This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. ...

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 is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ...

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. (In German, vier stands for four and viel stands for many)

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

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 the natural n-dimensional representation of SO(p, q). In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, the generalized orthogonal group, O(p, q) is the Lie group of all linear transformations of a p + q = n dimensional real vector space which leave invariant a nondegenerate, symmetric, bilinear form of signature (p, q). ... 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 and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ...

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. See: International System of Units, colloquially called the Metric System, and also metrication. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ... In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... 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, 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. ...

Constructions

A (pseudo-)Riemannian metric is defined over M as the pullback of η by e. To put it in other words, if we have two sections of TM, X and Y, In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... This article discusses the pullback in differential geometry. ...

g(X,Y) = η(e(X),e(Y)).

A connection over V is defined as the unique connection A satisfying these two conditions: In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ...

• 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 connection over the SO(p,q) principal bundle.
• d_Ae = 0. The quantity on the left hand side is called the torsion. 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: In differential geometry, the connection form describes connection on principal bundles (or vector bundles). ... 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 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, the term torsion has several meanings, mostly unrelated to each other. ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... Einstein-Cartan theory extends general relativity to correctly handle spin angular momentum. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

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. Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

See also connection form and curvature form. 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 differential geometry, the curvature form describes curvature of principal bundle with connection. ...

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.

The Palatini action

In the tetrad formulation of general relativity, the action, as a functional of the cotetrad e and a connection form A over a four dimensional differential manifold M is given by Two-dimensional visualisation of space-time distortion. ... In physics, the action principle is an assertion about the nature of motion, from which the trajectory of an object subject to forces can be determined. ... In mathematics, the term functional is applied to certain functions. ... 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 mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...

where F is the gauge curvature 2-form and ε is the antisymmetric intertwiner of four "vector" reps of SO(3,1) normalized by η. Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, the generalized orthogonal group, O(p, q) is the Lie group of all linear transformations of a p + q = n dimensional real vector space which leave invariant a nondegenerate, symmetric, bilinear form of signature (p, q). ...

Note that in the presence of spinor fields, the Palatini action implies that d_Ae is nonzero, that is, have torsion. See Einstein-Cartan theory. In mathematics, the term torsion has several meanings, mostly unrelated to each other. ... Einstein-Cartan theory extends general relativity to correctly handle spin angular momentum. ...

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