The space of physical events will be described by a real, smooth manifold of dimension coordinatised by local coordinates , and provided with smooth vector fields with components and linear forms , () in the local coordinate system, as well as further geometrical objects such as tensors, spinors, connections13. At each point, linearly independent vectors (linear forms) form a linear space, the tangent space (cotangent space) of . We will assume that the manifold is space- and time-orientable. On it, two independent fundamental structural objects will now be introduced.
The first is a prescription for the definition of the distance between two infinitesimally close points on , eventually corresponding to temporal and spatial distances in the external world. For , we need positivity, symmetry in the two points, and the validity of the triangle equation. We know that must be homogeneous of degree one in the coordinate differentials connecting the points. This condition is not very restrictive; it still includes Finsler geometry [281, 126, 224] to be briefly touched, below.
In the following, is linked to a non-degenerate bilinear form , called the first fundamental form; the corresponding quadratic form defines a tensor field, the metrical tensor, with components such that14. Besides the norm of a vector , the “angle” between directions , can be defined by help of the metric:
From this we note that an antisymmetric part of the metrical tensor does not influence distances and norms but angles.
With the metric tensor having full rank, its inverse is defined through15
We are used to being a symmetric tensor field, i.e., with and with only components; in this case the metric is called Riemannian if its eigenvalues are positive (negative) definite and Lorentzian if its signature is 16. In the following this need not hold, so that the decomposition obtains17:
For an asymmetric metric, the inverse , pp. 223–224)18 and also by Schrödinger . Eddington also calculated Equation (8); in his expression the term is missing (cf. , 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,