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Topic: Affine connection

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affine transformations

affine geometry

affine space

affine scheme

Affine representation

Affine cipher

affine varieties

affine maps

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affine function

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	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.
		The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand.
		The curvature of a Cartan connection is the
	en.wikipedia.org /wiki/Cartan_connection (1518 words)

	Levi-Civita connection - Wikipedia, the free encyclopedia
		In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free Riemannian connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric (or pseudo-Riemannian metric).
		In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.
		The components of this connection with respect to a system of local coordinates are called Christoffel symbols.
	en.wikipedia.org /wiki/Levi-Civita_connection (218 words)

	CSDC : Frame vs. Metric connections, and their curvatures
		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.
		The metric based connection need not be zero on the initial state, even though a metric based connection is zero on the final state.
	www22.pair.com /csdc/ed3/ed3fre26.htm (2632 words)

	2.1 Geometry
		The connection is a device introduced for establishing a comparison of vectors in different points of the manifold.
		In physical applications, a metric always seems to be needed; hence in affine geometry it must be derived solely by help of the connection or, rather, by tensorial objects constructed from it.
		Riemannian geometry is the further subcase with vanishing torsion of a metric-affine geometry with metric-compatible connection.
	relativity.livingreviews.org /Articles/lrr-2004-2/articlesu3.html (2680 words)

	[No title] (Site not responding. Last check: 2007-10-24)
		Namely, the Lorentzian analogue of an instanton is a metric compatible connection whose curvature is irreducible and simple ("pseudoinstanton").
		We prove that a pseudoinstanton is a solution of the Yang-Mills equation for the affine connection.
		In fact, we prove a much stronger result: a pseudoinstanton is a stationary point of any Lorentz-invariant quadratic action with respect to the independent variation of the metric and the connection.
	www.bath.ac.uk /~masdv/talks/2003/2003kiev/2003kiev.html (167 words)

	[No title] (Site not responding. Last check: 2007-10-24)
		Namely, the connection coefficients are assumed to be expressed via the components of the metric tensor in accordance with a certain explicit formula.
		Such a connection is called the Levi-Civita connection, and the connection coefficients in this case are called Christoffel symbols.
		In particular, the connection may not be metric compatible and (or) may have nonzero torsion.
	www.bath.ac.uk /~masdv/talks/2003/umist/umist.html (281 words)

	Vargas's Abstract. (Site not responding. Last check: 2007-10-24)
		of quantum mechanics, appears to be unrelated to the mathematics of affine connections.
		The stochasticity of the connection, and of the physics in general, is a necessary consequence of these more restrictive conditions.
		The Planck constant, whose annulment would also annul the stochasticity (read fluctuations) of quantum mechanics is thus shown to be intimately connected with the affine connection of spacetime, and with cosmology in particular.
	www.mindspring.com /~cerebroscopic/Vargas-a.html (402 words)

	The affine connection (Site not responding. Last check: 2007-10-24)
		With an affine connection, a path between two points of the manifold establishes a linear transformation between the tangent spaces at those points.
		A metric determines a symmetric affine connection, and a symmetric affine connection together with a sphere at some point of the manifold, determines a metric.
		The affine connection, on the other hand, speaks of no particular coordinate system but is, as any tensor representation, expressed in some coordinate system.
	www.cap-lore.com /MathPhys/Affine.html (479 words)

	Affine Connection (Site not responding. Last check: 2007-10-24)
		For the affine connection to be determined by the metric tensor only the following 2 cases arise:
		Symmetric metric and symmetric affine connection this leads to:
		The imaginary part of the affine connection is completely asymmetric.
	www.freenetpages.co.uk /hp/ph137/page1affine.htm (66 words)

	VANISHING VIERBEIN
		What is new in the paper is connected with this latter question, but I find the discussion misleading and in any case not sufficiently well developed to justify publication in the paper's present form.
		The principal connection chooses to live in the external bundle and the affine connection dies.
		It is interesting to notice that the principal connection continues smoothly over the bridge and need not be regularized as in the first example.
	quantumfuture.net /quantum_future/papers/vv (1917 words)

	The Affine Connection (Site not responding. Last check: 2007-10-24)
		Case I: The metric and affine connection are both symmetric
		Case 2: The metric and affine connection are both asymmetric
		With the conditions given by equation 1.2, and 1.5, the asymmetric affine connection is
	homepage.ntlworld.com /peter.hickman1/page1/page1.htm (124 words)

