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Topic: Tangent bundle


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  PlanetMath: section of a fiber bundle
its tangent bundle, a (smooth) section of this bundle is precisely a (smooth) tangent vector field.
The answer is yes, for example, in the case of a trivial vector bundle, but in general it depends on the topology of the spaces involved.
This is version 7 of section of a fiber bundle, born on 2003-02-10, modified 2004-06-19.
planetmath.org /encyclopedia/SectionOfAFiberBundle.html   (349 words)

  
 Station Information - Tangent bundle
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.
www.stationinformation.com /encyclopedia/t/ta/tangent_bundle.html   (136 words)

  
 Line bundle   (Site not responding. Last check: 2007-11-06)
For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these.
This reminds one of the orientation double cover on a differential manifold: indeed that's a special case in which the line bundle is the determinant bundle (top exterior power) of the tangent bundle.
It carries the universal real line bundle; in terms of homotopy theory that means that any real line bundle on a CW complex determines a classifying map from to, making a bundle isomorphic to the pullback of the universal bundle.
pedia.newsfilter.co.uk /wikipedia/l/li/line_bundle.html   (560 words)

  
 Tangent space - Wikipedia, the free encyclopedia
The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.
Once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold.
All the tangent spaces can be "glued together" to form a new differentiable manifold of twice the dimension, the tangent bundle of the manifold.
www.wikipedia.org /wiki/Tangent_space   (1214 words)

  
 Tangent bundle - Wikipedia, the free encyclopedia
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.
en.wikipedia.org /wiki/Tangent_bundle   (641 words)

  
 Talk:Tangent bundle - Wikipedia, the free encyclopedia
The tangent bundle to a circle is (isomorphic to) a cylinder, not just intuitively, but in actuality.
Comment on the conditions for the tangent bundle to be trivial (existence of a global frame) and the definition of parallizability.
Mention functorality of the tangent bundle construction and the pushforward map.
en.wikipedia.org /wiki/Talk:Tangent_bundle   (759 words)

  
 Encyclopedia: Tangent bundle   (Site not responding. Last check: 2007-11-06)
In mathematics, the tangent bundle of a manifoldIn mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension.
A typical example is the tangent bundle of a differentiable manifold: to every point of the manifold we attach the tangent space of the manifold at that point.
tangent bundle is the surface of an infinitely.
www.nationmaster.com /encyclopedia/Tangent-bundle   (454 words)

  
 Definition of Fiber bundle - Biocrawler   (Site not responding. Last check: 2007-11-06)
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.
Given a vector bundle E with a metric (such as the tangent bundle to a Riemannian manifold) one can construct the associated unit sphere bundle, for which the fiber over a point x is the set of all unit vectors in E
www.biocrawler.com /encyclopedia/Fiber_bundle   (1211 words)

  
 Encyclopedia topic: Tangent space   (Site not responding. Last check: 2007-11-06)
All the tangent spaces have the same dimension (The magnitude of something in a particular direction (especially length or width or height)), equal to the dimension of the manifold.
Once tangent spaces have been introduced, one can define vector field (additional info and facts about vector field) s, which are abstractions of the velocity field of particles moving on a manifold.
All the tangent spaces can be "glued together" to form a new differentiable manifold of twice the dimension, the tangent bundle (additional info and facts about tangent bundle) of the manifold.
www.absoluteastronomy.com /encyclopedia/t/ta/tangent_space.htm   (1315 words)

  
 On The Geometry Of The Tangent Bundle With The Cheeger-Gromoll Metric (ResearchIndex)   (Site not responding. Last check: 2007-11-06)
On The Geometry Of The Tangent Bundle With The Cheeger-Gromoll Metric (ResearchIndex)
On The Geometry Of The Tangent Bundle With The Cheeger-Gromoll Metric
1 the Geometry of the Tangent Bundle (context) - Dombrowski - 1962
citeseer.ist.psu.edu /449382.html   (262 words)

  
 Cotangent bundle - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-06)
In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold.
The cotangent bundle has a canonical symplectic 2-form on it, as an exterior derivative of a one-form.
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).
xahlee.org /_p/wiki/Cotangent_bundle.html   (274 words)

  
 Vector bundle   (Site not responding. Last check: 2007-11-06)
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 vectorspaces, "glued together", form another topological space (or manifold or variety).
As an example, sections of the tangent bundle of a differential manifold are nothing but vector fields on that manifold.
Smooth vector bundles are defined by requiring that E and X be smooth manifolds, π : E → X be a smooth map, and the local trivialization maps φ be diffeomorphisms.
www.sciencedaily.com /encyclopedia/vector_bundle   (989 words)

