## 2 Mathematical Preliminaries

For the convenience of the reader, some of the mathematical formalism given in the first part of this review is repeated in a slightly extended form: It is complemented by further special material needed for an understanding of papers to be described.

### 2.1 Metrical structure

First, a definition of the distance between two infinitesimally close points on a D-dimensional differential manifold is to be given, eventually corresponding to temporal and spatial distances in the external world. For , positivity, symmetry in the two points, and the validity of the triangle equation are needed. We assume to be homogeneous of degree one in the coordinate differentials connecting neighboring points. This condition is not very restrictive; it includes Finsler geometry [510, 199, 394, 4*] to be briefly discussed in Section 17.2.

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 neighboring points are labeled by and , respectively^{11}. 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’s tensor does not influence distances and norms but angles.

We are used to being a symmetric tensor field, i.e., with with only components; in this
case the metric is called Riemannian if its eigenvalues are positive (negative) definite and Lorentzian if its signature is
^{12}.
In this case, the norm is . In space-time, i.e., for , the Lorentzian signature is
needed for the definition of the light cone: . The paths of light signals through
the cone’s vertex are assumed to lie in this subspace. In unified field theory, the line element
(“metric”) is an asymmetric tensor, in general. When of full rank, its inverse is defined
through^{13}

^{14}: and have the same rank; also, and have the same signature [27]. Equation (2*) looks quite innocuous. When working with the decompositions (3*), (4*) however, eight tensors are floating around: and its inverse (indices not raised!); and its inverse ; and its inverse , and finally and its inverse .

With the decomposition of the inverse (4*) and the definitions for the respective inverses

the following relations can be obtained:^{15}and where . We also note: and Another useful relation is with . From (9*) we see that unlike in general relativity even invariants of order zero (in the derivatives) do exist: , and ; for the 24 invariants of the metric of order 1 in space-time cf. [512, 513, 514].

Another consequence of the asymmetry of is that the raising and lowering of indices with now becomes more complicated. For vector components we must distinguish:

where the dot as an upper index means that an originally upper index has been lowered. Similarly, for components of forms we have The dot as a lower index points to an originally lower index having been raised. In general, . Fortunately, the raising of indices with the asymmetric metric does not play a role in the following. An easier possibility is to raise and lower indices by the symmetric part of , i.e., by and its inverse
.^{16}
In fact, this is often seen in the literature; cf. [269*, 297*, 298*]. Thus, three new tensors (one symmetric, two
skew) show up:

The tensor density formed from the metric is denoted here by

The components of the flat metric (Minkowski-metric) in Cartesian coordinates is denoted by :

#### 2.1.1 Affine structure

The second structure to be introduced is a linear connection (affine connection, affinity) L with components ;
it is a geometrical object but not a tensor field and its components change inhomogeneously under local coordinate
transformations.^{17}
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 X
and linear forms it is defined by

We have adopted the notational convention used by Schouten [537*, 540*, 683]. Eisenhart and others [182*, 438*] change the order of indices of the components of the connection:

whence followsAs long as the connection is symmetric this does not make any difference because of

For both kinds of derivatives we have: Both derivatives are used in versions of unified field theory by Einstein and
others.^{18}

A manifold provided with only a linear (affine) connection L is called affine space. From the point of
view of group theory, the affine group (linear inhomogeneous coordinate transformations) plays
a special rôle: with regard to it the connection transforms as a tensor; cf. Section 2.1.5 of
Part I.

The covariant derivative with regard to the symmetrical part of the connection is denoted by
such that^{19}

^{20}In the following, always will denote a symmetric connection if not explicitly defined otherwise. To be noted is that: .

For a vector density of coordinate weight , the covariant derivative contains one more term (cf. Section 2.1.5 of Part I):

The metric density of Eq. (13*) has coordinate weight .^{21}For the concept of gauge weight cf. (491*) of Section 13.2.

A smooth vector field Y is said to be parallelly transported along a parametrized curve
with tangent vector X iff for its components holds along the curve. A
curve is called an autoparallel if its tangent vector is parallelly transported along it in each
point:^{22}

A transformation mapping auto-parallels to auto-parallels is given by:

The equivalence class of auto-parallels defined by (23*) defines a projective structure on [691], [690]. The particular set of connections with is mapped into itself by the transformation (23*), cf. [608].In Section 2.2.3, 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 it or to its symmetrized part:

By special choice of T or we can regain all connections used in work on unified field theories. One case is given by Schrödinger’s “star”-connection: for which or . The star connection thus shares the vanishing of the torsion vector with a symmetric connection. Further examples will be encountered in later sections; cf. (382*) of Section 10.3.3.

