Defines Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.
problem No specific problem stated; focuses on new structure definition.
method Definition and properties of Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.
result Defines a new structure on Jacobi-left-symmetric algebroids.
Classifies R-spaces with a specific symmetric structure.
problem Classifying R-spaces with a natural Γ-symmetric structure.
method Classification and determination of maximal antipodal sets.
result Classification of R-spaces with a natural Γ-symmetric structure.
We introduce and discuss (local) symmetries of geometric structures. These symmetries generalize the classical (locally) symmetric spaces to various other geometries. Our main tools are homogeneous Cartan geometries and their explicit description. This allows us to describe the structure of symmetric geometric structur…
The paper introduces new structures for left-symmetric algebroids.
problem Developing new mathematical structures for left-symmetric algebroids.
method Introducing Koszul-Vinberg-Nijenhuis structures and related concepts.
result Koszul-Vinberg-Nijenhuis structures provide a hierarchy of structures.
Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.
problem Understanding geometric structures related to geodesic foliations and dynamics.
method Introducing symmetric Poisson structures, proving correspondences with geodesic foliations and Jordan algebras.
result Symmetric Poisson structures correspond to totally geodesic foliations and Jacobi-Jordan algebras.
The paper defines left-symmetric bialgebroids and their Manin triples.
problem No specific problem stated; focuses on definitions and constructions.
method Introduced left-symmetric bialgebroids and Manin triples, constructed from pseudo-Hessian manifolds.
result Established a relation between Maurer-Cartan type equations and Dirac structures.
The paper describes Calabi-Yau structures and special Lagrangian submanifolds in complexified symmetric spaces.
problem Understanding Calabi-Yau structures and special Lagrangian submanifolds in complexified symmetric spaces.
method Using Schwarz's theorem and constructions based on the Stenzel metric.
result Constructs special Lagrangian submanifolds invariant under symmetric subgroups of isometry groups.
Characterizes invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
method Characterization through invariant Riemannian metrics and Killing vector fields.
result A family of invariant contact metric structures is obtained on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two.
The study classifies complex symplectic structures on 4D Lie algebras and constructs hypersymplectic structures.
problem Classifying and constructing complex symplectic structures on 4D Lie algebras.
method Interpreting complex symplectic and pseudo-Kähler structures, developing a method for constructing hypersymplectic structures.
result Obtained an example of a hypersymplectic structure on a 4-step nilmanifold.
Complete classification of quaternionic skew-Hermitian symmetric spaces found.
problem Classifying quaternionic skew-Hermitian symmetric spaces.
method Proving the existence of a torsion-free mSO∗(2n)mSp(1)-structure and showing that any homogeneous space is symmetric. result A complete classification of quaternionic skew-Hermitian symmetric spaces for arbitrary n>1. The paper defines symmetric brackets for skew-symmetric algebroids with totally skew-symmetric torsion.
problem Defining symmetric brackets for skew-symmetric algebroids.
method Using connections with totally skew-symmetric torsion and pseudo-Riemannian metrics.
result Explicit formula for the Levi-Civita connection and symmetric brackets on almost Hermitian manifolds.
Sasakian structures found on tangent sphere bundles of certain symmetric spaces.
problem Existence of Sasakian structures on tangent sphere bundles of compact rank-one symmetric spaces.
method Proof of existence using K-contact structures and induced structures from almost Hermitian structures.
result Tangent sphere bundles of compact rank-one symmetric spaces admit unique K-contact structures that are Sasakian.
Characterizes complex structures on hermitian symmetric spaces.
problem Understanding invariant complex structures on principal bundles.
method Using Jordan algebraic approach for curvature computations.
result Complete characterization of integrable complex structures.
Study spin structures on Kac-Moody symmetric spaces.
problem Existence of spin structures on affine Kac-Moody symmetric spaces.
method Sufficient conditions for spin structures existence.
result Obtained spin-c representation of certain Kac-Moody subgroups.
