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Topic: Connection (differential geometry)

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	Connection (mathematics) - Wikipedia, the free encyclopedia
		Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point to the local geometry at another point.
		An Ehresmann connection is a connection in a fibre bundle or a principal bundle using osculating spaces of the derivative of a field.
		A Koszul connection is a connection generalizing the derivative in a vector bundle.
	en.wikipedia.org /wiki/Connection_(mathematics) (1715 words)

	Category:Differential geometry - Wikipedia, the free encyclopedia
		In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.
		It arises naturally from the study of the theory of differential equations.
		Differential geometry is the study of geometry using calculus.
	en.wikipedia.org /wiki/Category:Differential_geometry (131 words)

	PlanetMath: connection (Site not responding. Last check: 2007-10-12)
		In fact, in differential geometry, the definition of a curved space is a space in which there exist two distinct curves with the same endpoints such that parallel transport along one curve is not the same as parallel transport along the other curve.
		Sometimes, as in the theory of embedded surfaces, there are two connections present so a semicolon is used to indicate covariant derivatives with repsect to one connection and a vertical bar or a colon is used to indicate covariant derivatives with respect to the other connection.
		By using the connection, we could generate a transformation of the fiber which is not described by the structure group of the bundle.
	planetmath.org /encyclopedia/Connection.html (3012 words)

	Differential geometry and topology
		Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions).
		The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent bundles, differential forms, exterior derivatives,integrals of p-forms over p-dimensional submanifolds and Stokes' theorem, wedge products, and Lie derivatives.
		A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point.
	www.knowledgefun.com /book/d/di/differential_geometry_and_topology.html (988 words)

	Geometry from a Differentiable Viewpoint
		Differential geometry is a subject of basic importance for all mathematicians, regardless of their special interests, and it also furnishes essential ideas and tools to physicists and engineers.
		The modern subject turns on problems that have emerged from the new foundations that are far removed from the ancient roots of geometry, and when we teach the new and cut off the past, students are left to find their own way to a meaning of "geometry" in differential geometry.
		In a very lively manner the spherical and hyperbolic geometries, the classical theory of curves and surfaces and a great part of Riemannian geometry are presented, as well as some applications (the tautochrone and accurate clock of Huygens, map projections and mathematical cartography, Lorentz manifolds as space-time models).
	math.vassar.edu /faculty/McCleary/Geom.page (966 words)

	5 Differential Geometry’s High Tide
		The generalisation of parallel transport in the sense of Levi-Civita and Weyl.
		In this geometry the paths are the shortest lines, and in that sense are a generalisation of geodesics.
		Other mathematicians were also stimulated by Einstein’s use of differential geometry in his general relativity and, particularly, by the idea of unified field theory.
	relativity.livingreviews.org /Articles/lrr-2004-2/articlese5.html (994 words)

	Manchester Geometry Seminar (Site not responding. Last check: 2007-10-12)
		Riemannian geometry is based on the assumption that the way we measure distances determines the way we define the concept of parallelism.
		Namely, the connection coefficients are assumed to be expressed via the components of the metric tensor in accordance with a certain explicit formula.
		Non-Riemannian geometry is based on the assumption that the way we measure distances is unrelated to the way we define the concept of parallelism.
	www.ma.umist.ac.uk /tv/Seminar/2003-2004/vassiliev.html (281 words)

	2.1 Geometry
		The connection is a device introduced for establishing a comparison of vectors in different points of the manifold.
		This is in stark contrast to Riemannian geometry where, vice versa, the connection is derived from the metric.
		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)

	Riemannian Geometry and General Relativity (Site not responding. Last check: 2007-10-12)
		Riemannian geometry is designed to describe the universe of creatures who live on a curved surface or in a curved space and do not know about the world of higher dimensions or do not have any access to it.
		The connection (or parallel transport) allows to compare what is happening at two distant points of a curved space, in spite of the fact that there is no direct and immediate way to communicate between these points.
		Recently connections appeared in the theory of gauge fields which is considered a basis of the modern physics of elementary particles.
	mystic.math.neu.edu /courses/diffgeom/intro.htm (356 words)

	Notes on Differential Geometry by B. Csikós (Site not responding. Last check: 2007-10-12)
		Gauss frame of a parameterized hypersurface, formulae for the partial derivatives of the Gauss frame vector fields, Christoffel symbols, Gauss and Codazzi-Mainardi equations, fundamental theorem of hypersurfaces, "Theorema Egregium", components of the curvature tensor, tensors in linear algebra, tensor fields over a hypersurface, curvature tensor.
		Vector fields and ordinary differential equations; basic results of the theory of ordinary differential equations (without proof); the Lie algebra of vector fields and the geometric meaning of Lie bracket, commuting vector fields, Lie algebra of a Lie group.
		Affine connection at a point, global affine connection, Christoffel symbols, covariant derivation of vector fields along a curve, parallel vector fields and parallel translation, symmetric connections, Riemannian manifolds, compatibility with a Riemannian metric, the fundamental theorem of Riemannian geometry, Levi-Civita connection.
	www.cs.elte.hu /geometry/csikos/dif/dif.html (588 words)

