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# Topic: Cartan connection

###### In the News (Sun 19 May 13)

 Cartan connection - Wikipedia, the free encyclopedia In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. Cartan formalism is an alternative approach to covariant derivatives and curvature, using differential forms and frames. The curvature of a Cartan connection is the en.wikipedia.org /wiki/Cartan_connection   (1328 words)

 PlanetMath: connection   (Site not responding. Last check: 2007-11-07) Since the notions of connection, parallel transport, and covariant derivative are so closely related, it is easy to translate propopsitions involving one of these terms into propsitions involving a different one of three terms. By using the connection, we could generate a transformation of the fiber which is not described by the structure group of the bundle. The notion of connection is intimately related to the notion of parallel transport, and indeed one can regard the former as the infinitesimal version of the latter. planetmath.org /encyclopedia/Connection.html   (2998 words)

 CSDC : Frame vs. Metric connections, and their curvatures Then by exterior differential and algebraic processes a right Cartan matrix of connection 1-forms is produced, and a second application of exterior differential and algbraic processes to the Cartan connection produces the concept of a Cartan matrix of curvature 2-forms. The induced metric on the initial state may be used to compute a connection on the initial state, and this connection is defined as the Christoffel, metric based, connection on the initial state. The Cartan connection is independent from the choice of metric on the final state, the Christoffel connection depends upon the metric. www22.pair.com /csdc/ed3/ed3fre26.htm   (2632 words)

 Cartan connection applications - Wikipedia, the free encyclopedia In any dimension, for a pseudo Riemannian geometry (with metric signature (p,q)), this Cartan connection theory is an alternative method in differential geometry. A connection over V is defined as the unique connection A satisfying these two conditions: In the tetrad formulation of general relativity, the action, as a functional of the cotetrad e and a connection form A over a four dimensional differential manifold M is given by en.wikipedia.org /wiki/Vierbein   (508 words)

 Re: Teleparallel Gravity AFAICT a "teleparallel > connection" is as I defined previously and a "Weitzenboeck connection" is > a teleparallel connection written in a coordinate basis. As you say, the Cartan connection does have both curvature and torsion, but is is _because_ of the way that absolute parallelism is defined in a Weitzenboeck spacetime that the curvature tensor becomes identically zero. I realize that there are some papers which do not identify the Cartan connection explicitly, and I have also seen reference, as have you, to a "Weitzenboeck connection," but I believe the proper mathematical identification is along the lines of what I have described. www.lns.cornell.edu /spr/2002-01/msg0038060.html   (383 words)

 CSDC : Frame and Metric Perturbations in 4D - A Schwarzshild Solution with Torsion   (Site not responding. Last check: 2007-11-07) The curvatures induced by the two connections are also identical and equal to zero. The first term will be defined as metric curvature relative to the Cartan connection [C], and the second term will be defined as inertial curvature relative to the Cartan connection [C]. The Cartan connection again will be compared to the Christoffel connection and the Ricci rotation terms [T] are not zero. www22.pair.com /csdc/ed3/ed3fre27.htm   (518 words)

 Cartan: a Mathematica package for tensor analysis   (Site not responding. Last check: 2007-11-07) Cartan was developed in part at the European Centre for Nuclear Research (CERN) in Geneva and at the Nordic Institute for Theoretical Physics (NORDITA) in Copenhagen. Cartan is an easy-to-use tensor component package for interactive tensor calculations in general Riemann-Cartan spaces of arbitrary dimensions and signatures. Cartan is presently available for all platforms running Mathematica and with the ability to down-load and unpack a gz-compressed tar file or a zip file. www.adinfinitum.no /cartan   (633 words)

 No Title For the Cartan connection (4), the deviation from the geodesics is caused by a torsion force in (7) coming from the symmetrical part of the contorsion tensor The induced connection is uniquely determined by the compatibility condition of the embedding of the tangent space with the parallel-transport law in the embedding space. The latter is sufficient to specify the metric and connection induced by the embedding. www.physik.fu-berlin.de /~kleinert/kleiner_re259/embed.html   (2902 words)

 Re: Teleparallelism; Einstein The curvature of the Cartan connection w^a_b is -defined- to be the gl(R,4)-valued two-form O^a_b = dw^a_b + w^a_m /\ w^m_b and the torsion of the connection is -defined- to be the vector-valued two-form t^a = do^a + w^a_b /\ w^b Notice that the coframe field o^a appears in these definitions! A subtle point: -locally-, Weitzenboeck connections always exist and indeed the conditions for a Weitzenboeck connection are "less stringent" than the requirements for a Levi-Civita connection, since in general a locally defined Weitzenboeck connection will not be -unique-. Here of course we intend the notion of covariant differentiation defined by the connection, in the manner which was explained by Cartan, and the frame field e_a is the dual of the coframe field o^a. www.lns.cornell.edu /spr/2002-01/msg0038083.html   (1494 words)