  
 PlanetMath: tangent bundle   (Site not responding. Last check: 2007-11-06)
forgetting the tangent vector and remembering the point, is a vector bundle.
Cross-references: fibers, base, obvious, section, vector field, vector bundle, tangent vector, projection, differentiable, bijective, isomorphism, derivative, map, diffeomorphism, neighborhood, isomorphic, structure, tangent spaces, disjoint union, differentiable manifold
This is version 2 of tangent bundle, born on 2003-10-06, modified 2003-10-06.
planetmath.org /encyclopedia/TangentBundle.html   (138 words)

  
 Cable Bundle --> Info and Comparisons   (Site not responding. Last check: 2007-11-06)
In political fundraising, bundling is when donations from many individuals are collected by one person and presented to the recipient, therfore maximizing the influence of the individual doing the presenting.
The bundle of His is a bundle of heart muscle cells specialized for electrical conduction that transmits the electrical impulses from the AV node to the cells of the ventricles, causing cardiac muscles in the ventricles to contract.
This implies, the bundle theorist maintains, that we cannot have any conception whatsoever of a "bare particular." As the English philosopher John Locke said, a substance by itself, apart from its properties, is "something, I know not what." The only way that we can conceive of an object is by conceiving of its properties.
www.crashdatabase.com /computers/14/cable-bundle.html   (1402 words)

  
 PlanetMath: vector field
The problem is that the tangent spaces form a fiber bundle, and this may not be trivial.
Sections of the cotangent bundle are one-forms, and since they are obtained by taking the dual, they transform according to the inverse matrix at each point.
This viewpoint on vector fields emphasizes the machinery of modern geometry, namely sheaves, local rings, and bundles; this machinery is useful in differential geometry, important in complex analtyic geometry, and foundational in algebraic geometry -- schemes cannot be described without it.
planetmath.org /encyclopedia/VectorField.html   (770 words)

  
 Tangent bundle   (Site not responding. Last check: 2007-11-06)
AMCA: Natural Poisson structures on the tangent bundle of a pseudo-Riemannian ma...
THE RESTRICTED TANGENT BUNDLE OF SMOOTH CURVES IN GRASSMANNIANS AND CURVES IN FL...
IngentaConnect Spectra of Unit Tangent Bundles of Compact Hyperbolic Riemann Sur...
www.scienceoxygen.com /math/705.html   (214 words)

  
 [No title]
The tangent bundle of LV is thus an (infinite-dimensional) c* *omplex T-equivariant vector bundle; in fact it is the free loopspace of T V.
The space ToeLV [0,1)spanned by eigen* *vectors of -iDoewith eigenvalues in [0, 1) is thus isomorphic to the tangent space Toe(* *0)V of V at the basepoint oe(0), by the map which assigns to an initial vector, its continuation by parallel transport around the loop.
SPn(R) is a section of the map e restricted to the zero-sum configurations on the circ* *le; it defines an analog of the logarithm.
hopf.math.purdue.edu /Morava/Looptangent.txt   (2625 words)

  
 Massless Quantum Fields in the Spacetime Tangent Bundle   (Site not responding. Last check: 2007-11-06)
Maximal-acceleration invariant quantum fields were recently formulated in terms of the maximal acceleration group and the differential geometric structure of the spacetime tangent bundle [1,2].
The simple case was addressed of a massive free scalar field in a flat Minkowski spacetime for which the bundle is also flat.
In the present work, massless quantum fields are considered as the zero-mass limit of massive fields and are shown to have an exponential spectral cutoff at the Planck frequency.
flux.aps.org /meetings/YR99/CENT99/abs/S7545009.html   (163 words)

  
 Universität Bayreuth
Recently I studied subsheaves in the tangent bundle in connection with the structure of the universal cover with F.Campana.
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.
In dimension 2, an infinitesimal quadric structure implies the splitting of the tangent bundle as a sum of line bundles (perhaps after some étale covering), and flatness means the universal covering space is indeed a product.
www.uni-bayreuth.de /forschungsberichte/03/1/1/01/01/engl.html   (1061 words)

