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# Topic: Cotangent bundle

###### In the News (Wed 19 Jun 19)

 Cotangent bundle - Wikipedia, the free encyclopedia Cotangent spaces possess a canonical symplectic 2-form out of which a non-degenerate volume form can be built for the cotangent bundle. The one-form assigns to a vector in the tangent bundle of the cotangent bundle the application of the element in the cotangent bundle (a linear functional) to the projection of the vector into the tangent bundle (the differential of the projection of the cotangent bundle to the original manifold). The cylinder is the cotangent bundle of the circle. en.wikipedia.org /wiki/Cotangent_bundle   (480 words)

 PlanetMath: cotangent bundle   (Site not responding. Last check: 2007-11-07) is the vector bundle dual to the tangent bundle The cotangent bundle to any manifold has a natural symplectic structure given in terms of the Poincaré 1-form, which is in some sense unique. This is version 13 of cotangent bundle, born on 2003-10-06, modified 2004-12-21. planetmath.org /encyclopedia/CotangentBundle.html   (295 words)

 Cotangent space - Wikipedia, the free encyclopedia All cotangent spaces have the same dimension, equal to the dimension of the manifold. Note that since the tangent space and the cotangent space at a point are both real vector spaces of the same dimension, they are isomorphic to each other. All the cotangent spaces of a manifold can be "glued together" to form a new differentiable manifold of twice the dimension, the cotangent bundle of the manifold. en.wikipedia.org /wiki/Cotangent_space   (772 words)

 Fiber bundle - Encyclopedia, History, Geography and Biography Fiber bundles generalize vector bundles of which the main example is the tangent bundle of a manifold. A sphere bundle is a fiber bundle whose fiber is an n-sphere. In the smooth category, a G-bundle is a smooth fiber bundle where G is a Lie group and the corresponding action on F is smooth and the transition functions are all smooth maps. www.arikah.com /encyclopedia/Fiber_bundle   (1226 words)

 Glossary of differential geometry and topology   (Site not responding. Last check: 2007-11-07) This is equivalent to the tangent bundle being trivial. A principal bundle is a fiber bundle P → B together with right action on P by a Lie group G that preverses the fibers of P and acts simply transitively on those fibers. Vector bundle, a fiber bundle whose fibers are vector spaces and whose transition functions are linear maps. www.factsite.co.uk /en/wikipedia/g/gl/glossary_of_differential_geometry_and_topology.html   (471 words)

 PlanetMath: proof that transition functions of cotangent bundle are valid   (Site not responding. Last check: 2007-11-07) In this entry, we shall verify that the transition functions proposed for the cotangent bundle the three criteria required by the classical definition of a manifold. "proof that transition functions of cotangent bundle are valid" is owned by rspuzio. This is version 10 of proof that transition functions of cotangent bundle are valid, born on 2004-12-10, modified 2004-12-20. planetmath.org /encyclopedia/ProofThatTransitionFunctionsOfCotangentBundleAreValid.html   (152 words)

 Cotangent space: Definition and Links by Encyclopedian.com - All about Cotangent space   (Site not responding. Last check: 2007-11-07) For example, if we have the cotangent bundle, it is easy to define a canonical symplectic form on it, as an exterior derivative of a one-form. The one form assigns to a vector in the tangent bundle to the cotangent bundle the application of the element in the cotangent bundle (a linear functional) to the projection of the vector into the tangent bundle (the differential of the projection of the cotangent bundle to the original manifold). If the original manifold was the set of possible positions, then the cotangent bundle can be thought of as the set of possible positions and speeds. www.encyclopedian.com /co/Cotangent-space.html   (307 words)

 Canonical bundle - Wikipedia, the free encyclopedia In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle The canonical class is the divisor class of a Cartier divisor K on V giving rise to the canonical bundle — it is an equivalence class for linear equivalence on V, and any divisor in it may be called a canonical divisor. The anticanonical bundle is the corresponding inverse bundle www.wikipedia.org /wiki/Canonical_class   (145 words)

