wo affine generalisations from and of the Ricci tensor in Riemannian geometry. We get2627. In order to shorten the presentation of affine geometry, we refrain from listing the corresponding set of equations for the other affine curvature tensor (cf., however, ).
For a symmetric affine connection, the preceding results reduce considerably due to . From Equations (29,30,32) we obtain the identities:independent trace tensor of the affine curvature tensor exists. For the antisymmetric part of the Ricci tensor holds. This equation will be important for the physical interpretation of affine geometry.
In affine geometry, the simplest way to define a fundamental tensor is to set , or . It may be desirable to derive the metric from a Lagrangian; then the simplest scalar density that could be used as such is given by 28.
As a final result in this section, we give the curvature tensor calculated from the connection (cf. Equation (20)), expressed by the curvature tensor of and by the tensor :, p. 141).
From the symmetric part of the first fundamental form , a connection may be constructed, often called after Levi-Civita ,Riemannian curvature tensor defined as in Equation (22) with (cf. Section 2.1.3); is called the Christoffel symbol. Thus, in metric-affine and in mixed geometry, two different connections arise in a natural way. In the remaining part of this section we will deal with a symmetric fundamental form only, and denote it by . With the help of the symmetric affine connection, we may define the tensor of non-metricity by29
Then the following identity holds:contorsion tensor , a linear combination of torsion , is defined by30
The inner product of two tangent vectors is not conserved under parallel transport of the vectors along if the non-metricity tensor does not vanish:
A connection for which the non-metricity tensor vanishes, i.e.,metric-compatible31. J. M. Thomas introduced a combination of the terms appearing in and to define a covariant derivative for the metric (, p. 188), . We will have to deal with Equation (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