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Connection (affine Bundle)
Let be an affine bundle modelled over a vector bundle . A connection on is called the affine connection if it as a section of the jet bundle of is an affine bundle morphism over . In particular, this is an affine connection on the tangent bundle of a smooth manifold . (That is, the connection on an affine bundle is an example of an affine connection; it is not, however, a general definition of an affine connection. These are related but distinct concepts both unfortunately making use of the adjective "affine".) With respect to affine bundle coordinates on , an affine connection on is given by the tangent-valued connection form : \begin\Gamma &=dx^\lambda\otimes \left(\partial_\lambda + \Gamma_\lambda^i\partial_i\right)\,, \\ \Gamma_\lambda^i&=_j\left(x^\nu\right) y^j + \sigma_\lambda^i\left(x^\nu\right)\,. \end An affine bundle is a fiber bundle with a general affine structure group of affine transformations of its typical fiber of dimension . Therefore, an affin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Bundle
In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.. (page 60) Formal definition Let \overline\pi:\overline Y\to X be a vector bundle with a typical fiber a vector space \overline F. An affine bundle modelled on a vector bundle \overline\pi:\overline Y\to X is a fiber bundle \pi:Y\to X whose typical fiber F is an affine space modelled on \overline F so that the following conditions hold: (i) Every fiber Y_x of Y is an affine space modelled over the corresponding fibers \overline Y_x of a vector bundle \overline Y. (ii) There is an affine bundle atlas of Y\to X whose local trivializations morphisms and transition functions are affine isomorphisms. Dealing with affine bundles, one uses only affine bundle coordinates (x^\mu,y^i) possessing affine transition functions : y'^i= A^i_j(x^\nu)y^j + b^i(x^\nu). There are the bundle morphisms : Y\times_X\overline Y\longrightarrow Y,\qquad (y^i, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautological One-form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagrangian mechanics with Hamiltonian mechanics (on the manifold Q). The exterior derivative of this form defines a symplectic form giving T^Q the structure of a symplectic manifold. The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle. To define the tautological one-form, select a coordinate chart U on T^*Q and a canonical coordinate system on U. Pick an arbitrary point m \in T^*Q. By definition of cotangent bundl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Gauge Theory
Affine gauge theory is classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold X. For instance, these are gauge theory of dislocations in continuous media when X=\mathbb R^3, the generalization of metric-affine gravitation theory when X is a world manifold and, in particular, gauge theory of the fifth force. Affine tangent bundle Being a vector bundle, the tangent bundle TX of an n-dimensional manifold X admits a natural structure of an affine bundle ATX, called the ''affine tangent bundle'', possessing bundle atlases with affine transition functions. It is associated to a principal bundle AFX of affine frames in tangent space over X, whose structure group is a general affine group GA(n,\mathbb R). The tangent bundle TX is associated to a principal linear frame bundle FX, whose structure group is a general linear group GL(n,\mathbb R). This is a subgroup of GA(n,\mathbb R) so that the latter is a semidirect product of G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a ''parallel'' and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an affine connection, the most elementary type of connection, gives a means for parallel transport of tangent vectors on a manifold from one point to another along a curve. An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation of a vector field from being parallel in a given direction. Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and the local geometry at another point. Differential geometry embraces severa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (vector Bundle)
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a '' covariant derivative'', an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear. Linear connections are also called Koszul connections after Jean-Louis Koszul, who g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with 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 affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space. On any manifold of positive dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (fibred Manifold)
In differential geometry, a fibered manifold is surjective submersion of smooth manifolds . Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manifolds provides a general framework of a connection on fiber bundles. Formal definition Let be a fibered manifold. A generalized ''connection'' on is a section , where is the jet manifold of . Connection as a horizontal splitting With the above manifold there is the following canonical short exact sequence of vector bundles over : where and are the tangent bundles of , respectively, is the vertical tangent bundle of , and is the pullback bundle of onto . A connection on a fibered manifold is defined as a linear bundle morphism over which splits the exact sequence . A connection always exists. Sometimes, this connection is called the Ehresmann connection because it yields the horizontal distribution : \mathrmY=\Gamma\left(Y\times_X \mathrmX \right) \subset \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion Tensor
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves (or rather the rotation of the Frenet–Serret frame about the tangent vector). In the geometry of surfaces, the ''geodesic torsion'' describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames "roll" along a curve "without twisting". More generally, on a differentiable manifold equipped with an affine connection (that is, a connection in the tangent bundle), torsion and curvature form the two fundamental invariants of the connection. In this context, torsion gives an intrinsic characterization of how tangent spaces twist about a curve when they are parallel transported; whereas curvature describes how the tangent spaces roll al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Connection
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces. The theory of Cartan connections was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames (''repère mobile''). The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand. In relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection. For Lie groups, Maurer–Cartan frames are used to view the Maurer–Cartan form of the group as a Car ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector-valued Differential Form
In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.) Definition Let ''M'' be a smooth manifold and ''E'' → ''M'' be a smooth vector bundle over ''M''. We denote the space of smooth sections of a bundle ''E'' by Γ(''E''). An ''E''-valued differential form of degree ''p'' is a smooth section of the tensor product bundle of ''E'' with Λ''p''(''T'' ∗''M''), the ''p''-th exterior power of the cotangent bundle of ''M''. The space of such forms is denoted by :\Omega^p(M,E) = \Gamma(E\otimes\Lambda^pT^*M). Because Γ is a strong monoidal functor, this can also be interpreted as :\Gamma(E\otimes\Lambd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (fibred Manifold)
In differential geometry, a fibered manifold is surjective submersion of smooth manifolds . Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manifolds provides a general framework of a connection on fiber bundles. Formal definition Let be a fibered manifold. A generalized ''connection'' on is a section , where is the jet manifold of . Connection as a horizontal splitting With the above manifold there is the following canonical short exact sequence of vector bundles over : where and are the tangent bundles of , respectively, is the vertical tangent bundle of , and is the pullback bundle of onto . A connection on a fibered manifold is defined as a linear bundle morphism over which splits the exact sequence . A connection always exists. Sometimes, this connection is called the Ehresmann connection because it yields the horizontal distribution : \mathrmY=\Gamma\left(Y\times_X \mathrmX \right) \subset \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (vector Bundle)
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a '' covariant derivative'', an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear. Linear connections are also called Koszul connections after Jean-Louis Koszul, who g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
