• vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields...
    45 KB (8,674 words) - 23:09, 1 January 2025
  • metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will...
    18 KB (3,283 words) - 23:21, 7 January 2024
  • Thumbnail for Affine connection
    values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of vector bundles. The notion of an...
    58 KB (7,693 words) - 14:11, 3 July 2024
  • Thumbnail for Vector bundle
    In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space...
    31 KB (4,092 words) - 13:27, 13 April 2025
  • connection is a connection which defines directional derivative for sections of a vector bundle more general than the tangent bundle. Connections also lead...
    19 KB (2,617 words) - 17:10, 15 March 2025
  • basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because under a change of basis, the connection form transforms...
    27 KB (4,630 words) - 05:01, 6 January 2025
  • (Ehresmann) connections on any fiber bundle associated to P {\displaystyle P} via the associated bundle construction. In particular, on any associated vector bundle...
    20 KB (3,436 words) - 15:33, 16 March 2025
  • In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and...
    13 KB (2,394 words) - 19:12, 28 January 2025
  • algebra-valued forms. (A connection form is an example of such a form.) Let M be a smooth manifold and E → M be a smooth vector bundle over M. We denote the...
    13 KB (2,332 words) - 07:37, 12 April 2025
  • bundle modelled over a vector bundle Y → X. A connection Γ on Y → X is called the affine connection if it as a section Γ : Y → J1Y of the jet bundle J1Y...
    6 KB (833 words) - 02:35, 14 March 2021
  • bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear connections may...
    23 KB (3,155 words) - 16:33, 10 January 2024
  • Yang–Mills connection (or Hermite–Einstein connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler...
    7 KB (1,048 words) - 21:04, 19 January 2025
  • Thumbnail for Vertical and horizontal bundles
    vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle π : E → B...
    11 KB (1,528 words) - 01:04, 20 December 2024
  • is a manifold. As such, the fiber is a vector space and the tensor bundle is a special kind of vector bundle. Lee, John M. (2012). Introduction to Smooth...
    1 KB (222 words) - 06:34, 6 April 2023
  • terms of modules and algebras. Connections on modules are generalization of a linear connection on a smooth vector bundle E → X {\displaystyle E\to X} written...
    6 KB (960 words) - 05:31, 2 November 2024
  • secondary vector bundle structure refers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smooth vector bundle...
    6 KB (1,018 words) - 19:54, 2 December 2018
  • physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be...
    72 KB (11,468 words) - 19:43, 14 May 2025
  • Thumbnail for Parallel transport
    with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold...
    20 KB (3,026 words) - 21:38, 30 March 2025
  • In mathematics, a Higgs bundle is a pair ( E , φ ) {\displaystyle (E,\varphi )} consisting of a holomorphic vector bundle E and a Higgs field φ {\displaystyle...
    4 KB (548 words) - 05:46, 10 July 2024
  • mathematics, an adjoint bundle is a vector bundle naturally associated with any smooth principal bundle. The fibers of the adjoint bundle carry a Lie algebra...
    5 KB (819 words) - 10:14, 8 February 2025
  • framework) Connection (mathematics), a way of specifying a derivative of a geometrical object along a vector field on a manifold Connection (affine bundle) Connection...
    3 KB (372 words) - 01:59, 17 December 2024
  • geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian...
    21 KB (3,432 words) - 05:24, 1 May 2025
  • Thumbnail for Yang–Mills equations
    a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations...
    24 KB (3,763 words) - 16:20, 7 February 2025
  • smooth vector bundle. Let E be a smooth vector bundle over a smooth manifold M, and let g be a smooth symmetric bilinear form on E. The Clifford bundle of...
    8 KB (1,203 words) - 01:20, 3 May 2025
  • Covariant derivative (category Connection (mathematics))
    notion of differentiation associated to a connection on a vector bundle, also known as a Koszul connection. Historically, at the turn of the 20th century...
    37 KB (6,453 words) - 09:24, 15 May 2025
  • can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric. If M is...
    6 KB (903 words) - 06:00, 1 November 2023
  • Kobayashi–Hitchin correspondence (category Vector bundles)
    Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence is named after Shoshichi...
    34 KB (4,434 words) - 02:43, 15 January 2025
  • Fiber bundle Principal bundle Frame bundle Hopf bundle Associated bundle Vector bundle Tangent bundle Cotangent bundle Line bundle Jet bundle Sheaf (mathematics)...
    9 KB (682 words) - 03:50, 5 December 2024
  • principal bundle is the frame bundle F ( E ) {\displaystyle F(E)} of a vector bundle E {\displaystyle E} , which consists of all ordered bases of the vector space...
    20 KB (3,361 words) - 22:19, 13 March 2025
  • A_{\Sigma }} . Given the composite bundle Y {\displaystyle Y} (1), there is the following exact sequence of vector bundles over Y {\displaystyle Y} : 0 →...
    7 KB (1,394 words) - 16:36, 27 December 2023