#### 2.1.2 Metric compatibility, non-metricity

We now assume that in affine space also a metric tensor exists. In the case of a symmetric connection the condition for metric compatibility reads:

In Riemannian geometry this condition guaranties that lengths and angles are preserved under parallel transport. The corresponding torsionless connection^{23}is given by: In place of (28*), for a non-symmetric connection the following equation was introduced by Einstein (and J. M. Thomas) (note the position of the indices!)

^{24}As we have seen in Section 2.1.1, this amounts to the simultaneous use of two connections: and .

^{25}We will name (30*) “compatibility equation” although it has lost its geometrical meaning within Riemannian geometry.

^{26}In terms of the covariant derivative with regard to the symmetric part of the connection, (30*) reduces to In the 2nd term on the r.h.s., the upper index has been lowered with the symmetric part of the metric, i.e., with . After splitting the metric into its irreducible parts, we obtain

^{27}

In place of (30*), equivalently, the -derivative of the tensor density can be made to vanish:

From (30*) or (33*), the connection may in principle be determined as a functional of the metric tensor, its first derivatives,
and of torsion.^{28}
After multiplication with , (33*) can be rewritten as , where
is formed with the Hermitian conjugate connection (cf. Section 2.2.2)
[396].^{29}

Remark:

Although used often in research on UFT, the -notation is clumsy and ambiguous. We apply the
-differentiation to (2*), and obtain: . While the l.h.s. of
the last equation is well defined and must vanish by definition, the r.h.s. is ambiguous and does not vanish:
in both cases , . Einstein had noted this when pointing out that only
but ([147*], p. 580). Already in 1926, J. M. Thomas had seen the
ambiguity of and defined a procedure for keeping valid the product rule for derivatives [607].
Obviously, .

A clearer presentation of (30*) is given in Koszul-notation:

The l.h.s. of (34*) is the non-metricity tensor, a straightforward generalization from Riemannian geometry: (34*) shows explicitly the occurrence of two connections; it also makes clear the multitude of choices for the non-metricity tensor and metric-compatibility. In principle, Einstein could have also used: and further combinations of the - and -derivatives. His adoption of (30*) follows from a symmetry demanded (Hermitian or transposition symmetry); cf. Section 2.2.2.An attempt for keeping a property of the covariant derivative in Riemannian geometry, i.e., preservation of the inner product under parallel transport, has been made by J. Hély [249]. He joined the equations to Eq. (30*). In the presence of a symmetric metric , in place of Eqs. (25*), (26*) a decomposition

with arbitrary can be made.^{30}Hély’s additional condition leads to a totally antisymmetric .

We will encounter another object and its derivatives, the totally antisymmetric tensor:

where is the totally antisymmetric tensor density containing the entries according to whether two indices are equal, or all indices forming an even or odd permutation. For certain derivatives and connections, the object can be covariantly constant [473*, 484*]:

### 2.2 Symmetries

#### 2.2.1 Transformation with regard to a Lie group

In Riemannian geometry, a “symmetry” of the metric with regard to a -generator of a Lie algebra (corresponding, locally, to a Lie-group)

is defined by The vector field is named a Killing vector; its components generate the infinitesimal symmetry transformation: . Equation (43*) may be expressed in a different form:In (44*), is the covariant derivative with respect to the metric [Levi-Civita connection; cf. (29*)]. A conformal Killing vector satisfies the equation:

#### 2.2.2 Hermitian symmetry

This is a generalization (a weakening) of the symmetrization of a real symmetric metric and
connection:^{31}
Hermitian “conjugate” metric and connection are introduced for a complex metric and connection by

The property “Hermitian” (or “self-conjugate”) can be generalized for any pair of adjacent indices of any tensor (cf. [149*], p. 122):

is called the (Hermitian) conjugate tensor. A tensor possesses Hermitian symmetry if . Einstein calls a tensor anti-Hermitian if As an example for an anti-Hermitian vector we may take vector torsion with . The compatibility equation (30*) is Hermitian symmetric; this is the reason why Einstein chose it.For real fields, transposition symmetry replaces Hermitian symmetry.

with .In place of (47*), M.-A. Tonnelat used

as the definition of a Hermitian quantity [627*]. As an application we find and .

