Related papers: Higher order Hessian structures on manifolds

This paper studies the magnitude homology groups of geodesic metric spaces. We start with a description of the second magnitude homology of a general metric space in terms of the zeroth homology groups of certain simplicial complexes. Then,…

We define a supercategorification of the $q$-Schur algebra of level two and an odd analogue of $\mathfrak{gl}_2$-foams. Using these constructions, we define a homological invariant of tangles, and show that it coincides with odd Khovanov…

The Hessian geometry is the real analogue of the K\"ahler one. Sasakian geometry is an odd-dimensional counterpart of the K\"ahler geometry. In the paper, we study the connection between projective Hessian and Sasakian manifolds analogous…

Vector bundles and double vector bundles, or $2$-fold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids, Courant algebroids and double Lie algebroids. It is known that all these…

Given a (genus 2) cube-with-holes M, i.e. the complement in S^3 of a handlebody H, we relate intrinsic properties of M (like its cut number) with extrinsic features depending on the way the handlebody H is knotted in S^3. Starting from a…

To each second-order ordinary differential equation $\sigma $ on a smooth manifold $M$ a $G$-structure $P^\sigma $ on $J^1(\mathbb{R},M)$ is associated and the Chern connection $\nabla ^\sigma $ attached to $\sigma $ is proved to be…

We describe conditions under which a spacetime connection and a scaled Lorentzian metric define natural symplectic and Poisson structures on the tangent bundle of the Einstein spacetime.

We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M. We suppose that the plane field defined by the complex lines is a contact plane and construct an…

In this paper we establish relations among the module of high-order derivations of the Hasse-Schmidt algebra and the module of high-order derivations of the base ring.

Several representations of geometric shapes involve quotients of mapping spaces. The projection onto the quotient space defines two sub-bundles of the tangent bundle, called the horizontal and vertical bundle. We investigate in these notes…

We define a class of metrics that extend the Sasaki metric of a tangent manifold of a Riemannian manifold. The new metrics are obtained by the transfer of the generalized (pseudo-)Riemannian metrics of the pullback of the big tangent bundle…

We construct a sequence of geodesics on the modular surface such that the complement of the canonical lifts to the unit tangent bundle are arithmetic 3-manifolds.

We construct examples of nonresolvable generalized $n$-manifolds, $n\geq 6$, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed $n$-manifold. We further investigate the structure of generalized…

In this PhD thesis, we have studied certain geometric structures over Lie groupoids and differentiable stacks. This thesis is based on the work [arXiv:2103.04560, arXiv:2012.08447, arXiv:2012.08442, arXiv:1907.00375]. In [arXiv:1907.00375],…

We introduce new families of quandles that serve as invariants for classifying closed orientable surfaces. These families generalize the classical Dehn quandle and are defined, respectively, on isotopy classes of unoriented closed curves…

In this note we examine a natural concept of a curve on a supermanifold and the subsequent notion of the jet of a curve. We then tackle the question of geometrically defining the higher order tangent bundles of a supermanifold. Finally we…

Quandles are self-distributive algebraic structures known as sources of strong knots invariants, but also appearing in other contexts. From any quandle, one can construct two invariants: the structure group and the second quandle homology…

We study the homotopy aspects of the twisted Chern classes of torsion bundle gerbe modules. Using Sullivan's rational homotopy theory, we realize the twisted Chern classes at the level of classifying spaces. The construction suggests a…

We show that on an arbitrary collection of objects there is a wide variety of higher order architectures governed by hyperstructures. Higher order gluing, local to global processes, fusion of collections, bridges and higher order types are…

In a previous paper we outlined how discrete torsion can be understood geometrically as an analogue of orbifold U(1) Wilson lines. In this paper we shall prove the remaining details. More precisely, in this paper we describe gerbes in terms…