 Research interests of Andreas Cap Parabolic geometries are Cartan geometries of type (G,P), where G is a (real or complex) semisimple Lie group and P is a parabolic subgroup of G. The homogeneous models G/P are the so-called generalized flag manifolds. Usually, this equivalence is not obvious be the result of a difficult theorem, a typical example being the canonical Cartan connection for CR structures, which has been found by Tanaka in [T1] and Chern-Moser in [CM]. In the process of working up Cartan's ideas in the 1950's, the main interest was not in Cartan connections (which are the basic ingredient for a Cartan geometry) but rather in principal connections. www.mat.univie.ac.at /~cap/research.html   (2536 words)

 Cartan connection   (Site not responding. Last check: 2007-11-07) In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, based on an understanding of the role of the affine group in the usual approach. In the tetrad formulation of general relativity, the action, as a functional of the cotetrad e and a connection A over a four dimensional differential manifold M is given by The main idea is to develop expressions for connectionss and curvature using orthogonal frames. www.sciencedaily.com /encyclopedia/cartan_connection   (887 words)

 Dr. Ilka Agricola They were extensively studied by Elie Cartan (see Part 2 and Part 3 of Cartan's "Oeuvres Completes", Gauthier-Villars, 1955) who developed a method, which in principle, enables one to solve equivalence problems for any given G-structure in a fairly algorithmic fashion. For example Cartan used it to find the invariants of Riemannian, conformal, projective and CR structures. Cartan's method may also be used to find all the G-structures with large symmetries and to determine the dimensions of the possible groups of symmetries of a given category of G-structures. www.mathematik.hu-berlin.de /~agricola/act-old.html   (423 words)

 Differential Geometry Of Cartan Connections - Alekseevsky, Michor (ResearchIndex)   (Site not responding. Last check: 2007-11-07) Abstract: this article a general theory of Cartan connections is developed and some applications are indicated. The starting idea is to consider a Cartan connection as a deformation of a local Lie group structure on the manifold, i.e. Alekseevsky, D.V.; Michor, P.W., Differential geometry of Cartan connections, ESI Preprint 39, Publ. citeseer.ist.psu.edu /49241.html   (498 words)

 Recent preprints by A. Rod Gover We show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tractor bundle and the normal standard tractor connection, which are equivalent to the Cartan bundle and the Cartan connection. In particular we characterize the normal Cartan connection from this induced bundle/connection perspective. For an important sub-class of parabolic geometries we explicitly and directly construct the tractor bundles, their canonical linear connections and the machinery for explicitly calculating via the tractor calculus. www.math.auckland.ac.nz /~gover/articles/preprints.html   (1097 words)

 Reduction, Symmetry and Phases in Mechanics   (Site not responding. Last check: 2007-11-07) We systematically exploit this point of view for fixed systems (for example with controls on the internal, or reduced, variables) and for slowly moving systems in an adiabatic context. For the latter, we obtain the phases as the holonomy for a connection which synthesizes the Cartan connection for moving mechanical systems with the Hannay-Berry connection for integrable systems. This synthesis allows one to treat in a natural way examples like the ball in the slowly rotating hoop and also non-integrable mechanical systems. www.cds.caltech.edu /~marsden/bib/1990/04-MaMoRa1990   (99 words)

 Invariants of elliptic and hyperbolic $CR$-structures of codimension 2   (Site not responding. Last check: 2007-11-07) The parallelism that we construct is defined on a sequence of two principal bundles over the manifold, takes values in the Lie algebra of infinitesimal automorphisms of the quadric corresponding to the Levi form of the manifold, and behaves almost'' like a Cartan connection. The construction is explicit and allows us to study the properties of the parallelism as well as those of its curvature form. It also leads to a natural class of semi-flat'' manifolds for which the two bundles reduce to a single one and the parallelism turns into a true Cartan connection. wwwmaths.anu.edu.au /research.reports/mrr/97.049/abs-plain.html   (177 words)

