### 2.1 Geometry

It is very easy to get lost in the many constructive possibilities underlying the geometry of unified field theories. We briefly describe the mathematical objects occurring in an order that goes from the less structured to the more structured cases. In the following, only local differential geometry is taken into account.

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, connections. 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.

#### 2.1.1 Metrical structure

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 [281126224] 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 that

where the neighbouring points are labeled by and , respectively. 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 through

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 . In the following this need not hold, so that the decomposition obtains:

An asymmetric metric was considered in one of the first attempts at unifying gravitation and electromagnetism after the advent of general relativity.

For an asymmetric metric, the inverse

is determined by the relations
and turns out to be [356]
where , , and are the determinants of the corresponding tensors , , and . We also note that
where , . The results (6,7,8) were obtained already by Reichenbächer ([273], pp. 223–224) and also by Schrödinger [320]. Eddington also calculated Equation (8); in his expression the term is missing (cf. [59], 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 equation

results an equivalence class of metrics with being an arbitrary smooth function. In view of the physical interpretation of the light cone as the locus of light signals, a causal structure is provided by the equivalence class of metrics [67]. 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 by

A geometrical characterization of Minkowski space as an uncurved, flat space is given below. Let be the Lie derivative with respect to the tangent vector X; then holds for the Lorentz group of generators .

The metric tensor may also be defined indirectly through vector fields forming an orthonormal -leg (-bein) with

where the hatted indices (“bein-indices”) count the number of legs spanning the tangent space at each point and are moved with the Minkowski metric. 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 :

Again, is required to be homogeneous of rank 1.

#### 2.1.2 Affine structure

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 transformations. 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

The expressions and are abbreviated by and , respectively, while for a scalar covariant and partial derivative coincide: .

We have adopted the notational convention used by Schouten [300310389]. Eisenhart and others [121234] change the order of indices of the components of the connection:

As long as the connection is symmetric, this does not make any difference as For both kinds of derivatives we have:
Both derivatives are used in versions of unified field theory by Einstein and others.

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 point:

By a particular choice of the curve’s parameter, may be imposed.

A transformation mapping autoparallels to autoparallels is given by:

The equivalence class of autoparallels defined by Equation (18) defines a projective structure on  [404403].

The particular set of connections

with is mapped into itself by the transformation (18[348].

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 part:

By special choice of we can regain all connections used in work on unified field theories. We will encounter examples in later sections. The antisymmetric part of the connection, i.e.,
is called 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

#### 2.1.3 Different types of geometry

##### 2.1.3.1 Affine geometry
Various subcases of affine spaces will occur, dependent on whether the connection is asymmetric or symmetric, i.e., with . In physical applications, a metric always seems to be needed; hence in affine geometry it must be derived solely by help of the connection or, rather, by tensorial objects constructed from it. This is in stark contrast to Riemannian geometry where, vice versa, the connection is derived from the metric. Such tensorial objects are the two affine curvature tensors defined by

respectively. In a geometry with symmetric affine connection both tensors coincide because of
In particular, in Riemannian geometry, both affine curvature tensors reduce to the one and only Riemann curvature tensor.

The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity:

For a vector density, the identity is given by
with the homothetic curvature to be defined below in Equation (31).

The curvature tensor (22) satisfies two algebraic identities:

where the curly bracket denotes cyclic permutation:
These identities can be found in Schouten’s book of 1924 ([300], p. 88, 91) as well as the additional single integrability condition, called Bianchi identity:
A corresponding condition obtains for the curvature tensor from Equation (23).

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 two affine generalisations from and of the Ricci tensor in Riemannian geometry. We get

and the following identities hold:
where . While is antisymmetric, has both tensorial symmetric and antisymmetric parts:
We use the notation in order to exclude the index from the symmetrisation bracket. 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, [356]).

For a symmetric affine connection, the preceding results reduce considerably due to . From Equations (29,30,32) we obtain the identities:

i.e., only one 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 .

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 :

where is the covariant derivative formed with the connection (cf. also [310], p. 141).

##### 2.1.3.2 Mixed geometry
A manifold carrying both structural elements, i.e., metric and connection, is called a metric-affine space. If the first fundamental form is taken to be asymmetric, i.e., to contain an antisymmetric part , we speak of a mixed geometry. In principle, both metric-affine space and mixed geometry may always be re-interpreted as Riemannian geometry with additional geometric objects: the 2-form field (symplectic form), the torsion , and the non-metricity (cf. Equation 41). It depends on the physical interpretation, i.e., the assumed relation between mathematical objects and physical observables, which geometry is the most suitable.

