In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field , that is, , where or for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators…
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Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
Study constructs transverse metrics using transformations commuting with elliptic operators.
problem Existence of transverse metrics in foliation theory.
method Applying the Average Method to construct a transverse metric.
result Pseudogroup of local transformations equicontinuous and quasi-analytic.
The paper finds commutator formulas for Gradient Ricci Shrinker metrics and applies them to linear stability.
problem Linear stability of Gradient Ricci Shrinker metrics.
method Found commutator formulas and generalized a stability theorem.
result Generalized a necessary condition for linear stability.
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.
Spirals are not shortest paths in certain sub-Riemannian geometries.
problem Nonminimality of spiral-like curves in sub-Riemannian manifolds.
method Construction of a competing curve to demonstrate non-minimality.
result Spiral-like curves are not length minimizing in sub-Riemannian manifolds.
New product structures encode superintegrable Hamiltonian systems in Euclidean spaces.
problem Encoding superintegrable Hamiltonian systems using product structures.
method Introducing commutative and associative product structures on Euclidean spaces of dimension at least three, satisfying specific conditions.
result All abundant superintegrable Hamiltonian systems on Euclidean space of dimension at least three arise from these product structures.
The paper examines conditions for commuting conjugates of finite-order mapping classes.
problem Conditions for commuting conjugates of finite-order mapping classes.
method Derived necessary and sufficient conditions for commuting conjugates.
result Conditions for the primitivity of finite-order mapping classes.
Study simplicial volume and stable commutator length for one-relator groups.
problem Understanding the relationship between simplicial volume and stable commutator length for one-relator groups.
method Introduced simplicial volume for one-relator groups and related it to stable commutator length. Analyzed the relationship for various cases and random elements.
result Often, the relationship between simplicial volume and stable commutator length is linear, with a multiplicative error of O(1/N).
New projection complex shows some surface homeomorphisms have positive commutator length.
problem Understanding commutator length in surface homeomorphisms.
method Constructing unbounded quasi-trees and a new projection complex.
result Some surface homeomorphisms have positive stable commutator length.
The paper connects disentanglement to manifold charts and commutativity.
problem Discovering local charts of the data manifold for disentanglement.
method Interpreting disentanglement as local charts of the data manifold and studying commutativity.
result Commutativity is a central property in disentanglement, as shown in manifold, group theoretic, and probabilistic frameworks.
Establishes a duality theorem connecting quasimorphisms and commutator lengths in group theory.
problem Connecting quasimorphisms and commutator lengths in group theory.
method Geometric interpretation and algebraic proof of -commutator lengths.
result Bi-Lipschitz equivalence of scl on under certain conditions.
Real hypersurfaces in complex two-plane Grassmannians with commuting restricted Jacobi operatorsmath.DG
In this paper, we have considered a new commuting condition, that is, \big(resp. $(\Bar{R}_Nφ) S = S (\Bar{R}_Nφ$)\big) between the restricted Jacobi operator~ (resp. $\Bar{R}_Nφ$), and the Ricci tensor for real hypersurfaces in . In terms of this condition we…
We study the extent to which the gauge symmetry of abelian Yang-Mills can be deformed under two conditions: first, that the deformation depend on a two-form scale. Second, that the deformation preserve supersymmetry. We show that (up to a single parameter) the only allowed deformation is the one determined by the star …
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.
No almost complex structures on the six-sphere satisfy certain commutation conditions.
problem Existence of almost complex structures on the six-sphere with specific properties.
method Refined differential inequalities and almost-complex analogue of the Gauss map.
result No integrable almost complex structures on the six-sphere satisfy the given conditions.
Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.
problem Conditions for differential operators on non-compact harmonic manifolds to have specific properties.
method Analyzing the algebra of differential operators, their commutation properties, and using geometric averages.
result Algebra of differential operators on non-compact harmonic manifolds has specific properties related to radial fundamental solutions and dense heat-semigroups.
