arXiv research

In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field $T$, that is, $R_ξφT=TR_ξφ$, where $T=A$ or $T=S$ for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators…

Study on deformation cohomology for braided commutative structures.

Study constructs transverse metrics using transformations commuting with elliptic operators.

The paper finds commutator formulas for Gradient Ricci Shrinker metrics and applies them to linear stability.

The paper studies Riemannian metrics on Lie groups with specific commutator subgroups.

Spirals are not shortest paths in certain sub-Riemannian geometries.

New product structures encode superintegrable Hamiltonian systems in Euclidean spaces.

New projection complex shows some surface homeomorphisms have positive commutator length.

The paper connects disentanglement to manifold charts and commutativity.

Establishes a duality theorem connecting quasimorphisms and commutator lengths in group theory.

In this paper, we have considered a new commuting condition, that is, $(R_ξφ) S = S (R_ξφ)$ \big(resp. $(\Bar{R}_Nφ) S = S (\Bar{R}_Nφ$)\big) between the restricted Jacobi operator~$R_ξφ$ (resp. $\Bar{R}_Nφ$), and the Ricci tensor $S$ for real hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$. 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.

Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.

We give a necessary and sufficient condition for orbits of commutative Hermann actions and actions of the direct product of two symmetric subgroups on compact Lie groups to be biharmonic in terms of symmetric triad with multiplicities. By this criterion, we determine all the proper biharmonic submanifolds in irreducibl…

Proves that emergent algebras right-distributivity implies left-distributivity.

The abstract defines and studies a Tits building for commutative rings and proves a Solomon-Tits theorem under certain conditions.

Study properties of orbits of Hermann actions without commutability assumptions.

Let $Γ$ be a finite index subgroup of the mapping class group $MCG(Σ)$ of a closed orientable surface $Σ$, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element $g\inΓ$ has positive stable commutator length. In addition, we show that in these situations th…

Study curvature operators in 4n-dimensional manifolds, finding new conformal invariants.

New projection operators for multipatch spaces with stable properties.

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 $C_0(M)|_L$ for a leaf $L \subset M$ in case such a connection exists. Finally we give some res…

We develop the formal analogue of the Morse theory for a pair of commuting gradient-like vector fields. The resulting algebraic formalism turns out to be very similar to the algebra of the infrared of Gaiotto, Moore and Witten (see [GMW], [KKS]): from a manifold M with the pair of gradient-like commuting vector fields,…

The paper extends quasimorphisms on subgroups to larger groups.

New algebraic structures on manifolds generalize supergeometry concepts.

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 …

A one-relator group is a group $G_r$ that admits a presentation $\langle S \mid r \rangle$ with a single relation $r$. One-relator groups form a rich classically studied class of groups in Geometric Group Theory. If $r \in F(S)'$, the commutator subgroup of $F(S)$, we introduce the simplicial volume of $\| G_r \|$. We …

Doodles link to commutator identities in a 2-sphere.

Examining singularities of commuting vector fields on submanifolds.

Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.

Embeddings preserve stable commutator length for surfaces.

New braided Frobenius algebras created from specific Hopf algebras.

We consider a family of $C^1$ 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.

Let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. In this paper, we derive necessary and sufficient conditions for two finite-order mapping classes to have commuting conjugates in $\text{Mod}(S_g)$. As an application of this result, we show that any finite-order…

The paper proves conditions for Darboux integrability in diagonal hydrodynamic systems.

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…

Formulae for non-symmetric connections derived from covariant derivatives.

For an $S^{1}$-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…

The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.

Homotopy commutativity in quasitoric manifolds depends on polytope structure and characteristic matrix type.

Study introduces dynamical ideals for non-commutative rings and classifies knots and links.

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.

Covariance is shown as a commutator in random variable calculus.

Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.

We show that solutions of Thurston equation on triangulated 3-manifolds in a commutative ring carry topological information. We also introduce a homogeneous Thurston equation and a commutative ring associated to triangulated 3-manifolds.

In this paper we use 3-manifold techniques to illuminate the structure of the string link monoid. In particular, we give a prime decomposition theorem for string links on two components as well as give necessary conditions for string links to commute under the stacking operation.