arXiv research

New connections found with specific torsion properties.

Classifies Riemannian manifolds with specific torsion properties.

We consider non-infinitesimal deformations of G2-structures on 7-dimensional manifolds and derive an exact expression for the torsion of the deformed G2-structure. We then specialize to a case when the deformation is defined by a vector v and we explicitly derive the expressions for the different torsion components of …

Homotopy braid groups are shown to be torsion-free.

On an open manifold, the spaces of metrics or connections of bounded geometry, respectively, split into an uncountable number of components. We show that for a pair of metrics or connections, belonging to the same component, relative $ζ$-functions, determinants, torsion for pairs of generalized Dirac operators are well…

The paper defines and calculates Reidemeister torsion for a specific class of representations.

Proves freely 2-periodic knots have two canonical components in their character variety.

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$-action on $X$. We introduce the Fourier components of the Ray-Singer analytic torsion on $X$ with respect to the $S^1$-action. We establish an asymptotic formula for the Fourier components of the an…

We define an (equivariant) quaternionic analytic torsion for antiselfdual vector bundles on quaternionic Kaehler manifolds, using ideas by Leung and Yi. We compute this torsion for vector bundles on quaternionic homogeneous spaces with respect to any isometry in the component of the identity, in terms of roots and Weyl…

The paper defines and analyzes the adjoint Reidemeister torsion for connected sums of knots.

Researchers compute twisted Reidemeister torsion for hyperbolic 3-manifolds.

We study lens space surgeries along two different families of 2-component links, denoted by $A_{m,n}$ and $B_{p,q}$, related with the rational homology 4-ball used in J.\ Park's (generalized) rational blow down. We determine which coefficient $r$ of the knotted component of the link yields a lens space by Dehn surgery.…

In this paper we define the Reidemeister torsion as a rational function on the geometric components of the character variety of a one-cusped hyperbolic manifold M. We study its poles and zeros, and we deduce sufficient conditions on the manifold M for this function being non-constant.

Researchers find limits on curvature of certain 3D solitons.

We prove that refined analytic torsion on a manifold with boundary is an analytic section of the determinant line bundle over the representation variety. As a fundamental application we establish a gluing formula for refined analytic torsion on connected components of the complex representation space which contain a un…

In the previous article "Refined Analytic Torsion on Manifolds with Boundary" we have presented a construction of refined analytic torsion in the spirit of Braverman and Kappeler, which does apply to compact manifolds with and without boundary. We now derive a gluing formula for our construction, which can be viewed as…

We overview the properties of non-infinitesimal deformations of G2-structures on seven-manifolds, and in particular, focus on deformations that lie in the seven-dimensional representation of G2 and are thus defined by a vector. We then consider deformations from G2-structures with the torsion class having one-dimension…

We analyse the relationship between the components of the intrinsic torsion of an SU(3) structure on a 6-manifold and a G_2 structure on a 7-manifold. Various examples illustrate the type of SU(3) structure that can arise as a reduction of a metric with holonomy G_2.

Researchers confirm integral formulas for $G_2$-structures in detail.

We construct a combinatorial invariant of 3-orbifolds with singular set a link that generalizes the Turaev torsion invariant of 3-manifolds. We give several gluing formulas from which we derive two consequences. The first is an understanding of how the components of the invariant change when we remove a curve from the …

This paper is devoted to a proof of a generalized Ray-Singer conjecture for a manifold with boundary (the Dirichlet and the Neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary). The Ray-Singer conjectur…

We extend the definition of analytic and Reidemeister torsion from closed compact Riemannian manifolds to compact Riemannian manifolds with boundary $(M, \partial M)$, given a flat bundle $\Cal F$ of $\Cal A$-Hilbert modules of finite type and a decomposition of the boundary $\partial M =\partial_- M \cup \partial_+ M$…

Bismut and Zhang computed the ratio of the Ray-Singer and the combinatorial torsions corresponding to non-unitary representations of the fundamental group. In this note we show that for representations which belong to a connected component containing a unitary representation the Bismut-Zhang formula follows rather easi…

We study the natural G_2 structure on the unit tangent sphere bundle SM of any given orientable Riemannian 4-manifold M, as it was discovered in \cite{AlbSal}. A name is proposed for the space. We work in the context of metric connections, or so called geometry with torsion, and describe the components of the torsion o…

Starting from the results in math.DG:1212.3161 we prove that for a given Bianchi group, certain natural coefficent modules and a lot of sequences of congruence subgroups of the size of the torsion subgroup of the first homology grows exponentially with the index (we give an explicit rate). We also prove limit multiplic…

The paper finds torsion in Johnson homomorphisms' cokernels for large genus surfaces.

