• 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) - 19:49, 24 October 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...
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  • 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 (7,683 words) - 14:11, 3 July 2024
  • 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) - 23:08, 31 October 2023
  • (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 (3,438 words) - 01:17, 19 December 2024
  • 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 (2,390 words) - 19:54, 26 May 2024
  • 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 (4,627 words) - 15:28, 2 October 2024
  • framework) Connection (mathematics), a way of specifying a derivative of a geometrical object along a vector field on a manifold Connection (affine bundle) Connection...
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  • 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
  • bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear connections may...
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  • In mathematics, a Hermitian connection ∇ {\displaystyle \nabla } is a connection on a Hermitian vector bundle E {\displaystyle E} over a smooth manifold...
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  • 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 (3,388 words) - 16:16, 22 December 2024
  • 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...
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  • physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be...
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  • Exterior covariant derivative (category Fiber bundles)
    differentiable principal bundle or vector bundle with a connection. Let G be a Lie group and P → M be a principal G-bundle on a smooth manifold M. Suppose...
    19 KB (2,810 words) - 06:18, 20 December 2024
  • 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 (2,253 words) - 22:50, 21 September 2021
  • Thumbnail for Parallel transport
    Parallel transport (category Connection (mathematics))
    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 (2,996 words) - 11:54, 5 November 2024
  • connection: see Ehresmann connection, connection (principal bundle) or connection (vector bundle). It is one of the numbers that are important in the Einstein...
    562 bytes (100 words) - 07:37, 14 November 2023
  • 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 (6,478 words) - 19:49, 24 October 2024
  • Thumbnail for Fiber bundle
    bundle I-bundle Natural bundle Principal bundle Projective bundle Pullback bundle Quasifibration Universal bundle Vector bundle Wu–Yang dictionary Seifert...
    29 KB (4,085 words) - 13:01, 12 September 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
  • frame bundle (principal bundle) of M (or equivalently, a connection on the tangent bundle (vector bundle) of M). A key aspect of the Cartan connection point...
    46 KB (6,745 words) - 22:53, 22 July 2024
  • 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...
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  • Fiber bundle Principal bundle Frame bundle Hopf bundle Associated bundle Vector bundle Tangent bundle Cotangent bundle Line bundle Jet bundle Sheaf (mathematics)...
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  • 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) - 22:46, 7 October 2024
  • mathematics, a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection. Let π : E →...
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  • In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space S of a family of algebraic varieties V s {\displaystyle...
    8 KB (1,093 words) - 15:37, 26 November 2024
  • differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be...
    5 KB (882 words) - 19:02, 2 November 2024
  • 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...
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  • 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