	Historical Remarks on Finsler Geometry
		Riemann:"Uber die Hypothesen, welche der Geometrie zugrnde liegen." In this memoir of 1854 Riemann discusses various possibilities by means of which an n-dimensional manifold may be endowed with a metric, and pays particular attention to a metric defined by the positive square root of a positive definite quadratic differential form.
		Thus the foundations of Riemannian geometry are laid; nevertheless, it is also suggested that the positive fourth root of a fourth order differential form might serve as a metric function.
		The significance of his work was enhanced by the advent of the general geometry of paths (a generalisation of the so-called Non-Riemannian geometry) due to Douglas and Knebelman, for the initial approach of Berwald was such as to establish a close affinity between these branches of metric and non-metric differential geometry.
	www.math.iupui.edu /~zshen/Finsler/history/rund.html (1032 words)

	Del - Biocrawler (Site not responding. Last check: 2007-10-12)
		In vector calculus, del is a vector differential operator represented by the symbol ∇.
		This symbol is sometimes called the nabla operator, after the Greek word for a kind of harp with a similar shape (with related words in Aramaic and Hebrew).
		In differential geometry, the nabla symbol is also used to refer to a connection.
	www.biocrawler.com /encyclopedia/Del (214 words)

	Introduction to Differential Geometry - Page 2
		i was trying to make the distinction between the language of differential forms and tangent bundles that is widely used in algebraic and differential topology, as opposed to the more specialized concept of curvature which is really the concept peculiar to differential geometry per se.
		Indeed vol2 of spivak is a good introduction to curvature and differential geometry which uses very little of the abstratc machinery of volume 1.
		Another plug for yet another fine geometry text: This may be a bit advanced but if anyone wants to get a truly broad spectrum of what differential geometry is all about and where it is going, I highly recommend Marcel Berger's A Panoramic View of Riemannian Geometry.
	www.physicsforums.com /showthread.php?p=903590#post903590 (1894 words)

	Principia Physica? II
		The latter leads to a nonlocal geometry, which is specified by a nonlocal connection (as opposed to classical, local connection in the sense of differential geometry).
		The seminal consequence of the principle is that it implies a nonlocal connection on spacetime.
		The classical connection, as conceived in differential geometry, is local in the sense that it is essentially a way of relating vectors at any point with those in its immediate (infinitesimal) vicinity.
	quantumfuture.net /quantum_future/principia1.htm (4777 words)

	Differential Geometry at Notre Dame
		The striking feature of modern Differential Geometry is its breadth, which touches so much of mathematics and theoretical physics, and the wide array of techniques it uses from areas as diverse as ordinary and partial differential equations, complex and harmonic analysis, operator theory, topology, ergodic theory, Lie groups, non-linear analysis and dynamical systems.
		Our work in complex geometry includes the affirmative solution of the Bochner Conjecture on the Euler number of ample Kaehler manifolds, a solution of Bloch's Conjecture (on the degeneracy of holomorphic curves in subvarieties of abelian varieties) and the classification of complex surfaces of positive bi-sectional curvature.
		In the past ten years it has been observed that there are profound connections between the existence of metrics with positive scalar curvature on a given compact space and the topological structure of the space.
	www.nd.edu /~jcao/dg.html (822 words)

	PlanetMath: Levi-Civita connection (Site not responding. Last check: 2007-10-12)
		On any Riemannian manifold, there is a unique connection
		Cross-references: local coordinates, torsion-free, metric, connection, Riemannian manifold
		This is version 2 of Levi-Civita connection, born on 2003-10-09, modified 2004-01-22.
	planetmath.org /encyclopedia/LeviCivitaConnection.html (48 words)

	basic tools for quantized differential geometry
		Yes GR is a subspecialty within the broad field of differential geometry and quantizing GR primarily means facing up to the question of how do you quantize differential geometry.
		And the specifically "loop" approach to quantizing GR simply means that you focus on one particular gadget, the connection---so that the quantum states are complex-valued functions defined on the space of all possible connections on the manifold you are studying.
		The connection is a very differential-geometry-type idea and all the other stuff you mentioned, that you use in quantizing geometry, are likewise at home here.
	www.physicsforums.com /showthread.php?t=7648 (583 words)