Proves that emergent algebras right-distributivity implies left-distributivity.
problem Proving the implication between emergent algebra distributivity conditions.
method Analyzing families of quasigroup operations indexed by commutative groups.
result Emergent algebras right-distributive imply left-distributive.
Develops Morse theory for commuting gradient-like vector fields.
problem Understanding the algebraic structure of the infrared data.
method Formal analogue of Morse theory for commuting vector fields, constructing L-infinity algebra and Maurer-Cartan elements.
result The algebraic formalism is similar to the algebra of the infrared of Gaiotto, Moore, and Witten.
The abstract defines and studies a Tits building for commutative rings and proves a Solomon-Tits theorem under certain conditions.
problem Defining and studying a Tits building for commutative rings.
method Proving a Solomon-Tits theorem for commutative rings under specific conditions, defining Steinberg modules, and computing ranks and lengths.
result Proves a Solomon-Tits theorem for commutative rings satisfying certain conditions.
Study properties of orbits of Hermann actions without commutability assumptions.
problem Investigate geometric properties of orbits of Hermann actions.
method Compute the second fundamental form and provide conditions for weak reflection and aridity.
result Sufficient conditions for weak reflection and aridity of orbits of Hermann action.
Let be a finite index subgroup of the mapping class group of a closed orientable surface , possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element has positive stable commutator length. In addition, we show that in these situations th…
The study identifies biharmonic submanifolds in compact symmetric spaces and Lie groups.
problem Characterizing biharmonic submanifolds in compact symmetric spaces and Lie groups.
method Using necessary and sufficient conditions derived from symmetric triad with multiplicities.
result Determined all biharmonic submanifolds in irreducible symmetric spaces of compact type and biharmonic hypersurfaces in compact Lie groups.
Study curvature operators in 4n-dimensional manifolds, finding new conformal invariants.
problem Analyzing curvature operators in oriented Riemannian 4n-manifolds.
method Examining finite systems of hafnian identities in eigenvalues, focusing on locally conformally flat cases.
result Discovering a new conformal invariant in dimensions 4n, related to nonnegativity of Euler characteristic.
New projection operators for multipatch spaces with stable properties.
problem Problems with non-matching interfaces in multipatch spaces.
method Construction of commuting projection operators on de Rham sequences of multipatch spaces with local tensor-product parametrization.
result Local and stable projection operators in any norm for shape-regular spline patches with different mappings and local refinements.
In this paper the author determines necessary and sufficient conditions for existence of the Ehresmann connection on a manifold foliated by locally free action of the commutative Lie group. Also here we describe structure of for a leaf in case such a connection exists. Finally we give some res…
The paper extends quasimorphisms on subgroups to larger groups.
problem Extending quasimorphisms from subgroups to larger groups.
method Provides a general sufficient condition for extendability of quasimorphisms on subgroups.
result New results for quasimorphisms on normal subgroups, including bi-Lipschitz equivalence of stable commutator length and group-theoretic Dehn filling.
New algebraic structures on manifolds generalize supergeometry concepts.
problem Developing algebraic structures for non-commutative manifolds.
method Introducing -commutative manifolds, Q-manifolds, and modular classes.
result Generalized modular classes for non-commutative spaces.
Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined as the supremum of commutator lengths of elements of its commutator subgroup. We …
Doodles link to commutator identities in a 2-sphere.
problem Understanding commutator identities in free groups via doodles.
method Analyzing doodles with proper noose systems and establishing bijections.
result A bijection between doodles and commutator identities.
Embeddings preserve stable commutator length for surfaces.
problem Stable commutator length in surfaces.
method Finding standard forms of admissible surfaces and proving homology vanishing conditions.
result Isometric embeddings preserve stable commutator length.
New braided Frobenius algebras created from specific Hopf algebras.
problem Creating new algebraic structures from Hopf algebras.
method Heap operation and Yang-Baxter operator on tensor product.
result Heap operation induces a braiding compatible with Frobenius operations.
