arXiv research

Paper compares absolute and relative real analytic torsion forms over fibrations.

New spectral torsion defined for rescaled Dirac operators.

Study of torsion forms for positive line bundles.

Study on instantons in $G_2$ and $Spin(7)$ manifolds.

We review the Reidemeister torsion, Ray-Singer's analytic torsion and the Cheeger-M"uller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differentia…

Paper studies the full asymptotic torsion forms of flat bundles.

Analytic surgery and gluing formula for torsion forms in fiber bundles.

Simplified proof of gluing formula for analytic torsion forms.

We introduce and study a canonical quadratic form, called the torsion quadratic form, of the determinant line of a flat vector bundle over a closed oriented odd-dimensional manifold. This quadratic form caries less information than the refined analytic torsion, introduced in our previous work, but is easier to construc…

We introduce multi-torsion, a spectral invariant generalizing Ray-Singer analytic torsion. We define multi-torsion for compact manifolds with a certain local geometric product structure that gives a bigrading on differential forms. We prove that multi-torsion is metric-independent in a suitable sense. Our definition of…

Paper introduces a new multilinear functional for spectral triples and computes its properties.

Formula for analytic torsion forms in fibrations by projective curves.

Study torsion and curvature in ACYT and AHKT 8-manifolds.

The paper studies curvature identities and solitons on Spin(7)-manifolds.

New insights on ACYT 6-manifolds reveal instanton conditions for curvature.

This paper continues math.DG/9903140. Here we construct a linking form on the torsion part of middle dimensional extended L^2 homology and cohomology of odd-dimensional manifolds. We give a geometric necessary condition when this linking form is hyperbolic. We compute this linking form in case when the manifold bounds.…

We study the intrinsic torsion of almost quaternion-Hermitian manifolds via the exterior algebra. In particular, we show how it is determined by particular three-forms formed from simple combinations of the exterior derivatives of the local Kaehler forms. This gives a practical method to compute the intrinsic torsion a…

Bi-forms extend contrast functions to handle torsion in information geometry.

We define analytic torsion for the twisted de Rham complex, consisting of the spaces of differential forms on a compact oriented Riemannian manifold X valued in a flat vector bundle E, with a differential given by a flat connection on E plus an odd-degree closed differential form H on X. The difficulty lies in the fact…

The study explores properties of metric connections with skew torsion and their curvature identities.

Characterizes curves in totally umbilical surfaces of space forms.

Analytic torsion expansions for symmetric and complex homogeneous spaces.

In this paper we define the adjoint Reidemeister torsion as a differential form on the character variety of a compact oriented 3-manifold with toral boundary, and prove it defines a regular volume form. Then we show that the torsion form can vanish only at singular points of the character variety. In fact, if the singu…

Torsion objects of von Neumann categories describe the phenomen "spectrum near zero" discovered by S. Novikov and M. Shubin. In this paper we classify Hermitian forms on torsion objects of a finite von Neumann category. We prove that any such form can be represented as a discriminant form of a degenerate Hermitian form…

In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both $L^2$ combinatorial and $L^2$ analytic torsion invariants …

The paper calculates spectral torsion for rescaled Dirac operators on manifolds.

Extends Gauduchon's result to higher dimensions, showing balanced metrics.

Identifies a mod-$p$ triple cup product for rational homology 3-spheres with specific first homology.

Spin(7) geometry linked to multisymplectic geometry.

Given a fiber bundle $Z \to M \to B$ and a flat vector bundle $E \to M$ with a compatible action of a discrete group $G$, and regarding $B / G$ as the non-commutative space corresponding to the crossed product algebra, we construct an analytic torsion form as a non-commutative deRham differential form. We show that our…

Formula derived for torsion of modified Dirac operator.

Study curvature properties of $G_2$ connections with skew-symmetric torsion.

Proves torsion and curvature are unique for smooth manifolds.

New theory allows simultaneous block-diagonalization of commuting operator fields.

The study examines hyperbolic 3-manifolds with uniform spectral gaps for coclosed 1-forms.

The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form itself is closely related to the shape map of the connection. The codimension one …

We introduce and study a notion of `Sasaki with torsion structure' (ST) as an odd-dimensional analogue of Kähler with torsion geometry (KT). These are normal almost contact metric manifolds that admit a unique compatible connection with 3-form torsion. Any odd-dimensional compact Lie group is shown to admit such a stru…

Paper connects homologically fibered links to solutions of equations.

The paper derives formulas for symplectic volume forms on surface representation varieties.

In this paper, we prove that the Reidemeister torsion twisted by the adjoint representation, which is considered as a 1-form, on the SU(2)-character variety of a knot exterior is invariant under mutation along a Conway sphere.

KT-geometry is the geometry of a Hermitian connection whose torsion is a 3-form. HKT-geometry is the geometry of a hyper-Hermitian connection whose torsion is a 3-form. We identify non-trivial conditions for a reduction theory for these types of geometry.

Study characterizes $G_2$-structures on specific Lie groups and identifies harmonic conditions.

It is shown that an HKT-space with closed parallel potential 1-form has $D(2,1;-1)$-symmetry. Every locally conformally hyperkähler manifold generates this type of geometry. The HKT-spaces with closed parallel potential 1-form arising in this way are characterized by their symmetries and an inhomogeneous cubic conditio…

The refined analytic torsion associated to a flat vector bundle over a closed odd-dimensional manifold canonically defines a quadratic form $τ$ on the determinant line of the cohomology. Both $τ$ and the Burghelea-Haller torsion are refinements of the Ray-Singer torsion. We show that whenever the Burghelea-Haller torsi…

We discuss the construction of Sp(2)Sp(1)-structures whose fundamental form is closed. In particular, we find 10 new examples of 8-dimensional nilmanifolds that admit an invariant closed 4-form with stabiliser Sp(2)Sp(1). Our constructions entail the notion of SO(4)-structures on 7-manifolds. We present a thorough inve…

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

Formula connects analytic torsion forms of fibration and its pieces.

Torsion invariants for manifolds which are not simply connected were introduced by K. Reidemeister and generalized to higher dimensions by W. Franz. The Reidemeister torsion, was the first invariant of manifolds which was not a homotopy invariant. The analytic counterpart of the combinatorial Reidemeister torsion was i…