 Vector bundle -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-07) The class of all vector bundles together with bundle morphisms forms a (A general concept that marks divisions or coordinations in a conceptual scheme) category. Two vector bundles on X, over the same field, have a Whitney sum, with fibre at any point the (A union of two disjoint sets in which every element is the sum of an element from each of the disjoint sets) direct sum of fibres. Vector bundles are special (A bundle of fibers (especially nerve fibers)) fiber bundles, loosely speaking those where the fibers are vector spaces. www.absoluteastronomy.com /encyclopedia/v/ve/vector_bundle.htm   (1345 words)

 Cotangent bundle   (Site not responding. Last check: 2007-11-07) In differential geometry, the cotangentbundle of a manifold is the vector bundle of all the cotangent spaces atevery point in the manifold. The cotangent bundle has a canonical symplectic 2-form on it, asan exterior derivative of a one-form. The one-form assigns to a vector in the tangent bundle of the cotangent bundle theapplication of the element in the cotangent bundle (a linear functional) to the projection of the vector into the tangent bundle(the differential of the projection of the cotangent bundle to the original manifold). www.therfcc.org /cotangent-bundle-210434.html   (242 words)

 Cotangent space -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-07) All cotangent spaces have the same (The magnitude of something in a particular direction (especially length or width or height)) dimension, equal to the dimension of the manifold. Note that since the tangent space and the cotangent space at a point are both real vector spaces of the same dimension, they are (Click link for more info and facts about isomorphic) isomorphic to each other. All the cotangent spaces of a manifold can be "glued together" to form a new differentiable manifold of twice the dimension, the (Click link for more info and facts about cotangent bundle) cotangent bundle of the manifold. www.absoluteastronomy.com /encyclopedia/c/co/cotangent_space.htm   (1055 words)

 Encyclopedia: Cotangent bundle   (Site not responding. Last check: 2007-11-07) In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. www.nationmaster.com /encyclopedia/Cotangent-bundle   (1056 words)

 Help : Tangent Bundles The kernel of the differential of the projection of a fibre bundle at a point p in the bundle is the tangent space to the fibre. Well, and if all parrallel transports defined from a connection on a vector bundle are linear maps of the fibres (which are *vector spaces*), then such a connection is called *linear*, and linear connections are in 1-1 correspondence to covariant derivatives in the vector bundle. And of course, LC induces a covariant derivative on cotangent vectors, thus induces a linear connection H on the vector bundle T*M -= M, thus induces a splitting. www.forum-one.org /new-6671452-4346.html   (710 words)

 Meningar.com om cotangent. bundle, space, manifold mm. All the cotangent spaces of a manifold can be "glued together" to form a new differentiable manifold of twice the dimension, the cotangent bundle In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotan.. The cotangent bundle as phase space Symplectic form The cotangent bundle has a canonical symplectic 2-form on it, as an exterior... The cotangent bundle as phase space Symplectic form The cotangent bundle has a canonical symplectic 2-form In mathematics, in particular in abstract formulations of classical mechanics and analytical mechanics, a symplectic manifold is a smooth manifold.. www.meningar.com /cotangent.html   (1366 words)

 Tangent bundle - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-07) In mathematics, the tangent bundle of a manifold is the union of all the tangent spaces at every point in the manifold. That is, the tangent bundle accounts for dimensions in the positions in the manifold as well as directions tangent to it. Since we can define a projection map, π for each element of the tangent bundle giving the element in the manifold whose tangent space the first element lies, tangent bundles are also fiber bundles. xahlee.org /_p/wiki/Tangent_bundle.html   (167 words)

 Cotangent space   (Site not responding. Last check: 2007-11-07) All cotangent spaces have the same dimension, equal to the dimension of themanifold. Note that since the tangent space and the cotangent space at a point are both real vector spaces of the same dimension, theyare isomorphic to each other. All the cotangent spaces of a manifold can be "glued together" to form a new differentiable manifold of twice the dimension,the cotangent bundle of the manifold. www.therfcc.org /cotangent-space-69132.html   (637 words)

 Cotangent bundle - Wikpedia   (Site not responding. Last check: 2007-11-07) Proving this form is, indeed, symplectic can be done by noting that being symplectic is a local property: since the cotangent bundle is locally trivial, this definition need only be checked on $\mathbb\left\{R\right\}^n \times \mathbb\left\{R\right\}^n.$ If the manifold $Mrepresents the set of possible positions in adynamical system, then the$cotangent bundle $\!\,T^\left\{*\right\}\!Mcan be thought of as the set of possible positions and momentums.$ For example, this is an easy way to describe the (non-trivial) phase space of a three-dimensional spherical pendulum: a weighted ball constrained to move along a 2-sphere. www.bostoncoop.net /~tpryor/wiki/index.php?title=Cotangent_bundle   (299 words)