	6.2 Further work on (metric-) affine and mixed geometry
		Research on affine geometry as a frame for unified field theory was also carried on by mathematicians of the Princeton school.
		By a remark of Straneo, that auto-parallels and geodesics have to be distinguished in an affine geometry, the Indian mathematician Kosambi
		This must be read in the sense that he could obtain the Einstein-Mayer equations from his formalism without introducing a connecting quantity leading from the space of 5-vectors to space-time [194].
	univie.ac.at /EMIS/journals/LRG/Articles/lrr-2004-2/articlesu13.html (904 words)

	Principia Physica? II
		This nonlocal connection explains entire quantum phe-nomenology on one hand while its local aspect, a classical connection, is the universal field which yields an integrated geometric description of all the forces, particles, fields, energy-momentum, charges, and other quantum numbers.
		The seminal consequence of the principle is that it implies a nonlocal connection on spacetime.
		Mathematically, this correspondence is an affine isomorphism from the tangent vector-space Tx at x to the tangent vector-space Ty at y; roughly speaking, this isomorphism, say !xy; is the ‘affine derivative’ of the correspondence referred to in the nonlocality principle.
	quantumfuture.net /quantum_future/principia1.htm (4777 words)

	4.3 Eddington’s affine theory
		In the first, shorter, part of two, Eddington describes affine geometry; in the second he relates mathematical objects to physical variables.
		He distinguishes the affine geometry as the “geometry of the world-structure” from Riemannian geometry as “the natural geometry of the world”.
		In principle, now a fictitious “Riemannian” connection (the Christoffel symbol) can be written down which, however, is a horribly complicated function of the affine connection - as the only fundamental geometrical quantity available.
	univie.ac.at /EMIS/journals/LRG/Articles/lrr-2004-2/articlesu11.html (2755 words)

	manoff-p
		On the symmetric energy-momentum tensor for theories in spaces with affine connection and metric.
		On the notion energy-momentum tensor for field theories in spaces with affine connection and metric.
		Differentiable manifolds with contravariant and covariant affine connections and metric [(Ln,g)-spaces].
	theo.inrne.bas.bg /~smanov/manoff-p.htm (1572 words)

	Decompositions for Control Systems on Manifolds with an Affine Connection - Lewis, Murray (ResearchIndex) (Site not responding. Last check: 2007-10-24)
		Abstract: In this letter we present a decomposition for control systems whose drift vector field is the geodesic spray associated with an affine connection.
		...connection control systems has a significant role to play in the field of mechanical control systems.
		1 Affine Connections and Distributions with Applications to Co..
	citeseer.ist.psu.edu /437173.html (502 words)

	Re: Affine Connection & Gauge Theory
		Generally the principal fiber bundle is defined as the quotient space of the direct product of local base space and transformation group based on an equivalence relations, while the space I've dealt with in the following webpage: http://139.134.5.123/tiddler2/gauge4/gauge.htm is the direct product of Lorentz space and inner space.
		ignoring the effect of gravity) and the inner space was supposed as the space with affine connection.
		Re: Affine Connection and Gauge Theory, Shmuel (Seymour J.) Metz
	www.usenet.com /newsgroups/sci.math/msg00869.html (306 words)

	Citebase - The Affine Connection of Supersymmetric SO(N)/Sp(N) Theories
		Citebase - The Affine Connection of Supersymmetric SO(N)/Sp(N) Theories
		We study the covariance properties of the equations satisfied by the generating functions of the chiral operators R and T of supersymmetric SO(N)/Sp(N) theories with symmetric/antisymmetric tensors.
		Furthermore, by means of the polynomial defining the Riemann surface, seen as quadratic-differential, one can construct an affine connection that added to T leads to a new generating function which can be consistently integrated.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0307285 (236 words)

	Torsion waves in metric-affine field theory (Site not responding. Last check: 2007-10-24)
		The approach of metric-affine field theory is to define spacetime as a real oriented 4-manifold equipped with a metric and an affine connection.
		The 10 independent components of the metric tensor and the 64 connection coefficients are the unknowns of the theory.
		We write the Yang-Mills action for the affine connection and vary it both with respect to the metric and the connection.
	stacks.iop.org /0264-9381/18/2317 (302 words)

	AMCA: Representation of two-dimensional stable planes by Riemannian metrics and affine connections by Gerhard Gerlich (Site not responding. Last check: 2007-10-24)
		The classical examples of plane topological geometries, the real projective, affine, and hyperbolic planes, have one thing in common: Their lines are geodesics of a Riemannian metric.
		Moreover, among the three families of non-classical affine planes with large collineation groups, only the Moulton planes admit affine connections that generate their line system.
		Among them is a Riemannian connection that is singular on one entire line and yields a Riemannian metric for two half-planes only.
	at.yorku.ca /c/a/f/v/09.htm (221 words)