Researchers found a maximal antipodal set of three elements in a 7x7 sphere space.
problem Determining the maximal antipodal set in the outer 3-symmetric space S7imesS7. method Investigated the polar and maximal antipodal set P for the given 3-symmetric space S7imesS7. result The maximal antipodal set P has three elements. It is shown that four-dimensional generalized symmetric spaces can be naturally equipped with some additional structures defined by means of their curvature operators. As an application, those structures are used to characterize generalized symmetric spaces.
Paper classifies 3D locally symmetric Riemannian Lie groups using Milnor bases.
problem Classifying 3D locally symmetric Riemannian Lie groups.
method Used Milnor bases to solve polynomial equations of structure constants.
result Identified E0(2) as the only non-symmetric locally symmetric Lie group. New symmetric quandles constructed from group elements and subgroups.
problem Understanding the structure of symmetric quandles.
method Constructing symmetric quandles from specific group elements and subgroups.
result Every symmetric quandle is isomorphic to the disjoint union of constructed quandles.
The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for ∣1∣--graded parabolic geometries and for almost Grassmannian structures, in particular.…
We develop various properties of symmetric generalized complex structures (in connection with their holomorphic space and B-field transformations), which are analogous to the well-known results of Gualtieri on skew-symmetric generalized complex structures. Given a symmetric or skew-symmetric generalized complex structu…
New Γ-structures found on symmetric spaces.
problem Existence of Γ-structures on manifolds. method Analyzing rational cohomology and geodesic symmetries.
result Geodesic symmetries define Γ-structures on symmetric spaces. Researchers found all invariant contact structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying all invariant contact metric structures on tangent sphere bundles of compact rank-one symmetric spaces.
method Explicitly obtained all structures, distinguishing K-contact, Sasakian, and 3-Sasakian structures.
result There is a unique Sasakian-Einstein metric on tangent sphere bundles of spheres and real projective spaces.
Study of complexified Hermitian symmetric spaces and their structures.
problem Understanding the hyperkähler structures and symplectic properties of complexified Hermitian symmetric spaces.
method Explicit diffeomorphisms and moment-critical subsets analysis.
result Almost all complex and symplectic structures are equivalent to the ones on G/K and $T^*\left(G_u/K_0
ight)$ respectively. Sprays on Frechet manifolds connect connections and tangent structures.
problem Characterizing linear symmetric connections on Frechet manifolds.
method Constructing connection maps and linear symmetric connections on tangent and second-order tangent bundles using sprays.
result A bijective correspondence exists between linear symmetric connections on tangent bundles and sprays.
The study finds conditions for quaternionic structures on symmetric spaces.
problem Conditions for quaternionic structures on symmetric spaces.
method Analysis of Lie group actions and representations.
result Symmetric spaces have invariant quaternionic structures under specific conditions.
Study the geometric properties of skew symmetric matrices and orthogonal groups.
problem Understanding the geometric properties of skew symmetric matrices and orthogonal groups.
method Investigate the differential-geometric properties of the exponential map and Riemannian structure.
result Connections between skew symmetric matrices and orthogonal groups are revealed.
In this article, we discuss which semisimple locally symmetric spaces admit an AHS--structure invariant to local symmetries. We classify them for all types of AHS--structures and determine possible equivalence classes of such AHS--structures.
Since the work of Henri Cartan finite dimensional Riemannian symmetric spaces are an important subject of mathematical interest. They are related in a natural way to semisimple Lie groups. In this work we introduce and study their infinite dimensional generalization: Affine Kac-Moody symmetric spaces. Affine Kac-Moody …
Characterizes real left symmetric algebras with positive definite Koszul form and related Kähler-Einstein structures.
problem Characterizing real left symmetric algebras with positive definite Koszul form.
method Analyzes the properties of left multiplication operators and symmetric bilinear forms.
result Provides a complete characterization of real left symmetric algebras with positive definite Koszul form.
In this work, we are interested in a non symmetric homogeneous space, namely SO(2m)/Sp(m). We show that this space admits a structure of Z22-symmetric space. We describe all the non degenerated metrics and classify the Riemannian and Lorentzian ones.