 Tangent Bundle Encyclopedia Article, Definition, History, Biography   (Site not responding. Last check: 2007-11-07) The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a manifold it its own right. The tangent bundle is an example of a more general construction called a vector bundle (which is itself a specific kind of fiber bundle). Explicitly, the tangent bundle to an n-dimensional manifold M may be defined as a rank n vector bundle over M whose transition functions are given by the Jacobian of the associated coordinate transformations. www.karr.net /encyclopedia/Tangent_bundle   (783 words)

 PlanetMath: cotangent bundle is a bundle   (Site not responding. Last check: 2007-11-07) The third criterion follows from the chain rule: "cotangent bundle is a bundle" is owned by rspuzio. This is version 3 of cotangent bundle is a bundle, born on 2004-12-21, modified 2004-12-21. planetmath.org /encyclopedia/CotangentBundleIsABundle.html   (66 words)

 Re: Geometric quantization It's easy to see that tensoring the tangent with the cotangent bundle gives the trivial bundle: all we're saying is that a cotangent vector paired with a tangent vector gives a number. So the answer for the cotangent bundle has to be minus the answer for the tangent bundle. And so the basis vector of the cotangent bundle will be: South: dz North: d(1/z) = -z^{-2} dz That minus in front is no problem, since all we care about is the homotopy class of the "gluing map". www.lns.cornell.edu /spr/2000-05/msg0025216.html   (531 words)

 Universität Bayreuth For example, if the tangent bundle of a Kähler manifold is trivial, then it is already a torus, i.e., the triviality of the tangent bundle already implies global flatness. We investigate the geometry of coverings of algebraic varieties which is to a large extent determined by a certain vector bundle on the base of the covering. Manifolds with nef subsheaves in the cotangent bundle (mit S.Kebekus, A.Sommese), in: Complex Geometry, Festschrift in honour of Hans Grauert, 157-163, Springer 2002 www.uni-bayreuth.de /forschungsberichte/03/1/1/01/01/engl.html   (1061 words)

 Brian C. Hall - Department of Mathematics - University of Notre Dame The cotangent bundle arises because Newton's equations are second order in time: a second-order equation on M becomes a first-order equation on the cotangent bundle of M. Segal and Bargmann themselves worked on the case in which the configuration space M is R_ and the phase space (cotangent bundle of M) is identified with C_. The complexification of K can also be identified with the cotangent bundle of K.) This work was extended by Stenzel to the case in which M is an arbitrary compact symmetric space; for example, M could be a sphere of arbitrary dimension. www.nd.edu /~bhall/research   (1873 words)

 Tangent bundle   (Site not responding. Last check: 2007-11-07) AMCA: Natural Poisson structures on the tangent bundle of a pseudo-Riemannian ma... A class of Poisson-Nijenhuis structures on a tangent bundle... LIE ALGEBROID tangent bundle along the leaves of a foliation, is also... www.scienceoxygen.com /math/705.html   (214 words)

 Poster.html   (Site not responding. Last check: 2007-11-07) Floer homology of the cotangent bundle of a closed Riemannian manifold M is naturally isomorphic to the homology of the loop space. An idea of proof is to relate the Floer equations in the cotangent bundle to the heat flow on the loop space. the open unit disk cotangent bundle, existence of a 1-periodic orbit representing the class, for every compactly supported time-dependent Hamiltonian which is sufficiently large over the zero section. www.math.tau.ac.il /~biranp/Seminars/GD/10.5.2004.Weber.html   (95 words)

 Geodesic - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-07) Geodesics can also be understood to be the Hamiltonian flows of a very special Hamiltonian defined on the cotangent space of the manifold. The Hamiltonian is constructed from the metric on the manifold, and is thus a quadratic form consisting entirely of the kinetic term. Thus, the geodesic lines are the integral curves of the geodesic flow onto the manifold M. www.bexley.us /project/wikipedia/index.php/Geodesics   (1184 words)

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