From the symmetric part of the first fundamental form , a connection may be constructed, often called after  [204],

and from it the 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 by

Then the following identity holds:

where the contorsion tensor , a linear combination of torsion , is defined by

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.,

holds, is called metric-compatible. introduced a combination of the terms appearing in and to define a covariant derivative for the metric ([346], p. 188),
and extended it for tensors of arbitrary rank . Einstein later used as a constraint on the metrical tensor
a condition that cannot easily be interpreted geometrically [97]. 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 [397395] we have

In case of such a relationship, the geometry is called semi-metrical [300310]. According to Equation (44), in Weyl’s theory the inner product multiplies by a scalar factor under parallel transport:
This means that the light cone is preserved by parallel transport.

We may also abbreviate the last term in the identity (42) by introducing

Then, from Equation (39), the curvature tensor of a torsionless affine space is given by
where is the covariant derivative formed with the Christoffel symbol.

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 form

is called semi-symmetric [300].

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:

This is an invariant characterisation irrespective of whether the Minkowski metric is given in Cartesian coordinates as in Equation (10), or in an arbitrary coordinate system. We also have where is the Lie-derivative (see below under “symmetries”), and stands for the generators of the Lorentz group.

In Riemanian geometry, the so-called geodesic equation,

determines the shortest and the straightest curve between two infinitesimally close points. However, in metric affine and in mixed geometry geodesic and autoparallel curves will have to be distinguished.

A conformal transformation of the metric,

with a smooth function changes the components of the non-metricity tensor,
as well as the Levi-Civita connection,
with As a consequence, the Riemann curvature tensor is also changed; if, however, can be reached by a conformal transformation, then the corresponding space-time is called conformally flat. In , for , the vanishing of the curvature tensor
is a necessary and sufficient condition for to be conformally flat ([397], p. 404, [300], 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 ([196]; see also [401], Appendix 1; for a modern approach, cf. [67]).

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. [6566]). In the following we shall deal only with relativistic unified field theories.

##### 2.1.3.3 Projective geometry
Projective geometry is a generalisation of Riemannian geometry in the following sense: Instead of tangent spaces with the light cone , where is the Minkowski metric, in each event now a tangent space with a general, non-degenerate surface of second order will be introduced. This leads to a tangential cone in the origin (cf. Equation (9)), and to a hyperplane, the polar plane, formed by the contact points of the tangential cone and the surface . In place of the inhomogeneous coordinates of , homogeneous coordinates () are defined such that they transform as homogeneous functions of first degree:

The connection to the inhomogeneous coordinates is given by homogeneous functions of degree zero, e.g., by . Thus, the themselves form the components of a tangent vector. Furthermore, the quadratic form is adopted with being a homogeneous function of degree . A tensor field (cf. Section 2.1.5) depending on the homogeneous coordinates with contravariant (upper) and covariant (lower) indices is required to be a homogeneous function of degree .

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):

The covariant derivative of the quantity interconnecting both spaces is given by

#### 2.1.4 Cartan’s method

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,

where is the connection-1-form, and is the torsion-2-form, . We have The link to the components of the affine connection is given by . The covariant derivative of a tangent vector with bein-components then is

By further external derivation on we arrive at the second structure relation of Cartan,

In Equation (65) the curvature-2-form appears, which is given by
is the homothetic curvature.

#### 2.1.5 Tensors, spinors, symmetries

##### 2.1.5.1 Tensors
Up to here, no definitions of a tensor and a tensor field were given: A tensor of type at a point on the manifold is a multi-linear function on the Cartesian product of cotangent- and tangent spaces in . A tensor field is the assignment of a tensor to each point of . Usually, this definition is stated as a linear, homogeneous transformation law for the tensor components in local coordinates:

where with smooth functions on the r.h.s. are taken from the set (“group”) of coordinate transformations (diffeomorphisms). Strictly speaking, tensors are representations of the abstract group at a point on the manifold.

A relative tensor of type and of weight at a point on the manifold transforms like

An example is given by the totally antisymmetric object with , or depending on whether is an even or odd permutation of (0123), or whether two indices are alike. for ; in this case, the relative tensor is called tensor 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 Relativity is restricted to a representation of the Lorentz group; however, as soon as the theory is to be given a coordinate-independent (“generally covariant”) form, then the full tensor concept comes into play.

