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In comparison with General Relativity, dynamic variables of metric-affine gravitation theory are both a pseudo-Riemannian metric and a general linear connection on a world manifold X {\displaystyle X}. Metric-affine gravitation theory has been suggested as a natural generalization of Einstein–Cartan theory of gravity with torsion where a linear connection obeys the condition that a covariant derivative of a metric equals zero.
Metric-affine gravitation theory straightforwardly comes from gauge gravitation theory where a general linear connection plays the role of a gauge field. Let T X {\displaystyle TX} be the tangent bundle over a manifold X {\displaystyle X} provided with bundle coordinates {\displaystyle }. A general linear connection on T X {\displaystyle TX} is represented by a connection tangent-valued form
It is associated to a principal connection on the principal frame bundle F X {\displaystyle FX} of frames in the tangent spaces to X {\displaystyle X} whose structure group is a general linear group G L {\displaystyle GL} . Consequently, it can be treated as a gauge field. A pseudo-Riemannian metric g = g μ ν d x μ ⊗ d x ν {\displaystyle g=g_{\mu \nu }dx^{\mu }\otimes dx^{\nu }} on T X {\displaystyle TX} is defined as a global section of the quotient bundle F X / S O → X {\displaystyle FX/SO\to X} , where S O {\displaystyle SO} is the Lorentz group. Therefore, one can regard it as a classical Higgs field in gauge gravitation theory. Gauge symmetries of metric-affine gravitation theory are general covariant transformations.
It is essential that, given a pseudo-Riemannian metric g {\displaystyle g} , any linear connection Γ {\displaystyle \Gamma } on T X {\displaystyle TX} admits a splitting