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1 Vierbeins, et cetera

The vierbein or tetrad theory is the special case of a four-dimensional manifold. It applies to metrics of any signature. In any dimension, for a pseudo Riemannian geometry (with metric signature (p,q)), this Cartan connection theory is an alternative method in differential geometry. In different contexts it has also been called the orthonormal frame, repère mobile, soldering form or orthonormal nonholonomic basis method.

This section is an approach to tetrads, but written in general terms. In dimensions other than 4, words like triad, pentad, funfbein, elfbein etc. have been used. Vielbein covers all dimensions.

If you're looking for a basis-dependent index notation, see tetrad (index notation) .

1.1 The basic ingredients

Suppose given a differential manifold M of dimension n, and fixed natural numbers p and q with p + q = n. Further, we suppose given a SO(p, q) principal bundle B over M (called the frame bundle), and a vector SO(p, q)-bundle V associated to B by means of with the natural n-dimensional representation of SO(p, q).

Suppose given also a SO(p, q)-invariant metric η of signature (p, q) over V; and an invertible linear map between vector bundles over M, , where TM is the tangent bundle of M.

1.2 Constructions

A ( pseudo-) Riemannian metric is defined over M as the pullbackThis article discusses the pullback in differential geometry. For the pullback in category theory see pullback (category theory). In mathematics, the pullback of smooth map f : M → N between differentiable manifolds is a smooth vector bundle morphism of η by e. To put it in other words, if we have two sections of TM, X and Y,

g(X,Y) = η(e(X),e(Y)).

A connectionIn differential geometry, the connection form describes connection on principal bundles (or vector bundles). It can be considered as an generalization/alternative for Christoffel symbols. Principal bundles For a principal G bundle , for each let denote th over V is defined as the unique connection A satisfying these two conditions:

• dη(a,b) = η(d_Aa,b) + η(a,d_Ab) for all differentiable sections a and b of V (i.e. d_Aη = 0) where d_A is the covariant exterior derivative. This implies that A can be extended to a connectionIn differential geometry, the connection form describes connection on principal bundles (or vector bundles). It can be considered as an generalization/alternative for Christoffel symbols. Principal bundles For a principal G bundle , for each let denote th over the SO(p,q) principal bundle.
• d_Ae = 0. This basically states that ∇ defined below is torsion-free. This condition is dropped in the Einstein-Cartan theory, but then we can't define A uniquely anymore.

Now that we've specified A, we can use it to define a connection ∇ over TM via the isomorphism e:

e(∇X) = d_Ae(X) for all differentiable sections X of TM.

Since what we now have here is a SO(p,q) gauge theory, the Riemann curvature F defined as is pointwise gauge covariant. This is simply the Riemann tensor in a different guise.

See also connection form and curvature form.

Side note: the e here is often written as θ, the A here as ω and the F here as Ω and d_{A as D.}