Examining singularities of commuting vector fields on submanifolds.
problem Understanding singularities of commuting vector fields on submanifolds.
method Analyzing real submanifolds within Kähler manifolds.
result Characterized singularities of commuting vector fields.
Two commutators generate mapping class groups of surfaces with genus ≥5.
problem Generating mapping class groups efficiently.
method Analyzing commutators in mapping class groups of surfaces.
result Mapping class groups of surfaces with genus ≥5 are generated by two commutators.
Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.
problem Understanding stable commutator length on infinite-type surfaces.
method Analyzing mapping class groups of infinite-type surfaces, showing continuity and openness of commutator subgroups.
result Stable commutator length defines a continuous function on commutator subgroups of infinite-type mapping class groups.
We consider a family of vector fields satisfying a suitable higher order involutivity condition. We discuss the definition of commutators, the regularity of Sussmann's orbits and the Poincaré inequality.
New proof and description of commutator subgroups for free and surface groups.
problem Understanding commutator subgroups of free and surface groups.
method Geometric proof and representation-theoretic description.
result New free generating sets and structure descriptions for commutator subgroups.
The paper proves conditions for Darboux integrability in diagonal hydrodynamic systems.
problem Conditions for Darboux integrability in diagonal hydrodynamic systems.
method Proof of conditions using Laplace transformation sequences and geometric interpretations.
result Diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable.
Simple proof of Hardy inequality on Carnot groups and hypoelliptic vector fields.
problem Proving Hardy inequality on Carnot groups and hypoelliptic vector fields.
method Integration by parts and analysis of commutator structure.
result Elementary proof of Hardy inequality on Carnot groups.
Study on commutator subgroups of welded braid groups, proving their finiteness and perfection.
problem Investigating the structure of commutator subgroups in welded braid groups.
method Proved finiteness and Hopfian property, showed perfection for , computed finite presentations.
result Commutator subgroups of welded braid groups are finitely generated, Hopfian, and perfect for .
We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action…
The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.
problem Generalizing differential geometric structures to algebroids.
method Introducing statistical, conjugate connection, and Hessian structures on anti-commutable pre-Leibniz algebroids.
result Statistical and conjugate connection structures are equivalent for admissible connections.
Formulae for non-symmetric connections derived from covariant derivatives.
problem Deriving commutation formulae for non-symmetric affine connections.
method Covariant derivatives of tensors with respect to symmetric and non-symmetric affine connections.
result Formulae for non-symmetric connections derived from covariant derivatives.
For an -manifold with boundary, we prove a localization formula applying to any equivariant cohomology theory satisfying a certain algebraic condition. We show how the localization result of Kalkman and a case of the quantization commutes with reduction theorem follow easily from the localization formula.
Noncommutatively deformed geometries, such as the noncommutative torus, do not exist generically. I showed in a previous paper that the existence of such a deformation implies compatibility conditions between the classical metric and the Poisson bivector (which characterizes the noncommutativity). Here I present anothe…
Analytic torsion form constructed for non-commutative spaces.
problem Constructing analytic torsion form on non-commutative spaces.
method Using fiber bundles, flat vector bundles, and crossed product algebras, we define a non-commutative deRham differential form.
result The constructed torsion form appears in a transgression formula and leads to an index formula.
We investigate the role of Hertling-Manin condition on the structure constants of an associative commutative algebra in the theory of integrable systems of hydrodynamic type. In such a framework we introduce the notion of F-manifold with compatible connection generalizing a structure introduced by Manin.
Homotopy commutativity in quasitoric manifolds depends on polytope structure and characteristic matrix type.
problem Conditions for homotopy commutativity in quasitoric manifolds.
method Analyzing characteristic matrices and polytope structures.
result Homotopy commutativity is determined by specific polytope and matrix conditions.