p. 233).
The manifold is called space-time if and the metric is symmetric and Lorentzian, i.e., symmetric and with signature . Nevertheless, sloppy contemporaneaous usage of the term “space-time” includes arbitrary dimension, and sometimes is applied even to metrics with arbitrary signature.
In a manifold with Lorentzian metric, a non-trivial real conformal structure always exists; from the equationcausal structure is provided by the equivalence class of metrics . For an asymmetric metric, this structure can exist as well; it then is determined by the symmetric part of the metric alone taken to be Lorentzian.
A special case of a space with a Lorentzian metric is Minkowski space, whose metrical components, in Cartesian coordinates, are given by19; then holds for the Lorentz group of generators .
The metric tensor may also be defined indirectly through vector fields forming an orthonormal -leg (-bein) with20. From the geometrical point of view, this can always be done (cf. theories with distant parallelism). By introducing 1-forms , Equation (11) may be brought into the form .
A new physical aspect will come in if the are considered to be the basic geometric variables satisfying field equations, not the metric. Such tetrad-theories (for the case ) are described well by the concept of fibre bundle. The fibre at each point of the manifold contains, in the case of an orthonormal -bein (tetrad), all -beins (tetrads) related to each other by transformations of the group , or the Lorentz group, and so on.
In Finsler geometry, the line element depends not only on the coordinates of a point on the manifold, but also on the infinitesimal elements of direction between neighbouring points :
The second structure to be introduced is a linear connection with components ; it is a geometrical object but not a tensor field and its components change inhomogeneously under local coordinate transformations21. The connection is a device introduced for establishing a comparison of vectors in different points of the manifold. By its help, a tensorial derivative , called covariant derivative is constructed. For each vector field and each tangent vector it provides another unique vector field. On the components of vector fields and linear forms it is defined by
We have adopted the notational convention used by Schouten [300, 310, 389]. Eisenhart and others [121, 234] change the order of indices of the components of the connection:symmetric, this does not make any difference as For both kinds of derivatives we have: 22.
A manifold provided with only a linear connection L is called affine space. From the point of view of group theory, the affine group (linear inhomogeneous coordinate transformations) plays a special role: With regard to it the connection transforms as a tensor (cf. Section 2.1.5).
For a vector density (cf. Section 2.1.5), the covariant derivative of contains one more term:
A smooth vector field is said to be parallely transported along a parametrised curve with tangent vector if for its components holds along the curve. A curve is called an autoparallel if its tangent vector is parallely transported along it at each point23:
A transformation mapping autoparallels to autoparallels is given by:projective structure on [404, 403].
The particular set of connections.
In Part II of this article, we shall find the set of transformations playing a role in versions of Einstein’s unified field theory.
From the connection further connections may be constructed by adding an arbitrary tensor field to its symmetrised part24:antisymmetric part of the connection, i.e., torsion; it is a tensor field. The trace of the torsion tensor is called torsion vector; it connects to the two traces of the affine connection ; , as
The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity:
The curvature tensor (22) satisfies two algebraic identities:
From both affine curvature tensors we may form two different tensorial traces each. In the first case , and is called homothetic curvature, while is the first of the t