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- ...Vect<sub>''n''</sub> is the [[stack (mathematics)|stack]] parametrizing [[vector bundle]]s (or [[locally free sheaf|locally free sheaves]]) of rank ''n'' ov ...rvnb|Behrend|2002|loc=Example 20.2.}}</ref> Moreover, viewing a rank-''n'' vector bundle as a principal <math>GL_n</math>-bundle, Vect<sub>''n''</sub> is iso ...2 KB (330 words) - 03:56, 22 May 2024
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- Any [[vector bundle]] over a [[compact space|compact]] [[Hausdorff space|Hausdorff]] bas ...n | last=Hatcher | first=Allen | author-link=Allen Hatcher | title=Vector Bundles & K-Theory | url=http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html | ...859 bytes (121 words) - 03:13, 28 September 2024
- ...structures, which contains in particular the tangent <math>TE</math> of a vector bundle <math>E</math> and the [[double tangent bundle]] <math>T^2M</math>. A '''double vector bundle''' consists of <math>(E, E^H, E^V, B)</math>, where ...2 KB (395 words) - 01:01, 15 February 2024
- ...Vect<sub>''n''</sub> is the [[stack (mathematics)|stack]] parametrizing [[vector bundle]]s (or [[locally free sheaf|locally free sheaves]]) of rank ''n'' ov ...rvnb|Behrend|2002|loc=Example 20.2.}}</ref> Moreover, viewing a rank-''n'' vector bundle as a principal <math>GL_n</math>-bundle, Vect<sub>''n''</sub> is iso ...2 KB (330 words) - 03:56, 22 May 2024
- ...aic geometry]], '''Lange's conjecture''' is a theorem about stability of [[vector bundle]]s over curves, introduced by {{ill|Herbert_Lange_(mathematician)|lt ...try|genus]] greater or equal to 2. For [[generic property|generic]] vector bundles <math>E_1</math> and <math>E_2</math> on ''C'' of ranks and degrees <math>( ...3 KB (403 words) - 00:16, 10 November 2024
- {{distinguish|text=a tensor bundle, a vector bundle whose section is a [[tensor field]]}} ...over a paracompact base, first note the construction is clear for trivial bundles. For the general case, if the base is compact, choose {{mvar|E{{'}}}} such ...2 KB (360 words) - 22:10, 13 February 2025
- ...e to [[André Weil]] and will be needed to define essentially finite vector bundles: ==Finite vector bundles== ...4 KB (577 words) - 07:36, 30 September 2022
- ...al linear group]]; i.e., the moduli stack was the [[moduli stack of vector bundles]], but, today, the term refers to any of generalizations. The scheme-theore ...<math>\operatorname{Bun}_G^d</math> locally closed substack consisting of bundles having degree <math>d: X(G) \to \mathbb{Z}</math>. ...2 KB (381 words) - 02:09, 23 April 2024
- ...mes I</math>, and the Möbius band is a non-trivial or twisted bundle. Both bundles are [[2-manifold]]s, but the annulus is an [[orientable manifold]] while th ...three I-bundles: the trivial bundle <math>K\times I</math> and two twisted bundles. ...3 KB (462 words) - 11:43, 8 February 2025
- ...logy]], the '''Milnor–Wood inequality''' is an obstruction to endow circle bundles over surfaces with a flat structure. It is named after [[John Milnor]] and == Flat bundles == ...2 KB (347 words) - 08:50, 15 October 2024
- ...alence relation]] on the families of [[Stable vector bundle|semistable]] [[vector bundle]]s on an [[algebraic curve]]. ...be a [[projective curve]] over an [[algebraically closed field]] ''k''. A vector bundle on ''X'' can be considered as a locally free [[sheaf (mathematics)|s ...1 KB (158 words) - 22:01, 5 June 2023
- ...ctor bundle]] is said to be ''flat'' if it is endowed with a [[connection (vector bundle)|linear connection]] with vanishing [[curvature]], i.e. a [[flat con ==de Rham cohomology of a flat vector bundle== ...3 KB (420 words) - 23:26, 21 September 2021
- ...ogical perspective, there is no difference between sphere bundles and disk bundles: this is a consequence of the [[Alexander trick]], which implies <math>\ope ...vector bundle on a space ''X'' and if ''E'' is given an [[orientation of a vector bundle|orientation]], then a sphere bundle formed from ''E'', Sph(''E''), i ...3 KB (461 words) - 17:47, 28 June 2022
- ...s |issn=0024-6115 |volume=11 |pages=623–640}}</ref> The jumping lines of a vector bundle form a proper closed subset of the [[Grassmannian]] of all lines of ...of powers of the [[Tautological bundle]], or in fact of [[line bundle|line bundles]] in general. Still one can gain information of this type by using the foll ...3 KB (514 words) - 02:29, 12 July 2024
- ...etry (mathematics)|gauge transformations]] whose parameter functions are [[vector field]]s on <math>X</math>. From the physical viewpoint, general covariant ...ameter [[Lie group]] of automorphisms of <math>Y</math> is a projectable [[vector field]] ...5 KB (777 words) - 15:58, 23 July 2022
- {{Short description|Type of vector bundle}} ...ins the [[Tannakian formalism|Tannakian category]] of [[essentially finite vector bundle]]s, whose naturally associated group scheme is the [[fundamental gro ...4 KB (512 words) - 13:48, 14 December 2024
- {{Short description|Type of vector bundle}} ...f name="simpson">{{cite journal |last1=Simpson |first1=Carlos |title=Higgs bundles and local systems |journal=Publications Mathématiques de l'IHÉS |date=1992 ...4 KB (614 words) - 06:46, 10 July 2024
- ...nger India.</ref><ref>Ramanathan, A., 1996, November. Moduli for principal bundles over algebraic curves: II. In Proceedings of the Indian Academy of Sciences ...d Biswas, I., 2001. Einstein-Hermitian connections on polystable principal bundles over a compact Kähler manifold. American Journal of Mathematics, 123(2), pp ...8 KB (1,309 words) - 22:21, 10 January 2024
- ...try]], especially [[Moduli space#Moduli of vector bundles|Moduli of Vector Bundles]] on curves.<ref>[http://math.tufts.edu/people/facultyTeixidor.htm People M ...that "If <math>0<s\le n'(n-n')(g-1)</math>, then there exist stable vector bundles with <math>s_{n'}(E)=s</math>." They also clarified what happens in the int ...7 KB (884 words) - 04:07, 14 August 2024
- ...tes the [[moduli stack of vector bundles|moduli stack of rank-''n'' vector bundles]], then there is a presentation <math>\operatorname{Spec}(k) \to \operatorn ...2 KB (243 words) - 19:31, 1 October 2024
- ...tor space]] <math>\overline F</math>. An '''affine bundle''' modelled on a vector bundle <math>\overline\pi:\overline Y\to X</math> is a fiber bundle <math>\ ...ace modelled over the corresponding fibers <math>\overline Y_x</math> of a vector bundle <math>\overline Y</math>. ...5 KB (813 words) - 15:19, 2 November 2021