Related papers: Noncommutative integrability and action-angle vari…
The geometric approach to mechanics based on the Jacobi metric allows to easily construct natural mechanical systems which are integrable (actually separable) at a fixed value of the energy. The aim of the present paper is to investigate…
Following the guidelines of classical differential geometry the `building material' for the tensor calculus in non-commutative geometry is suggested. The algebraic account of moduli of vectors and covectors is carried out.
We present a short introductory overview of the non-commutative extensions of several classical physical theories. After a general discussion of the reasons that suggest that the non-commutativity is a major issue that will eventually lead…
A notion of fundamental group of spectral triples has been introduced. The notion uses a noncommutative analogue of unramified coverings. It was shown that in commutative case this fundamental group is a profinite completion of fundamental…
We initiate a study of projections and modules over a noncommutative cylinder, a simple example of a noncompact noncommutative manifold. Since its algebraic structure turns out to have many similarities with the noncommutative torus, one…
This is a compilation of some well known propositions of Alain Connes concerning the use of noncommutative geometry in mathematical physics.
This is an expository article on the noncommutative singularity theory of power series in noncommuting variables, its motivation from deformation theory, and its applications to contractibility of curves and the classification of smooth…
Coordinate noncommutativity, rather than being introduced through deformations of operator products, is achieved by coupling an auxiliary system with large energy excitations to the one of interest. Integrating out the auxiliary dynamics,…
We describe the gauge-theoretic approach to transformations in integrable geometry through discussion of two classical examples: surfaces of constant negative Gauss curvature and isothermic surfaces. These are purely expository notes…
The restrictions imposed on the strong force in the `non-commutative standard model' are examined. It is concluded that given the framework of non-commutative geometry and assuming the electroweak sector of the standard model many details…
We introduce a category of noncommutative bundles. To establish geometry in this category we construct suitable noncommutative differential calculi on these bundles and study their basic properties. Furthermore we define the notion of a…
For a noncommutative space X, we study Inj(X), the set of isomorphism classes of indecomposable injective X-modules. In particular, we look at how this set, suitably topologized, can be viewed as an underlying "spectrum" for X. As…
A new framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully flat quantum homogeneous…
A Riemannian geometry of noncommutative n-dimensional surfaces is developed as a first step towards the construction of a consistent noncommutative gravitational theory. Historically, as well, Riemannian geometry was recognized to be the…
A noncommutative algebra corresponding to the classical catenoid is introduced together with a differential calculus of derivations. We prove that there exists a unique metric and torsion-free connection that is compatible with the complex…
We define a relative version of contact homology for contact manifolds with convex boundary, and prove basic properties of this relative contact homology. Similar considerations also hold for embedded contact homology.
A motivation of using noncommutative and nonarchimedean geometry on very short distances is given. Besides some mathematical preliminaries, we give a short introduction in adelic quantum mechanics. We also recall to basic ideas and tools…
Nowadays, noncommutative geometry is a growing domain of mathematics, which can appear as a promising framework for modern physics. Quantum field theories on "noncommutative spaces" are indeed much investigated, and suffer from a new type…
In this paper we present the theorem on Lie integrability by quadratures for time-independent Hamiltonian systems on symplectic and contact manifolds, and for time-dependent Hamiltonian systems on cosymplectic and cocontact manifolds. We…
We study the geometry of null hypersurfaces in indefinite complex contact manifolds. We prove several classification results for a variety of well-known null hypersurfaces, including the totally umbilic, totally screen umbilic, and the…