#### 2.2.3 -transformation

In (23*) of Section 2.1.1, we noted that transformations of a symmetric connection which preserve auto-parallels are given by:

where is a real 1-form field. They were named projective by Schouten ([537*], p. 287). In later versions of his UFT, Einstein introduced a “symmetry”-transformation called -transformation [156*]: Einstein named the combination of the “group” of general coordinate transformations and -transformations the “extended” group . For an application cf. Section 9.3.1. After gauge- (Yang–Mills-) theory had become fashionable, -transformations with were also interpreted as gauge-transformations [702*, 23*]. According to him the parts of the connection irreducible with regard to diffeomorphisms are “mixed” by (52*), apparently because both will then contain the 1-form . Under (52*) the torsion vector transforms like , i.e., it can be made to vanish by a proper choice of . The compatibility equation (30*) is not conserved under -transformations because of
. The same holds for the projective transformations (51*), cf. ([430*], p. 84). No
generally accepted physical interpretation of the -transformations is known.

### 2.3 Affine geometry

We will speak of affine geometry in particular if only an affine connection exists on the 4-manifold, not a
metric. Thus the concept of curvature is defined.

#### 2.3.1 Curvature

In contrast to Section 2.1.3 of Part I, the two curvature tensors appearing there in Eqs. (I,22) and (I,23) will now be denoted by the -sign written beneath a letter:

Otherwise, this “minus”-sign and the sign for complex conjugation could be mixed up.Trivially, for the index pair , . The curvature tensors (53*), (54*) are skew-symmetric only in the second pair of indices. A tensor corresponding to the Ricci-tensor in Riemannian geometry is given by

On the other hand, Note that the Ricci tensors as defined by (55*) or (56*) need not be symmetric even if the connection is symmetric, and also that when denotes the Hermitian (transposition) conjugate. Thus, in generalIf the curvature tensor for the symmetric part of the connection is introduced by:

then The corresponding expression for the Ricci-tensor is: whence follows: with . Also, the relations hold (for (63*) cf. [549*], Eq. (2,12), p. 278)): A consequence of (62*), (63*) is: Another trace of the curvature tensor exists, the so-called homothetic curvature^{32}: Likewise, such that . For the curvature tensor, the identities hold: where the bracket denotes cyclic permutation while the index does not take part.

Equation (68*) generalizes Bianchi’s identity. Contraction on leads to:

or for a symmetric connection (cf. Section 2.1.3.1 of Part I, Eq. (38)):Finally, two curvature scalars can be formed:

For a symmetric connection, an additional identity named after O. Veblen holds:

The integrability condition for (30*) is ([399*], p. 225), [51]:

For a complete decomposition of the curvature tensor (53*) into irreducible parts with regard to the permutation group further objects are needed, as e.g., ; cf. [348].

#### 2.3.2 A list of “Ricci”-tensors

In many approaches to the field equations of UFT, a generalization of the Ricci scalar serves as a Lagrangian.
Thus, the choice of the appropriate “Ricci” tensor plays a distinct role. As exemplified by Eq. (64*), besides
and there exist many possibilities for building 2-rank tensors which could form a substitute for
the unique Ricci-tensor of Riemannian geometry. In ([150*], p. 142), Einstein gives a list of 4 tensors
following from a “single contraction of the curvature tensor”. Santalò derived an 8-parameter set of
“Ricci”-type tensors constructed by help of
([524*], p. 345). He discusses seven of them used by Einstein, Tonnelat, and
Winogradzki.^{33}
The following collection contains a few examples of the objects used as a Ricci-tensor in
variational principles/field equations of UFT besides and of the previous
section.^{34}
They all differ in terms built from torsion. Among them are:

One of the puzzles remaining in Einstein’s research on UFT is his optimism in the search for a preferred Ricci-tensor although he had known, already in 1931, that presence of torsion makes the problem ambiguous, at best. At that time, he had found a totality of four possible field equations within his teleparallelism theory [176]. As the preceding list shows, now a 6-parameter object could be formed. The additional symmetries without physical support suggested by Einstein did not help. Possibly, he was too much influenced by the quasi-uniqueness of his field equations for the gravitational field.

#### 2.3.3 Curvature and scalar densities

From the expressions (73*) to (81*) we can form scalar densities of the type: to etc. As the preceding formulas show, it would be sufficient to just pick and add scalar densities built from homothetic curvature, torsion and its first derivatives in order to form a most general Lagrangian. As will be discussed in Section 19.1.1, this would draw criticism to the extent that such a theory does not qualify as a unified field theory in a stronger sense.

#### 2.3.4 Curvature and -transformation

The effect of a -transformation (52*) on the curvature tensor is:

In case the curvature tensor is used, instead of (52*) we must take the form for the -transformation:^{35}

Then

also holds. Application of (52*) to , or (85*) to results in many more terms in on the r.h.s. For the contracted curvatures a -transformation leads to (cf. also [430]): If , the curvature tensors and their traces are invariant with regard to the -transformations of Eq. (52*). Occasionally, is interpreted as a gravitational gauge transformation.