The construction of invariants of three-dimensional manifolds with a triangulated boundary, proposed earlier by the author for the case when the boundary consists of not more than one connected component, is generalized to any number of components. These invariants are based on the torsion of acyclic complexes of geome…

The aim of this paper is to describe the local Bianchi identities for an $h$-normal $Γ$-linear connection of Cartan type $\nablaΓ$ on the first-order jet space $J^1(R,M)$. In this direction, we present the local expressions of the adapted components of the torsion and curvature d-tensors produced by $\nablaΓ$ and we gi…

We consider the Yang-Mills flow on hyperbolic 3-space. The gauge connection is constructed from the frame-field and (not necessarily compatible) spin connection components. The fixed points of this flow include zero Yang-Mills curvature configurations, for which the spin connection has zero torsion and the associated R…

If $M$ is a finite volume complete hyperbolic 3-manifold with one cusp and no 2-torsion, the geometric component $X_M$ of its $\SL(2,\BC)$-character variety is an affine complex curve, which is smooth at the discrete faithful representation $ρ_0$. Porti defined a non-abelian Reidemeister torsion in a neighborhood of $ρ…

Study primes dividing torsion in homology of commuting elements in Lie groups.

Study pressure metrics for cusped Hitchin representations.

Unified CR-twistor spaces for $G_2$ and $Spin(7)$ structures.

The exterior derivative $d θ$ of the Lee form $θ$ of almost Hermitian manifolds is studied. If $ω$ is the Kähler two-form, it is proved that the $\mathbb{R}ω$-component of $dθ$ is always zero. expressions for the other components, in $[λ_0^{1,1}]$ and in $[[ λ^{2,0} ]]$, of $dθ$ are also obtained. They are given in ter…

The paper introduces the notion of Γ-linear connection \nabla on the 1-jet fibre bundle J^1(T,M), and presents its local components. We also describe the local Ricci and Bianchi identities of $\nabla$.

Study geometric flows of G2-structures, determining curvature and torsion invariants.

We prove a conjecture of K. Schmidt in algebraic dynamical system theory on the growth of the number of components of fixed point sets. We also generalize a result of Silver and Williams on the growth of homology torsions of finite abelian covering of link complements. In both cases, the growth is expressed by the Mahl…

Paper proves non-triviality of Johnson kernel torsion subgroup.

We give definitions of cohomology determinants for compact, connected, orientable 3-manifolds. We also give formulae relating cohomology determinants before and after gluing a solid torus along a torus boundary component. Cohomology determinants are related to Turaev torsion, though the author hopes that they have othe…

Study $p$-brane Galilean and Carrollian geometries via intrinsic torsion.

The paper proves new curvature estimates in quaternionic contact geometry.

New result on finite gerbes simplifies classification of certain bundles.

Formula derived for Spin(7)-structures on 8D manifolds.

The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle $E$ we define the generalized component $\gencomp (E)$ as the set of Clifford bundles $E'$ which have finite distance to $E$. If $D$, $D'$ are the associated generalized Dirac ope…

Geometric derivation of quantum dynamics from Lie group actions.

Let l be a link of d components. For every finite-index lattice in Z^d there is an associated finite abelian cover of S^3 branched over l. We show that the order of the torsion subgroup of the first homology of these covers has exponential growth rate equal to the logarithmic Mahler measure of the Alexander polynomial …

The aim of this paper is to present the main geometrical objects on the dual 1-jet bundle $J^{1*}(\cal{T},M)$ (this is the polymomentum phase space of the De Donder-Weyl covariant Hamiltonian formulation of field theory) that characterize our approach of multi-time Hamilton geometry. In this direction, we firstly intro…

Extends Coulomb gauge existence to non-associative gauge theory.