Related papers: On Duality, Legendre Bundles and Deformations
In $(2n+1)$-dimensional non-Sasakian contact metric manifolds, we consider Legendre curves whose mean curvature vector fields are $\mathcal{C}$-parallel or $\mathcal{C}$-proper in the tangent or normal bundles. We obtain the curvature…
In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map…
A basis for the space of generalized theta functions of level one for the spin groups, parameterized by the theta characteristics (the even theta characteristcs for the odd spin groups) on a curve, is shown to be projectively flat over the…
Although perturbative quantum field theory is highly successful, it possesses a number of well-known analytic problems, from ultraviolet and infrared divergencies to the divergence of the perturbative expansion itself. As a consequence, it…
Results on derivations and automorphisms of some quantum and classical Poisson algebras, as well as characterizations of manifolds by the Lie structure of such algebras, are revisited and extended. We prove in particular somehow unexpected…
We define a natural concept of duality for the h-Hopf algebroids introduced by Etingof and Varchenko. We prove that the special case of the trigonometric SL(2) dynamical quantum group is self-dual, and may therefore be viewed as a…
The Legendre curve in the unit tangent bundle over Euclidean plane is a plane curve with a moving frame. We have the (Legendre) curvature of the Legendre curve, and the existence and uniqueness theorems for the curvature are valid. In this…
This review paper is concerned with the generalizations to field theory of the tangent and cotangent structures and bundles that play fundamental roles in the Lagrangian and Hamiltonian formulations of classical mechanics. The paper…
It is shown that a canonical geometric setting of the integrable TED equation is a Kahlerian tangent bundle of an affine manifold. The remarkable multi-dimensional consistency of this 4+4-dimensional dispersionless partial differential…
A noncommutative-geometric generalization of the classical formalism of frame bundles is developed, incorporating into the theory of quantum principal bundles the concept of the Levi-Civita connection. The construction of a natural…
In this paper we construct arithmetic analogs of the Riemann-Roch theorem and Serre's duality for line bundles. This improves on the works of Tate and van der Geer - Schoof. We define $H^0(L)$ and $H^1(L)$ as some convolution of measures…
In this note we give a complete description of all the hyperplane section of the projective bundle associated to the tangent bundle of $\mathbb{P}^2$ under its natural embedding in $\mathbb{P}^7.$ As an application one obtains a description…
In this paper, motivated by the singularity formation of ASD connections in gauge theory, we study an algebraic analogue of the singularity formation of families of rank two holomorphic vector bundles over surfaces. For this, we define a…
We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic,…
Noncommutative K\"ahler structures were recently introduced by the second author as a framework for studying noncommutative K\"ahler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector…
A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle.…
A duality theory of bundles of C$^*$-algebras whose fibres are twisted transformation group algebras is established. Classical T-duality is obtained as a special case, where all fibres are commutative tori, i.e. untwisted group algebras for…
The first goal of the article is to solve several fundamental problems in the theory of holomorphic bundles over non-algebraic manifolds: For instance we prove that stability and semi-stability are Zariski open properties in families when…
This text describes the fiber bundle structure and shows its universality for writing the laws of classical physics: newtonian, relativistic and quantum mechanics.
We prove that the canonical bundle of any holomorphic family of compact complex algebraic manifolds carries a singular Hermitian metric having non-negative curvature current and such that every holomorphic section of the canonical bundle of…