  
 [No title]
The tangent bundle constructed by Cheeger does not have as fiber the metric tangent space, in the case of regular sub-Riemannian manifolds.
The bundle structure of $WG$ is given by the map $$[c]_{s} \mapsto x_{c} \ = \ \lim_{\varepsilon \rightarrow 0} \delta_{\varepsilon} c\left(\frac{1}{\varepsilon}\right)$$ $W_{0}G$ is the fiber of the neutral element and it is a group.
Here $d_{N}$ is the distance on the tangent cone at the neutral element (remember that, as in Vodop'yanov \& Greshnov \cite{vodopis2}, we have identified a neighbourhood of the neutral element of the tangent cone with a neighbourhood of the neutral element of the group $G$; on this neighbourhood we have also the distance $d_{N}$).
irmi.epfl.ch /cag/tangent_bundle_hotfile.tex   (11697 words)

  
 LMS Proceedings Abstract, paper PLMS 1485   (Site not responding. Last check: 2007-11-06)
A line bundle over a complex projective variety is called big and 1-ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1-dimensional fibers.
The main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more, whose tangent bundle is big and 1-ample, is equal either to a projective space or to a smooth quadric.
Since big and 1-ample bundles are 'almost' ample, the present result is yet another extension of the celebrated Mori paper 'Projective manifolds with ample tangent bundles' (Ann.
www.lms.ac.uk /publications/proceedings/abstracts/p1485a.html   (179 words)

  
 Tangent bundle
The Tangent Bundle of a manifold is the union of all the tangent spaces at every point in the manifold.
Suppose M is a C^k manifold, and \phi : U \rightarrow \mathbb{R}^n , where U is an open subset of M, and n is the the dimension of the manifold, in the chart \phi(\circ); furthermore suppose T_{p}M is the tangent space at a point p in M .
The text of this article is licensed under the GFDL.
www.ebroadcast.com.au /lookup/encyclopedia/ta/Tangent_bundle.html   (141 words)

  
 Line bundle   (Site not responding. Last check: 2007-11-06)
In fact the topology of the 1x1 invertible real matricesand complex matrices is entirely different: the first of those is a space homotopy equivalent to a discrete two-point space(positive and negative reals contracted down), while the second has the homotopy type of a circle.
A real line bundle is therefore in the eyes of homotopy theory asgood as a fiber bundle with a two-point fiber - a double covering.
There are theories of holomorphic line bundles on complexmanifolds, and invertible sheaves in algebraic geometry, that work out a line bundle theory in thoseareas.
www.therfcc.org /line-bundle-218358.html   (485 words)

  
 Meningar.com om cotangent. bundle, space, manifold mm.
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..
"...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..
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 In mathematics, an isomorphism is a kind of interesting mapping between objects...
www.meningar.com /cotangent.html   (1366 words)

  
 HogBlog: Vector Bundles
Since all the tangent lines fit together, all the other lines have to move, too; the effect sort of ripples along the circle, leaving all the tangent lines vertical in its wake.
None of these tangent planes are supposed to overlap; I imagine them sort of fitting together like the petals of a rose, and again I use a sort of 'twistiness' to imagine their necessary extension into higher dimensions.
Just as there was a circular hole in the centre of the tangent bundle to the circle, there is a spherical hole in the centre of this bundle.
www.koschei.net /blog/archives/000807.html   (1092 words)

  
 Fibre bundles   (Site not responding. Last check: 2007-11-06)
A typical fibre bundle associated to any manifold M is the tangent bundle TM, which is the bundle formed by all the tangent vectors.
is the bundle of all covectors, that is the fibres of this bundle are dual to the fibres of the tangent bundle.
A principal fibre bundle P with structure group G is defined as a fibre bundle with a free and transitive right G-action on the fibres.
www.phys.uu.nl /~hofman/scriptie/duality/node50.html   (738 words)

  
 DC MetaData pour: Differential geometry over general base fields and rings. Part I: First and second order geometry
We establish a natural version of this procedure in the context of general differential calculus: the tangent functor is the functor of scalar extension by dual numbers; iterating this, we get as powerful tool the interpretation of jet functors as functors of scalar extensions by "jet rings".
The main topic of the present work is the definition and study of linear connections on vector bundles, of their curvature- and torsion forms and the application of the theory to symmetric spaces and Lie groups.
The basic idea is very simple: as is well-known, the tangent bundle $T F$ of a linear (i.e., vector) bundle $p : F \longrightarrow M$, seen as a bundle over $M$, is no longer a linear bundle.
www.iecn.u-nancy.fr /Preprint/publis/2003_bertram.Mon_Nov_17_11_34_34_CET_2003   (452 words)

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