(47) in Section 6.1 and, more intensively, in Part II of this review.
Connections that are not metric-compatible have been used in unified field theory right from the beginning. Thus, in Weyl’s theory [397, 395] we havesemi-metrical [300, 310]. According to Equation (44), in Weyl’s theory the inner product multiplies by a scalar factor under parallel transport:
We may also abbreviate the last term in the identity (42) by introducingtorsionless affine space is given by
Riemann–Cartan geometry is the subcase of a metric-affine geometry in which the metric-compatible connection contains torsion, i.e., an antisymmetric part ; torsion is a tensor field to be linked to physical observables. A linear connection whose antisymmetric part has the formsemi-symmetric .
Riemannian geometry is the further subcase with vanishing torsion of a metric-affine geometry with metric-compatible connection. In this case, the connection is derived from the metric: , where is the usual Christoffel symbol (40). The covariant derivative of with respect to the Levi-Civita connection is abbreviated by . The Riemann curvature tensor is denoted by
An especially simple case of a Riemanian space is Minkowski space, the curvature of which vanishes:
In Riemanian geometry, the so-called geodesic equation,geodesic and autoparallel curves will have to be distinguished.
A conformal transformation of the metric,conformally flat. In , for , the vanishing of the Weyl curvature tensor , p. 404, , p. 170).
Even before Weyl, the question had been asked (and answered) as to what extent the conformal and the projective structures were determining the geometry: According to Kretschmann (and then to Weyl) they fix the metric up to a constant factor (; see also , Appendix 1; for a modern approach, cf. ).
The geometry needed for the pre- and non-relativistic approaches to unified field theory will have to be dealt with separately. There, the metric tensor of space is Euclidean and not of full rank; time is described by help of a linear form (Newton–Cartan geometry, cf. [65, 66]). In the following we shall deal only with relativistic unified field theories.
If we define , with , then transforms like a tangent vector under point transformations of the , and as a covariant vector under homogeneous transformations of the . The may be used to relate covariant vectors and by Thus, the metric tensor in the space of homogeneous coordinates and the metric tensor of are related by with The inverse relationship is given by with The covariant derivative for tensor fields in the space of homogeneous coordinates is defined as before (cf. Section 2.1.2):
In this section, we briefly present Cartan’s one-form formalism in order to make understandable part of the literature. Cartan introduces one-forms () by The reciprocal basis in tangent space is given by . Thus, . The metric is then given by . The covariant derivative of a tangent vector with bein-components is defined via Cartan’s first structure equations,34. The covariant derivative of a tangent vector with bein-components then is
By further external derivation35 on we arrive at the second structure relation of Cartan,
A relative tensor of type and of weight at a point on the manifold transforms liketensor density. We can form a tensor from by introducing , where is a Lorentz-metric. Note that The dual to a 2-form (skew-symmetric tensor) then is defined by
In connection with conformal transformations , the concept of the gauge-weight of a tensor is introduced. A tensor is said to be of gauge weight if it transforms by Equation (56) as
Objects that transform as in Equation (67) but with respect to a subgroup, e.g., the linear group, affine group , orthonormal group , or the Lorentz group , are tensors in a restricted sense; sometimes they are named affine or Cartesian tensors. All the subgroups mentioned are Lie-groups, i.e., continuous groups with a finite number of parameters. In general relativity, both the “group” of general coordinate transformations and the Lorentz group are present. The concept of tensors used in Special Relati