### 2.4 Differential forms

In this section, we repeat and slightly extend the material of Section 2.1.4, Part I, concerning Cartan’s
one-form formalism in order to make understandable part of the literature. Cartan introduced one-forms
() by . The reciprocal basis in tangent space is given by .
Thus, . An antisymmetric, distributive and associative product, the external or
“wedge”()-product is defined for differential forms. Likewise, an external derivative can be
introduced.^{36}
The metric (e.g., of space-time) is given by , or . The covariant derivative of a
tangent vector with bein-components is defined via Cartan’s first structure equations,

^{37}. 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 Eq. (90*) the curvature-2-form appears, which is given by is the homothetic curvature.A p-form in n-dimensional space is defined by

^{38}

### 2.5 Classification of geometries

A differentiable manifold with an affine structure is called affine geometry. If both, a (possibly non-symmetric) “metric” and an affine structure, are present we name the geometry “mixed”. A subcase, i.e., metric-affine geometry demands for a symmetric metric. When interpreted just as a gravitational theory, it sometimes is called MAG. A further subdivision derives from the non-metricity tensor being zero or . Riemann–Cartan geometry is the special case of metric-affine geometry with vanishing non-metricity tensor and non-vanishing torsion. Weyl’s geometry had non-vanishing non-metricity tensor but vanishing torsion. In Sections 2.1.3 and 4.1.1 of Part I, these geometries were described in greater detail.

#### 2.5.1 Generalized Riemann-Cartan geometry

For the geometrization of the long-range fields, various geometric frameworks have been
chosen. Spaces with a connection depending solely on a metric as in Riemannian geometry
rarely have been considered in UFT. One example is given by Hattori’s connection, in
which both the symmetric and the skew part of the asymmetric metric enter the
connection^{39}
[240*]:

#### 2.5.2 Mixed geometry

Now, further scalars and scalar densities may be constructed, among them curvature scalars (Ricci-scalars):

Here, and come from the decomposition into irreducible parts of the inverse of the non-symmetric metric . Both parts on the r.h.s. could be taken as a Lagrangian, separately. The inverse of the symmetric part, i.e., of could be used as well to build a scalar: . Mixed geometry is the one richest in geometrical objects to be constructed from the asymmetric metric and the asymmetric connection. What at first may have appeared as an advantage, turned out to become an ‘embarras de richesses’: defining relations among geometric objects and physical observables abound; cf. Section 9.7.Whenever a symmetric tensor appears which is independent of the connection and of full rank, it can play the role of a metric. The geometry then may be considered to be a Riemannian geometry with additional geometric objects: torsion tensor, non-metricity tensor, skew-symmetric part of the “metric” etc. These might be related to physical observables. Therefore, it is moot to believe that two theories are different solely on the basis of the criterion that they can be interpreted either in a background of Riemannian or mixed geometry. However, by a reduction of the more general geometries to a mere Riemannian one plus some additional geometric objects the very spirit of UFT as understood by Einstein would become deformed; UFT explicitly looks for fundamental geometric objects representing the various physical fields to be described.

#### 2.5.3 Conformal geometry

This is an “angle preserving” geometry: in place of a metric a whole equivalence class with a function obtains. Geometrical objects of interest are those invariant with regard to the transformation: . One such object is Weyl’s conformal curvature tensor:

where is the dimension of the manifold (: space-time). is trace-free. For is a necessary and sufficient condition that the space is conformally flat, i.e., ([191], p. 92).If is a Killing vector field for , then is a conformal Killing vector field for ; cf. Eq. (45*) in Section 2.1.2.

A particular sub-case of conformal geometry is “similarity geometry”, for which the restricted group of transformations acts , with a constant , cf. Section 3.1.

### 2.6 Number fields

In Section 2.3 of Part I, the possibility of choosing number fields different from the real numbers for the field variables was stated. Such field variables then would act in a manifold with real coordinates. A more deeply going change is the move to an underlying manifold with coordinates taken from another number field, e.g., complex spaces. The complex number field was most often used in connection with unified field theory in both roles. cf. A. Einstein, (complex space, Section 7.2), J. Moffat, (complex field on real space, Section 13) and A. Crumeyrolle, (hypercomplex manifold, Section 11.2.2).

As hypercomplex numbers are less well known, we briefly introduce them here. Let real and consider the algebra with two elements , where is the unit element and . is called a hypercomplex number. A function will be differentiable in if

The product of two identical real manifolds of dimension n can be made into a manifold with hypercomplex structure.