	NullG.html (Site not responding. Last check: 2007-10-24)
		The affine connection supports comparing vectors along a path and in particular along a geodesic.
		In a space-time manifold the metric tensor produces an affine connection which allows comparison of vectors along even a null geodesic.
		"λ" is known as the "affine parameter" with which the photon keeps its phase.
	www.cap-lore.com /MathPhys/NullG.html (343 words)

	DC MetaData for: Topological Affine Planes with Affine Connections (Site not responding. Last check: 2007-10-24)
		Abstract:We consider the question whether the system of lines of a two-dimensional topological plane can be described as the system of geodesics of a Riemannian metric or an affine connection.
		However, for the Moulton planes affine connections do exist and we determine all of them.
		Moreover, we derive a characterization of the classical affine plane in the case that R^2 acts by vector space translations as a subgroup of the collineation group.
	www.mathematik.tu-bs.de /preprints/shadow/200305_shadow.html (176 words)

	Optimal Control of Affine Connection Control Systems: A Variational Approach (Site not responding. Last check: 2007-10-24)
		The formalism of the affine connection can be used to describe geometrically the dynamics of me chanical systems, including those with nonholonomic constraints.
		An alternative approach, which we develop in this paper, is to include the system dynamics as second order constraints of the optimization, and optimize relative to variations in the configuration space.
		Using the affine connection, its associated tensors, and the notion of covariant differentiation, we show how variations in the configuration space induce variations in the tangent space.
	www.cds.caltech.edu /~murray/papers/2000b_fm00-cdc.html (174 words)

	The Category of Affine Connection Control Systems (ResearchIndex)
		Abstract: The category of affine connection control systems is one whose objects are control systems whose drift vector field is the geodesic spray of an affine connection, and whose control vector fields are vertical lifts to the tangent bundle of vector fields on configuration space.
		1 Affine submersions (context) - Blumenthal - 1985
		1 Affine connections and distributions with applications to no..
	citeseer.ist.psu.edu /421238.html (369 words)

	The flat generalized affine connection and twistors for the Kerr solution (Site not responding. Last check: 2007-10-24)
		The flat generalized affine connection and twistors for the Kerr solution
		Certain natural affine and twistor structures for the Kerr solution suggested by a paper of Bergqvist and Ludvigsen (1991) are developed.
		The momentum quantities constructed are shown to arise from a certain linear approximation to the Kerr curvature.
	stacks.iop.org /0264-9381/10/407 (201 words)

	Find in a Library: On manifolds with an affine connection and the theory of general relativity
		Find in a Library: On manifolds with an affine connection and the theory of general relativity
		On manifolds with an affine connection and the theory of general relativity
		WorldCat is provided by OCLC Online Computer Library Center, Inc. on behalf of its member libraries.
	worldcatlibraries.org /wcpa/ow/edef0fdeeedcb21aa19afeb4da09e526.html (82 words)

	Nonholonomic Mapping Principle
		Parallel transport in q-spacetime is performed with an affine connection
		Recall the way in which the affine connection
		If we lower the last index of the affine connection by a contraction,
	www.physik.fu-berlin.de /~kleinert/kleiner_re261/node2.html (121 words)

	Shima: Vanishing theorems for compact Hessian manifolds
		A manifold is said to be Hessian if it admits a flat affine connection
		We study cohomology for Hessian manifolds, and prove a duality theorem and vanishing theorems.
		YAGI, On Hessian structures on an affine manifold, in Manifolds and Lie groups.
	www.numdam.org /numdam-bin/item?id=AIF_1986__36_3_183_0 (231 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations (Site not responding. Last check: 2007-10-24)
		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.
		nent parts of the spin affine connection is presented.
		It is shown that this method is independent of the assumption of symmetry of the affine connection.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=4805021 (120 words)

	Citebase - Deviation equations in spaces with affine connection
		Citebase - Deviation equations in spaces with affine connection
		Authors: Iliev, Bozhidar Z. Manoff, Sawa S. Connections between Lie derivatives and the deviation equation has been investigated in spaces with affine connection.
		The deviation equations of the geodesics as well as deviation equations of non-geodesics trajectories have been obtained on this base.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0512008 (162 words)

	DC MetaData for: Twistor Theory, Complex Homogeneous Manifolds and G-Structures (Site not responding. Last check: 2007-10-24)
		Abstract: One of the most useful characteristics of an affine connection on a manifold
		general linear group can be realised as a holonomy of some affine connection
		posed in the class of {\em torsion-free}\, (non-locally symmetric) affine
	www.esi.ac.at /Preprint-shadows/esi158.html (281 words)

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