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In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. In a certain sense, it captures the idea of Christoffel symbols on a Riemannian manifold and re-expresses this idea in a more general way, so that it is applicable on a principle bundle. See also Cartan connection for a more abstract treatment. Mathematics is the study of quantity, structure, space and change. ...
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. ...
In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves...
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 vector spaces, glued together, form another topological space (or manifold or variety). ...
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ...
In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...
In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...
In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. ...
Principal bundles
For a principal G-bundle , for each let Tx(E) denote the tangent space at x and Vx the vertical subspace tangent to the fiber . Then connection is an assignment of a horizontal subspace Hx of Tx(E) such that - Tx(E) is direct sum of Vx and Hx,
- The distribution of Hx is invariant with respect to the G-action on E, i.e. Hax = Dx(Ra)Hx for any and , here Dx(Ra) denotes the differential of the group action by a at x.
- The distribution Hx depends smoothly on x.
This can be recast more elegantly using the jet bundle . The assignment of a horizontal subspace at each point is none other than a smooth section of this jet bundle. In mathematics, groups are often used to describe symmetries of objects. ...
In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. ...
The subspace Vx can be naturally identified with the Lie algebra g of group G, say by map . Then the connection form is a form ω on E with values in g defined by where v denotes projection at of to Vx with kernel Hx.
Given a local trivialization one can reduce ω to the horizontal vector fields (in this trivialization). It defines form say ω' on B. The form ω' defines ω completely, but it depends on the choice of trivialization. (This form is often also called connection form and denoted also by ω.)
Related definitions
Exterior covariant derivative The exterior covariant derivative is a very useful notion which makes possible to simplify formulas in using connection. Given a tensor-valued differential k-form φ its exterior covariant derivative defined by - Dφ(X0,X1,...,Xk) = dφ(h(X0),h(X1),...,h(Xk))
where h denotes the projection to the horizontal subspace, Hx with kernel Vx and Xi are arbitrary vector fields on E.
The curvature form Ω, a g-valued 2-form, can be defined by In differential geometry, the curvature form describes curvature of principal bundle with connection. ...
where [ * , * ] denotes the Lie bracket. This equation is also called the second structure equation. A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
Torsion For the connection on a frame bundle, the curvature is not the only invariant of connection since the additional structure should be taken into account. Namely one has an extra canonical Rn-valued form θ = θi on E defined by identity In mathematics, the term torsion has several meanings, mostly unrelated to each other. ...
. Then the torsion form, an Rn-valued 2-form can be defined by This equation is also called the first structure equation.
Vector bundles The connection form for the vector bundle is the form on the total space of associated principal bundle, but it can be completely described by the following form (on the base in a NOT invariant way). This subsection can be considered as a smoother but bit wrong introduction to connection form. In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . ...
A covariant derivative on a vector bundle is a way to "differentiate" bundle sections along tangent vectors, it is also sometimes called connection. Let be a vector bundle over a smooth manifold B with a n-dimensional vector space F as a fiber. Let us denote by a section of the vector bundle, the result of differentiation of the section of vector bundle v along tangent vector field u. In order to be a covariant derivative must satisfy the following identities: In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...
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 vector spaces, glued together, form another topological space (or manifold or variety). ...
In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. ...
- (i) and (linearity)
- (ii) and for any smooth function f.
The simplest example: if is the projection, i.e. ζ is a trivial vector bundle, then any section can be described by a smooth map . Therefore one can consider the trivial covariant derivative defined by partial derivatives:
If one has two connections and on the same vector bundle then the difference depends only on values of u and v at a point, ω is a 1-form on B with values in Hom(F,F); i.e. and ω can be described as an -matrix of one-forms. In particular one can choose a local trivialization of the vector bundle and take to be correspondent trivial connection, then ω gives a complete local description of . A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...
The choice of trivialization is equivalent to choice of frames in each fiber, that explains the reason for the name Method of moving frames. Let us choose (a local smooth section of) basis frames ei in fibers. Then the matrix of 1-forms is defined by the following identity:
If is the structure group of the vector bundle and connection respects... the group then the form ω is a 1-form with values in g, the Lie algebra of G. In particular for the tangent bundle of a Riemannian manifold we have O(n) as the structure group and for the form ω for the Levi-Civita connection is a form with values in so(n), the Lie algebra of O(n) (which can be thought of as antisymmetric matrices in an orthonormal basis). In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ...
In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...
In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ...
In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...
In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ...
Related definitions
Curvature The connection form (ω) describes connection () in a non-invariant way; it depends on the choice of local trivialization. The following construction extracts invariant information out of ω.
The following 2-form with values in Hom(F,F) is called curvature form In differential geometry, the curvature form describes curvature of principal bundle with connection. ...
where d stands for exterior derivative and is the wedge product. This equation also called the second structure equation. In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...
In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...
Torsion For the connection on tangent bundle the curvature is not the only invariant of connection since the additional structure should be taken into account. Namely one has an extra canonical Rn-valued form θ = θi on B defined by identity Then the torsion, an of Rn-valued 2-form can be defined by This equation is also called the first structure equation.
References - Kobayashi, Shoshichi; Nomizu, Katsumi; Foundations of differential geometry Vol. I. Reprint of the 1963 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. xii+329 pp. ISBN 0-471-15733-3
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