##### 2.1.5.2 Spinors
Spinors are representations of the Lorentz group only; as such they are related strictly to the tangent space of the space-time manifold. To see how spinor representations can be obtained, we must use the 2–1 homomorphism of the group and the proper orthochronous Lorentz group, a subgroup of the full Lorentz group. Let ; then is a complex (2-by-2)-matrix with . By picking the special Hermitian matrix

where is the (2 by 2)-unit matrix and are the Pauli matrices satisfying

Then, by a transformation from ,

where is the Hermitian conjugate matrix. Moreover, which, according to Equation (70), expresses the invariance of the space-time distance to the origin:
The link between the representation of a Lorentz transformation in space-time and the unimodular matrix mapping spin space (cf. below) is given by
Thus, the map is two to one: and give the same .

Now, contravariant 2-spinors () are the elements of a complex linear space, spinor space, on which the matrices are acting. The spinor is called elementary if it transforms under a Lorentz-transformation as

Likewise, contravariant dotted spinors are those transforming with the complex-conjugate matrix :
Covariant and covariant dotted 2-spinors correspondingly transform with the inverse matrices,
and
The space of 2-spinors can be used as a representation space for the (proper, orthochronous) Lorentz group, with the 2-spinors being the elements of the most simple representation .

Higher-order spinors with dotted and undotted indices transform correspondingly. For the raising and lowering of indices now a real, antisymmetric -matrix with components is needed, such that

Next to a spinor, bispinors of the form , etc. are the simplest quantities (spinors of 2nd order). A vector can be represented by a bispinor ,

where () is a quantity linking the tangent space of space-time and spinor space. If numerates the matrices and , designate rows and columns, then we can chose to be the unit matrix while for the other three indices are taken to be the Pauli matrices. Often the quantity is introduced. The reciprocal matrix is defined by
whereas

In order to write down spinorial field equations, we need a spinorial derivative,

with The simplest spinorial equation is the equation:
The next simplest spinor equation for two spinors would be
where is a mass. Equation (85) is the 2-spinor version of Dirac’s equation.

Dirac- or 4-spinors with 4 components , , may be constructed from 2-spinors as a direct sum of contravariant undotted and covariant dotted spinors and : For , we enter and ; for , we enter and In connection with Dirac spinors, instead of the Pauli-matrices the Dirac -matrices (-matrices) appear; they satisfy

The Dirac equation is in 4-spinor formalism [5354]:
with the 4-component Dirac spinor . In the first version of Dirac’s equation, - and -matrices were used, related to the ’s by
where the matrices and are given by , .

The generally-covariant formulation of spinor equations necessitates the use of -beins , whose internal “rotation” group, operating on the “hatted” indices, is the Lorentz group. The group of coordinate transformations acts on the Latin indices. In Cartan’s one-form formalism (cf. Section 2.1.4), the covariant derivative of a 4-spinor is defined by

where .

Equation (89) is a special case of the general formula for the covariant derivative of a tensorial form , i.e., a vector in some vector space , whose components are differential forms,

where is a particular representation of the corresponding Lie algebra in with basis vectors For the example of the Dirac spinor, the adjoint representation of the Lorentz group must be used.

##### 2.1.5.3 Symmetries
In Section 2.1.1 we briefly met the Lie derivative of a vector field with respect to the tangent vector defined by . With its help we may formulate the concept of isometries of a manifold, i.e., special mappings, also called “motions”, locally generated by vector fields satisfying

The generators solving Equation (91) given some metric, form a Lie group , the group of motions of . If a group is prescribed, e.g., the group of spatial rotations , then from Equation (91) the functional form of the metric tensor having as a symmetry group follows.

A Riemannian space is called (locally) stationary if it admits a timelike Killing vector; it is called (locally) static if this Killing vector is hypersurface orthogonal. Thus if, in a special coordinate system, we take then from Equation (91) we conclude that stationarity reduces to the condition . If we take to be the tangent vector field to the congruence of curves , i.e., if , then a necessary and sufficient condition for hypersurface-orthogonality is

A generalisation of Killing vectors are conformal Killing vectors for which with an arbitrary smooth function holds. In purely affine spaces, another type of symmetry may be defined: ; they are called affine motions [425].