On Curvature in Noncommutative Geometry

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.Comment: 16 pages, PlainTe

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

Non Commutative Differential Geometry, and the Matrix Representations of Generalised Algebras

The underlying algebra for a noncommutative geometry is taken to be a matrix algebra, and the set of derivatives the adjoint of a subset of traceless matrices. This is sufficient to calculate the dual 1-forms, and show that the space of 1-forms is a free module over the algebra of matrices. The concept of a generalised algebra is defined and it is shown that this is required in order for the space of 2-forms to exist. The exterior derivative is generalised for higher order forms and these are also shown to be free modules over the matrix algebra. Examples of mappings that preserve the differential structure are given. Also given are four examples of matrix generalised algebras, and the corresponding noncommutative geometries, including the cases where the generalised algebra corresponds to a representation of a Lie algebra or a $q$-deformed algebra.Comment: 16 pages Latex, No figures. Accepted for publication: Journal of Physics and Geometry, March 199

On the first order operators in bimodules

We analyse the structure of the first order operators in bimodules introduced by A. Connes. We apply this analysis to the theory of connections on bimodules generalizing thereby several proposals.Comment: 13 pages, AMSLaTe

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

How Did E. M. Walker Measure the Length of the Labium of Nymphs of \u3ci\u3eAeshna\u3c/i\u3e and \u3ci\u3eRhionaeschna\u3c/i\u3e (Odonata: Aeshnidae)?

The exhaustive studies of nymphs of Aeshna Fabricius and Rhionaeschna Förster by E. M. Walker (1912-1958) have long guided the taxonomy of these groups and formed the basis for keys still in use today. However, uncertainty about how he measured the length of the labium, including the varied terminology he used over the duration of his career concerning this structure, has led to confusion about application of his taxonomic recommendations. We recalculated ratios of the maximum width/length [W(max)/L] by measuring the illustration dimensions of folded labia and prementums in publications throughout his career and compared these data with the ratios he stated in those publications and with ratios derived from measurements of specimens in our collections. Our results show that from 1912 to 1941, Walker restricted length measurement to the prementum proper (which he called the “mentum of the labium”), exclusive of the ventrally visible portion of the postmental hinge. However, in 1941 he reported ratios from length measurements done two ways, excluding the postmental hinge in his description of the nymph of A. verticalis Hagen, but including the hinge in his description of the nymph of A. septentrionalis Burmeister (Whitehouse 1941). In Walker’s most recent and influential work (1958), he included the postmental hinge in labium length measurements of nine species, but restricted length measurements to the prementum for five others. He was consistent with the use of terms, using both “folded labium” by which he meant the prementum plus the postmental hinge, and “prementum” by which he meant only that structure. However, Walker’s descriptions of the labium in his latest work are buried in long, frequently punctuated sentences that for most species include the terms “folded labium” and “prementum” in the same sentence, so careful reading is required to know which term is intended in the width/length ratio. Width/length ratios we each calculated independently were invariably similar for a given species and were usually similar to Walker’s stated ratio for that species. These similarities affirm our conclusion that while labium measurements must be done with care, they are closely repeatable among workers and will consistently lead to correct determinations in properly designed couplets of dichotomous keys to these genera. We recommend measuring the length of the prementum proper in future studies of these genera when labium ratios are calculated because we found less variability in those cases than when the measurements included the postmental hinge. An approximate conversion between the two methods of calculating W(max)/L ratios can be made as follows: ratio calculated when the length of the prementum excluding the postmental hinge is used x 0.88 is approximately equal to the ratio when the postmental hinge is included for species of Aeshna and Rhionaeschna in North America

Classical Gravity on Fuzzy Space-Time

A review is made of recent efforts to find relations between the commutation relations which define a noncommutative geometry and the gravitational field which remains as a shadow in the commutative limit.Comment: Lecture given at the 30th International Symposium Ahrenshoop on the Theory of Elementary Particles, Buckow, Germany, August 27-31, 1996; 11 Pages LaTe