Linear Connections on Fuzzy Manifolds

Linear connections are introduced on a series of noncommutative geometries which have commutative limits. Quasicommutative corrections are calculated.Comment: 10 pages PlainTex; LPTHE Orsay 95/42; ESI Vienna 23

Linear connections on matrix geometries

A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique metric connection.Comment: 14p, LPTHE-ORSAY 94/9

Linear Connections in Non-Commutative Geometry

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of $\Omega^1$. The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois- Violette and then a generalisation to the framework proposed by Connes as well as other non-commutative differential calculi is suggested. The covariant derivative obtained admits an extension to the tensor product of several copies of $\Omega^1$. These constructions are illustrated with the example of the algebra of $n \times n$ matrices.Comment: 15 pages, LMPM ../94 (uses phyzzx

Fuzzy Surfaces of Genus Zero

A fuzzy version of the ordinary round 2-sphere has been constructed with an invariant curvature. We here consider linear connections on arbitrary fuzzy surfaces of genus zero. We shall find as before that they are more or less rigidly dependent on the differential calculus used but that a large number of the latter can be constructed which are not covariant under the action of the rotation group. For technical reasons we have been forced to limit our considerations to fuzzy surfaces which are small perturbations of the fuzzy sphere.Comment: 11 pages, Late

On Finite 4D Quantum Field Theory in Non-Commutative Geometry

The truncated 4-dimensional sphere $S^4$ and the action of the self-interacting scalar field on it are constructed. The path integral quantization is performed while simultaneously keeping the SO(5) symmetry and the finite number of degrees of freedom. The usual field theory UV-divergences are manifestly absent.Comment: 18 pages, LaTeX, few misprints are corrected; one section is remove

Chirality and Dirac Operator on Noncommutative Sphere

We give a derivation of the Dirac operator on the noncommutative $2$-sphere within the framework of the bosonic fuzzy sphere and define Connes' triple. It turns out that there are two different types of spectra of the Dirac operator and correspondingly there are two classes of quantized algebras. As a result we obtain a new restriction on the Planck constant in Berezin's quantization. The map to the local frame in noncommutative geometry is also discussed.Comment: 24 pages, latex, no figure

On the Informational Comparison of Qualitative Fuzzy Measures

International audienceFuzzy measures or capacities are the most general representation of uncertainty functions. However, this general class has been little explored from the point of view of its information content, when degrees of uncertainty are not supposed to be numerical, and belong to a finite qualitative scale, except in the case of possibility or necessity measures. The thrust of the paper is to define an ordering relation on the set of qualitative capacities expressing the idea that one is more informative than another, in agreement with the possibilistic notion of relative specificity. To this aim, we show that the class of qualitative capacities can be partitioned into equivalence classes of functions containing the same amount of information. They only differ by the underlying epistemic attitude such as pessimism or optimism. A meaningful information ordering between capacities can be defined on the basis of the most pessimistic (resp. optimistic) representatives of their equivalence classes. It is shown that, while qualitative capacities bear strong similarities to belief functions, such an analogy can be misleading when it comes to information content