Symmetric TSP is structurally equivalent to a constrained Group Steiner Tree Problem.
problem Finding the shortest tour in a symmetric TSP.
method Structural equivalence between symmetric TSP and constrained Group Steiner Tree Problem.
result Maximizing net weight in the cGSTP is equivalent to minimizing the TSP tour length.
The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.
problem Constructing invariant Calabi-Yau structures on complexified symmetric spaces.
method Solutions of a Monge-Ampère type equation.
result Existence of solutions to the Monge-Ampère type equation.
We study Spin(9)-structures on 16-dimensional Riemannian manifolds and characterize the geometric types admitting a connection with totally skew-symmetric torsion.
Flag manifolds are in general not symmetric spaces. But they are provided with a structure of Z2k-symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. We detail for the flag manifold SO(5)/SO(2)×SO(2)×SO(1) what are the conditions…
We construct series of examples of exotic smooth structures on compact locally symmetric spaces of noncompact type. In particular, we obtain higher rank examples, which do not support Riemannian metric of nonpositive curvature. The examples are obtained by taking the connected sum with an exotic sphere. To detect the c…
We present some basic results on a natural Poisson structure on any compact symmetric space. The symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold.
The paper studies geometric structures of wormholes using a new connection.
problem Exploring new geometric properties of wormholes.
method Extended Levi-Civita connection to semi-symmetric non-metric connections.
result Morris-Thorne wormholes exhibit specific geometric properties.
The paper studies geometries with parallel skew-symmetric torsion and their submersions.
problem Understanding geometries with parallel skew-symmetric torsion.
method Analyzing metric connections and submersions.
result Complete local classification of geometries with parallel skew-symmetric torsion in principal bundle cases.
Fried's theorem proven for symmetric space boundaries.
problem Characterizing manifolds with similarity structures.
method General proof for all rank one symmetric space boundary geometries.
result Closed manifolds are either complete or develop onto Heisenberg-type spaces.
Study shows lightlike hypersurfaces in indefinite Sasakian manifolds are not symmetric.
problem Characterizing curvature symmetries in lightlike hypersurfaces of indefinite Sasakian manifolds.
method Analyzing properties of lightlike hypersurfaces in indefinite Sasakian manifolds.
result Lightlike hypersurfaces are not locally symmetric, semi-symmetric, or semi-parallel.
New connections on symmetric spaces with invariant properties.
problem Understanding invariant connections on hermitian symmetric spaces.
method Introduced a class of G-invariant connections on homogeneous bundles over hermitian symmetric spaces. result Parameter space of connections is a normal variety with a canonical anti-holomorphic involution.
The notion of Γ-symmetric space is a natural generalization of the classical notion of symmetric space based on Z2-grading of Lie algebras. In our case, we consider homogeneous spaces G/H such that the Lie algebra $\g$ of G admits a Γ-grading where Γ is a finite abelian group. In this work we study Rieman…
Study finds only two CROSSes can have specific quaternionic structure.
problem Characterizing compact rank one symmetric spaces with quaternionic structures.
method Analyzing properties of CROSSes (compact rank one symmetric spaces).
result Only HPn and CP2 admit almost quaternionic structures. The paper studies geometric structures in Sol_3 with two connections.
problem Analyzing geometric structures in Sol_3 using specific connections.
method Used Levi-Civita and semi-symmetric non-metric connections to study Sol_3.
result Concluded geometric structures in Sol_3 with both connections.
Study of colored triangulations linked to symmetric groups.
problem Enumeration of permutations up to conjugation.
method Analysis of checker triangulated surfaces and their Belyi data.
result Links between triangulations and infinite symmetric groups.
We prove that compact quaternionic-Kähler manifolds of positive scalar curvature admit no almost complex structure, even in the weak sense, except for the complex Grassmannians Gr2(Cn+2). We also prove that irreducible inner symmetric spaces M4n of compact type are not weakly complex, except for spheres and …
Symmetrizes 4d and 3d BPS quivers for Argyres-Douglas theories.
problem Understanding the relationship between 4d and 3d BPS quivers.
method Analyzes geometric backgrounds and uses skein modules to derive quiver partition functions.
result Proves isomorphism between 4d wall-crossing and unlinking of symmetric quivers.
We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties.