface group is maximal in some S(U(p,p)×U(q−p))⊂SU(p,q) (resp. SO∗(2n−2)×SO(2)⊂SO∗(2n))…
Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest paths on Lie groups with sub-Riemannian metrics.
method Analytical and geometric methods to find geodesics and shortest arcs.
result Found geodesics, shortest arcs, distances, and conjugate loci for specified metrics.
Existence of Kähler-Einstein metrics on compactifications of Lie groups.
problem Existence of Kähler-Einstein metrics on Q-Fano compactifications of Lie groups. method Proving existence through compactifications of Lie groups.
result Classification of Q-Fano compactifications of SO4(C) with Kähler-Einstein metrics. The authors found geodesics, shortest arcs, cut loci, and conjugate sets for left-invariant sub-Riemannian matric on the Lie group SL(2), which is right-invariant relative to the Lie subgroup SO(2)⊂SL(2) (in other words, for invariant sub-Riemannian metric on weakly symmetric space $(SL(2)\times SO(2))/SO(2)…
Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest arcs in sub-Riemannian metrics on Lie groups.
method Investigated left-invariant sub-Riemannian metrics on SU(1,1)imesR and SO0(2,1)imesR. result Found geodesics, shortest arcs, cut loci, and conjugate loci.
The paper finds a contact form on SL(2p) for p > 1.
problem Left invariant Pfaffian forms on SL(2p) are not contact forms for p > 1.
method Constructed a contact form invariant under SO(2p).
result Found a contact form on SL(2p) for p > 1.
New biharmonic functions created on Lie groups.
problem Constructing explicit biharmonic functions on Lie groups.
method Developed a new scheme for constructing complex-valued biharmonic functions on Riemannian Lie groups.
result Manufactured infinite series of new solutions on SU(n) and showed applicability to SO(n) and Sp(n). It is well known that every compact simple Lie group G admits an Einstein metric that is invariant under the independent left and right actions of G. In addition to this bi-invariant metric, with G x G symmetry, it was shown by D'Atri and Ziller that every compact simple Lie group except SU(2) and SO(3) admits at least…
The authors compute distances between arbitrary elements of Lie groups SU(2) and SO(3) for special left-invariant sub-Riemannian metrics ρ and d. To compute distances for the second metric, we essentially use the fact that canonical two-sheeted covering epimorphism Ω of the Lie group SU(2) onto the Lie group SO(3…
Current Lie groupoids generalize mappings to Lie groupoids.
problem Generalizing Lie group properties to mappings into Lie groupoids.
method Study of superposition operators and Lie groupoid properties.
result Current Lie groupoids inherit properties from underlying maps.
Study conformal foliations on Lie groups, finding new families and harmonic morphisms.
problem Classifying conformal foliations on Lie groups with minimal leaves.
method Analyzing left-invariant foliations generated by specific subgroups.
result New multi-dimensional families of Lie groups with conformal foliations.
We show the local rigidity of complex hyperbolic lattices in classical Hermitian semisimple Lie groups, SU(np,p),Sp(2n+2,R),SO∗(2n+2),SO(2n,2). This reproves or generalizes some results in \cite{GM, KKP, Klingler-inv, Pozzetti}.
New minimal submanifolds found in Lie group duals.
problem Finding minimal submanifolds in Lie group duals.
method Constructing families of complete minimal submanifolds of codimension two.
result New families of complete minimal submanifolds of codimension two in Lie group duals constructed.
In this paper we introduce a new method for manufacturing harmonic morphisms from semi-Riemannian manifolds. This is employed to yield a variety of new examples from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with their standard Riemannian metrics. We develop a duality principle and show how this can be use…
Lie group integrators improve global error estimates.
problem Global error estimates for Lie group integrators.
method Relate local error to global error, derive from bounds.
result Lie-Butcher theory proves global error estimates for Lie group integrators.
We construct the first known complex valued harmonic morphisms from the non-compact Lie groups SL(n,R), SU*(2n) and Sp(n,R) equipped with their standard Riemannian metrics. We then introduce the notion of a bi-eigenfamily and employ this to construct the first known solutions on the non-compact Riemannian SO*(2n), SO(p…
A left-invariant sub-Riemannian metric d on the shortened Lorentz group SO0(2,1) under the condition that d is right-invariant relative to the orthogonal Lie subgroup 1⊗SO(2) is studied. The distance between arbitrary two elements, the cut locus (as the union of the subgroup 1⊗SO(2) with the an…
New method constructs Ricci-flat metrics on Lie groups.
problem Finding Ricci-flat metrics on Lie groups.
method Combinatorial method to construct indefinite Ricci-flat metrics.
result Infinite families of Ricci-flat nilmanifolds constructed.
Study on representation varieties of RAAGs for different Lie groups.
problem Investigating connectedness of representation varieties for RAAGs under various Lie groups.
method Examined G-representation varieties of RAAGs for SU(n),Sp(n),U(n) and compared to other Lie groups. result Connectedness of representation varieties for SU(n),Sp(n),U(n), but not for SO(n),Spin(n) for n≥3. Introduces Θ-positivity in Lie groups, linking to surface group representations.
problem Understanding positivity in Lie groups and its relation to surface group representations.
method Introduces Θ-positivity as a new concept and shows its applicability to specific Lie groups. result Identifies new families of Lie groups (SO(p,q) for p<q and exceptional Lie groups) with Θ-positive structures. New method constructs complex-valued r-harmonic functions on Riemannian manifolds.
problem Constructing complex-valued r-harmonic functions on Riemannian manifolds.
method Introducing a new method for constructing complex-valued r-harmonic functions on Riemannian manifolds and applying it to specific semisimple Lie groups.
result The method successfully constructs complex-valued r-harmonic functions on various Riemannian manifolds, including specific Lie groups.
Generalizes shape analysis to Lie groups and homogeneous spaces.
problem Shape analysis on non-Euclidean spaces.
method Square Root Velocity Transform (SRVT) generalization to Lie groups and homogeneous manifolds.
result Shape analysis on Lie groups and homogeneous spaces.
The study proves that for certain Lie groups, eigenspaces are irreducible under group actions.
problem Irreducibility of Laplace eigenspaces on compact Lie groups.
method Established algebraic criteria for existence of left invariant metrics making eigenspaces irreducible.
result Generic left invariant metrics on specific Lie groups make eigenspaces irreducible.
The paper explores metrics on Lie groups and their connections to dual quaternions.
problem Understanding metrics on Lie groups and their geometric properties.
method Analyzing Cartan-Schouten metrics on perfect Lie groups and their connections to dual quaternions.
result Biinvariant metrics on perfect Lie groups are shown to be Cartan-Schouten metrics.
A classical theorem of Drinfel'd states that the category of simply connected Poisson Lie groups H is isomorphic to the category of Manin triples (d, g, h), where h is the Lie algebra of H. In this paper, we consider Dirac Lie groups, that is, Lie groups H endowed with a multiplicative Courant algebroid A and a Dirac s…
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…
Poisson sigma models represent an interesting use of Poisson manifolds for the construction of a classical field theory. Their definition in the language of fibre bundles is shown and the corresponding field equations are derived using a coordinate independent variational principle. The elegant form of equations of mot…
The Lie group SO_0(n, 1) has the left-invariant metric coming from the Killing-Cartan form. The maximal compact subgroup SO(n) of the isometry group acts from the left. The geometry of the quotient space of the homogeneous submersion SO_0(n, 1) -> SO(n)\SO_0(n, 1) is investigated. The space is expressed as a warped pro…
Introduces Lie groupoids and their applications.
problem Analyzing Jordan-Hölder resolutions and integrating partial differential equations.
method Discussing properties and techniques of Lie groupoids, algebroids, and pseudo-groups.
result Just the first Lie Theorem holds for Lie groupoids.
Study on 3D Lie groups' symmetry index, proving positive index for all.
problem Determining the index of symmetry in 3D Lie groups.
method Analyzing left-invariant metrics and quotient geometry.
result Positive index of symmetry for all 3D unimodular Lie groups.
Study on CR structures on 3D Lie groups, focusing on equivalence and closed chains.
problem Characterizing CR structures on 3D Lie groups and their chains.
method Analyzing left-invariant CR structures on 3D Lie groups and their equivalence.
result All chains on G are closed if and only if G is CR equivalent to specific spherical CR structures.
Study on deformation of affine structures on Lie groups using cohomology.
problem Understanding the gap between global topological invariants and left-invariant affine structures.
method Comparing De Rham and Koszul-Vinberg cohomology groups on Lie groups SO(2), H3(R), and SGal(3).
result Vanishing theorem for the second KV-cohomology group, proving structural rigidity of orbits.
The cohomology of affine structures on Lie groups is compared with that of Koszul-Vinberg algebras.
problem Comparing De Rham cohomology and KV-cohomology on Lie groups SO(2), H3(R), and Galilei group SGal(3).
method Constructing a three vertex directed graph connecting associative algebras, KV-cohomology, and Lie groups.
result Evaluating the algebraic quotient of the cohomology groups.
Constructs Lie groups with negative Ricci curvature.
problem Finding Lie groups with negative Ricci curvature.
method Constructs Lie groups with specific algebraic structures and representations.
result Proves existence of Lie groups with negative Ricci curvature for various Levi factors.
Recent work Bobienski-Nurowski on 5-dimensional Riemannian manifolds with an SO(3) structure prompts us to investigate which Lie groups admit such a geometry. The case in which the SO(3) structure admits a compatible connection with torsion is considered. This leads to a classification under special behaviour of the on…
We show that the canonical central extension of the group of sections of a Lie group bundle over a compact manifold, constructed in [NW09], is universal. In doing so, we prove universality of the corresponding central extension of Lie algebras in a slightly more general setting.
New Einstein metrics found on orthogonal groups without natural reductivity.
problem Finding non-naturally reductive Einstein metrics on orthogonal groups.
method Using real flag manifolds and symmetry assumptions on left-invariant metrics.
result Obtained new invariant Einstein metrics on $\SO(n)$.
Study on symplectic half-flat 6-manifolds and their automorphism groups.
problem Characterizing the automorphism group of symplectic half-flat 6-manifolds.
method Proved the Lie algebra of the automorphism group has an abelian structure with dimension bounded by min{5, b1(M)}. Analyzed properties of the automorphism group action.
result Found new complete examples on TS3 invariant under a cohomogeneity one action of SO(4). New cohomology groups generalize Euler number for Lie superalgebras.
problem Generalizing cohomology groups for Lie superalgebras.
method Abstracted Poisson cohomology groups to Poisson-like cohomology groups for general Lie superalgebras.
result De Rham cohomology groups match Poisson-like cohomology groups for differential forms.
We develop techniques for classifying the nonnegatively curved left-invariant metrics on a compact Lie group G. We prove rigidity theorems for general G and a partial classification for G=SO(4). Our approach is to reduce the general question to an infinitesimal version; namely, to classify the directions one can move a…
Researchers extend reparameterization to Lie groups for better probability modeling.
problem Lack of reparameterization for distributions on Lie groups.
method Developed a general framework for creating reparameterizable densities on Lie groups.
result Demonstrated complex and multimodal distributions on SO(3) for pose estimation.