 Physics Help and Math Help - Physics Forums - Levi-Civita Connection Decomposition in GR? Comments would be helpful.\n\n"The Question is: What is The Question?" J.A. Wheeler\n\nThe {LC} connection field in GR curved spacetime is analogous to the EM\nvector potential A connection in internal space.\n\nThe tidal stretch-squeeze GCT tensor field = {LC} curl of itself is\nanalogous to the Maxwell EM Fuv field tensor.\n\n{LC} and A are both Cartan 1-forms. The {LC} connection field in GR curved spacetime is analogous to the EM vector potential A connection in internal space. {LC} the connection of 1916 GR is a 1-form. www.physicsforums.com /printthread.php?t=57287   (661 words)

 Functional Analysis and Applications: Seminars   (Site not responding. Last check: 2007-11-07) Further, we proof that there is a canonical Cartan connection associated with this structure. The talk is devoted to the local equivalence problem for rank 2 distributions on an $n$-dimensional manifold (or shortly $(2,n)$-distributions) and it is based on the joint work with Boris Doubrov.  In 1910 for maximally nonholomomic $(2,5)$-distributions E. Cartan constructed the canonical coframe and found the most symmetric case. To each subriemannian manifold $(M, D, g)$ whose distribution $D$ is bracket-generating and whose subriemannian symbol is isomorphic to a constant one $(\mathfrak m,\sigma)$, one can canonically associate a Cartan connection $(P, M, G_0, \omega)$ on a principal fiber bundle $P$ on $M$, where the structure group $G_0$ is the automorphism group of $(\mathfrak m,\sigma)$. www.sissa.it /fa/att/seminar.html   (3713 words)

 [No title]   (Site not responding. Last check: 2007-11-07) cartan: considering what nationality the dead white oppressor people who founded the country considered themselves to be, yes. cartan: If you take dislike of government to it's logical extreme then patriotism doesn't make sense: patriotism is believing in something attributed to geographical borders of governmental control. cartan: A poor man in the US would be -lucky- to have a house and a car. tunes.org /~nef/logs/lisp/04.03.13   (17088 words)

 Grassmannian Structures on Manifolds (ResearchIndex)   (Site not responding. Last check: 2007-11-07) It is shown that such a structure is the prolongation of a subbundle of the first order framebundle. A canonical normal connection is constructed from a Cartan connection on the bundle and a Grassmannian curvature tensor for the structure is derived. 1 Introduction The theory of Cartan connections has lead S. Kobayashi and T. Nagano, in... citeseer.ist.psu.edu /419279.html   (330 words)

 Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations Energy Citations Database (ECD) Document #5182821 - Gravity as a gauge theory with Cartan connection Availability information may be found in the Availability, Publisher, Research Organization, Resource Relation and/or Author (affiliation information) fields and/or via the "Full-text Availability" link. Gravity as a gauge theory with Cartan connection www.osti.gov /energycitations/product.biblio.jsp?osti_id=5182821   (98 words)

 DC MetaData for: Parabolic Geometries and Canonical Cartan Connections   (Site not responding. Last check: 2007-11-07) DC MetaData for: Parabolic Geometries and Canonical Cartan Connections geometrical meaning of Cartan connections corresponding to the pair geometrically) to determine the bundle and the Cartan connection. www.esi.ac.at /Preprint-shadows/esi450.html   (223 words)

 FUB-HEP/94-5 Euler Equations for Rigid-Body -- a Case for Autoparallel Trajectories in Spaces with Torsion To calculate this we observe that the only nonzero components of the affine-flat Cartan connection are is a constant, the space has obviously a vanishing Riemann connection. Because of (15), it has a nonvanishing torsion. www.physik.fu-berlin.de /~kleinert/kleiner_re224/euler.html   (2135 words)

 [No title]   (Site not responding. Last check: 2007-11-07) 06:01:51 cartan: duels with people who are any good at fighting at least (-: 06:02:31 cartan: is there any correct return value for modifying a literal? cartan: I think it should drop to the debugger on account of attempting to write to read-only memory, but that's just me. 13:04:48 tunes.org /~nef/logs/lisp/04.08.17   (16083 words)

 Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 16, pp. 33-63, 2000   (Site not responding. Last check: 2007-11-07) A new look at Finsler connections and special Finsler manifolds Abstract: Continuing Grifone's pioneering work, we present a systematic treatment of some distinguished Finsler connections and some special Finsler manifolds, built on three pillars: the theory of horizontal endomorphisms, the calculus of vector-valued forms and a "tangent bundle version" of the method of moving frames. Keywords: Horizontal endomorphisms; Cartan tensors; Berwald connection, Cartan connection, Chern-Rund connection; Berwald manifold; locally Minkowski manifold. www.emis.de /journals/AMAPN/vol16/5.html   (104 words)

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