Cartan Connection

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In mathematics, the Cartan connection construction in differential geometry is a broad generalization of the connection concept, based on an understanding of the role of the affine group in the usual approach. It was developed by Élie Cartan, as part of (and as a way of formulating) his moving trihedron method. See also Cartan formalism

Almost formal definitions for vector bundles

A connection in a vector bundle is a way to "distinguish" sections of the bundle along tangent vectors. Let ζ: E →→ B be a vector bundle over a differentiable manifold B with a vector space F of dimension n as the fiber. Let us denote by ∇_uv a section of a vector bundle, the result of the differentiation of the section of the vector bundle v along the field tangent vector u. To be a connection ∇ must satisfy the following identities:

(i) Linearity

{displaystyle nabla _{u}(v_{1}+v_{2})=nabla _{u}v_{1}+nabla _{u}v_{2}}

and

{displaystyle nabla _{u_{1}+u_{2}}v=nabla _{u_{1}}v+nabla _{u_{2}}v}

(ii) Leibniz Rule

{displaystyle nabla _{u}(fv)=df(u)v+fnabla _{u}v}

and

{displaystyle nabla _{fu}v=fnabla _{u}v}

for any differential function

f

the simplest example: if the ζ: E = F × B → B is the projection, ie ζ is a trivial vector bundle, then any section can be described by a differentiable function v: B → F. Therefore one can consider the trivial connection ∇_uv = ∂v/∂u. If one has two connections ∇ and ∇' in the same vector bundle then the difference ω(u, v) = ∇_uv-∇'_uv depends only on the values of u and v at a point, a 1-form in B to values at the Hom(F, F); that is, the ω(u, -) ∈ Hom(F, F) and ω can be described as an n × n matrix of one -shapes. In particular one can choose a local trivialization of the vector bundle and take ∇' as a corresponding trivial connection, then ω gives a complete local description of ∇.

Yeah. G certified GL(F) is the structural group of vector fiber then form ω is a 1-form with values ${mathfrak {g}}$ , the algebra of Lie G. In particular for the tangent fiber of a variety of Riemann we have O(n) as a structural group and for the form ω for the connection of Levi-Civita is a form with values in ${displaystyle {mathfrak {so}}}$ (n), the algebra of Lie O(n) (which can be thought of as antisymmetric matrices in an orthonormal base, or 2-vctors of tangent fiberdo). This way, ω, describe ► in a non-invariant way; it depends on the choice of local trivialization. The following construction extracts invariant information ω.

The following 2-form with values in Hom(F, F) is called curvature form Ω = dω + ω ∧ ω,

where d is the outer derivative and ∧ is the outer (wedge) product (it may seem a bit strange to apply the outer product to forms with values in Hom(F, F), but works the same way). The curvature shape provides the complete local description of the connection up to a gauge transformation.

Once again, if the G certified GL(F) is the structure group of a vector fiber then form Ω is a 2-form with values ${mathfrak {g}}$ , the algebra of Lie G. For tangent fiber from a differentiable variety of Riemann we have O(n) like the structure group and Ω is a 2-form with values ${displaystyle {mathfrak {so}}}$ (n) (which can be thought of as antisymmetric matrices in an orthonormal base). This way Ω is an equivalent description of the curvature tensor.

Aspects of theory

It was developed by Élie Cartan, as part of (and a way of formulating) his movable trihedron method. He works with differential shapes and so on, which are of a computational nature, but have two other important aspects, both of which are more geometric.

A General Theory of Frames

The first of these looks first at the theory of principal bundles (which one might call the general theory of frameworks). The ideal of a connection in a main bundle for a Lie group G is relatively easy to formulate, because in the vertical direction it can be seen that the required data is given by translating all tangent vectors back to the identity element (in Lie algebra), and the definition of the connection must simply add a horizontal component, compatible with that. If G is a kind of affine group with respect to another Lie group H - meaning that G is a semidirect product of H With a vector translation group T on which H acts, an H-bundle can become a G-bundled by the construction of an associated bundle. There is T-associated bundle, too: a vector bundle, in which H acts by automorphisms that become inner automorphisms in G.

The first type of definition in this provision is that a Cartan connection for H is a specific type of G-main connection.

Identifying the tangent bundle

The second type of definition points directly to the tangent bundle T(M) of the differentiable manifold M assumed as the basis. Here the data is a certain type of identification of the T(M), as bundled, as the 'vertical ' tangents in the T-bundling mentioned above (where M is naturally identified as the null section). This is called the weld (the weld): now we have T(M) inside a richer picture, expressed by the transition data H-rated. An important point here, as with the previous discussion, is that H is not assumed to act faithfully on T. That immediately allows spin bundles to take their place in the theory, with H a spin group rather than simply an orthogonal group.

General theory

Cartan reformulated (pseudo) Riemannian differential geometry; but not only for such (metric) differentiable manifolds, but he made the theory for an arbitrary differentiable manifold, including the differentiable manifolds given by Lie groups. This was in terms of mobile frames (repère mobile) as an alternative reformulation of general relativity.

The main idea is to develop the expressions for connections and curvature using orthogonal frames.

The Cartan formalism is an alternative approach to covariant derivative and curvature, with differential forms and frames. Although it is framework dependent, it is very well suited to computations. It can also be understood in terms of base bundles and allows generalizations like spinor bundles.

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