	Differential Geometry: A Geometric Introduction - Publications - Maplesoft
		Appropriate for introductory undergraduate courses in Differential Geometry with a prerequisite of multivariable calculus and linear algebra courses.
		This is the only text that introduces differential geometry by combining an intuitive geometric foundation, a rigorous connection with the standard formalisms, computer exercises with Maple, and a problems-based approach.
		Starting with basic geometric ideas and proceeding to the analytic and algebraic formalisms, this text provides a common and accessible foundation on which all of the various formalisms of differential geometry can be based and from which they can be assessed.
	www.maplesoft.com /books/books_detail.aspx?isbn=0-135-69963-0&L=E (134 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		Title : The Differential Geometry of Partial Differential Equations Abstract : Abstract Proposal: DMS-9870164 Principal Investigator: Robert Bryant The principal investigator plans to apply the theory of differential systems and the method of equivalence to problems in differential geometry and mathematical physics that have resisted more traditional approaches, emphasizing two main problems.
		Bryant has already done the classification in various low dimensions and is ready to study the intermediate dimensions (six through twenty-six) that are of physical interest, using the techniques of exterior differential systems that contributed to the solution of the holonomy problem in the classical case (in which the three-form was identically zero).
		Bryant also plans to continue his collaboration with Griffiths and Hsu on the geometry of PDE and their conservation laws and to generalize his recent structure theorems for harmonic morphisms.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9870164.txt (484 words)

	Natural operations in differential geometry
		A systematic treatment of naturality in differential geometry requires to describe all natural bundles, and this is also one of the undertakings of this book.
		Even though Ehresmann in his original papers from 1951 underlined the conceptual meaning of the notion of an $r$-jet for differential geometry, jets have been mostly used as a purely technical tool in certain problems in the theory of systems of partial differential equations, in singularity theory, in variational calculus and in higher order mechanics.
		Further, some functors of modern differential geometry are defined on the category of fibered manifolds and their local isomorphisms, the bundle of general connections being the simplest example.
	www.emis.de /monographs/KSM (1563 words)

	Amazon.com: Differential Geometry : Cartan's Generalization of Klein's Erlangen Program (Graduate Texts in ... (Site not responding. Last check: 2007-10-12)
		Cartan geometries were the first examples of connections on a principal bundle.
		The aim of the present book is to fill the gap in the literature on differential geometry by the missing notion of Cartan connections.
		Sharpe's book is a detailed argument supporting the assertion that most of differential geometry can be considered the study of principal bundles and connections on them, disguised as an introductory differential geometry textbook.
	www.amazon.com /exec/obidos/tg/detail/-/0387947329?v=glance (862 words)

	Amazon.ca: Elementary Differential Geometry: Books (Site not responding. Last check: 2007-10-12)
		In chapter two, the author presents the Frenet frame formulas, covariant derivatives, connection forms, and Cartan's structural equations, which are generalizations of the Frenet frame formulas for surfaces.
		Geometry of surfaces is the subject of chapter six, where the crucial Gauss' egregium theorem and some global theorems are also discussed, and in chapter seven students are introduced to the basics of the Riemannian geometry, culminating in the famous Gauss-Bonnet theorem.
		In fact, all the subsequent geometry is based on pullbacks and pushforwards.This itself motivates the more abstract definition of a differentiable manifold with its coordinate charts.
	www.amazon.ca /exec/obidos/ASIN/0125267452 (1562 words)

	Taylor & Francis Online (Site not responding. Last check: 2007-10-12)
		Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential forms, and integration on manifolds provide the foundation for many applications in dynamical systems and mechanics.
		The construction of the de Rham cohomology builds further arguments for the strong connection between the differential structure and the topological structure.
		Differential geometry is an actively developing area of modern mathematics.
	www.crcpress.com /shopping_cart/products/product_detail.asp?sku=C2530&parent_id=&pc= (408 words)

	Synthetic Differential Geometry and Surface Holonomy \| The String Coffee Table
		Breen and W. Messing in their famous math.AG/0106083 had noted that what is called synthetic differential geometry with its use of combinatorial differential forms is naturally suited for talking about connections on higher order structures such as gerbes in terms of ‘finite’ morphisms between these structures.
		One can safely include synthetic/combinatorial differential geometry in the list of concepts which are very simple and easy to handle in their pedestrian version, but which are powerful and far-reaching enough to admit mind-bogglingly complex generalizations.
		The result is the exponential of the curvature of the connection evaluated on the two ‘vectors’ which span the simplex.
	golem.ph.utexas.edu /string/archives/000655.html (1473 words)

	Projective Differential Geometry Old and New - Cambridge University Press
		The authors' main goal is to emphasize connections between classical projective differential geometry and contemporary mathematics and mathematical physics.
		Related topics include differential operators, the cohomology of the group of diffeomorphisms of the circle, and the classical four-vertex theorem.
		'¿ this is an introduction to global projective differential geometry offering felicitous choice of topics, leading from classical projective differential geometry to current fields of research in mathematics and mathematical physics.
	www.cambridge.org /catalogue/catalogue.asp?isbn=0521831865 (340 words)

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