In his assessment, Eisenhart [121] adds to this all
the geometries whose metric is
“based upon an integral whose integrand is
homogeneous of the first degree in the differentials. Developments
of this theory have been made by Finsler, Berwald, Synge, and
J. H. Taylor. In this geometry the paths are the shortest
lines, and in that sense are a generalisation of geodesics. Affine
properties of these spaces are obtained from a natural
generalisation of the definition of Levi-Civita for Riemannian
spaces.” ([121], p. V)
In fact, already in May 1921 Jan Arnoldus Schouten in Delft had submitted
two papers classifying all possible connections [297, 296
]. In the first he
wrote:
“Motivated by relativity theory, differential
geometry received a totally novel, simple and satisfying
foundation; I just refer to G. Hessenberg’s ‘Vectorial
foundation...’, Math. Ann. 78, 1917, S. 187-217 and
H. Weyl, Raum-Zeit-Materie, 2. Section, Leipzig
1918 (3. Aufl. Berlin 1920) as well as ‘Reine
Infinitesimalgeometrie’ etc.107
. [...] In the
present investigation all 18 different linear connections are
listed and determined in an invariant manner. The most general
connection is characterised by two fields of third degree, one
tensor field of second degree, and a vector field [...].” 108 ([297
], p. 57)
The fields referred to are the torsion tensor
, the tensor of non-metricity
, the metric
, and the tensor
which, in unified field theory, was rarely used. It
arose because Schouten introduced different
linear connections for tangent vectors and linear forms. He defined
the covariant derivative of a 1-form not by the connection
in Equation (13
), but by
Furthermore, on p. 57 of [297] we read:
“The general connection for at least theoretically opens the door for an
extension of Weyl’s theory. For such an
extension an invariant fixing of the connection is needed, because
a physical phenomenon can correspond only to an invariant
expression.” 110
Through footnote 5 on the same page we learn the pedagogical reason why Schouten did not use the ‘direct’ method [294, 337] in his presentation, but rather a coordinate dependent formalism111 :
“As the results of the present investigation
might be of interest for a wider circle of mathematicians, and also
for a number of physicists [...].” 112
At the end of the first paper we can find a
section “Eventual importance of the present investigation for
physics” (p. 79-81) and the confirmation that during the
proofreading Schouten received Eddington’s paper ([58],
accepted 19 February 1921). Thus, while Einstein and Weyl influenced Eddington, Schouten apparently did his
research without knowing of Eddington’s idea. Einstein, perhaps, got to know Schouten’s work only later through
the German translation of Eddington’s book where it is
mentioned ([60], p. 319), and
to which he wrote an addendum, or, more directly, through Schouten’s book on the Ricci
calculus, Die Grundlehren der Mathematischen
Wissenschaften in Einzeldarstellungen, in the same famous
yellow series of Springer Verlag [300
]. On the other hand,
Einstein’s papers following Eddington’s [77
, 74
] inspired Schouten to publish on a theory
with vector torsion that tried to remedy a problem Einstein had noted in his papers,
i.e., that no electromagnetic field
could be present in regions of vanishing electric current density.
According to Schouten
“[...] we see that the electromagnetic field
only depends on the curl of the electric current vector, so that
the difficulty arises that the electromagnetic field cannot exist
in a place with vanishing current density. In the following pages
will be shown that this difficulty disappears when the more general
supposition is made that the original deplacement is not
necessarily symmetrical.” ([300], p. 850)
Schouten criticised Einstein’s argument for using a
symmetric connection113
as unfounded (cf. Equation (15)). He then restricted
the generality of his approach; in modern parlance, he did allow
for vector torsion only:
“We will not consider the most general case, but the semi-symmetric case in which the alternating part of the parameters has the form:
The affine connection can then be
decomposed as follows:
“Einstein has said (in Meaning of Relativity) that ‘a theory of
relativity in which the gravitational field and the electromagnetic
field enter as an essential unity’ is desirable and recently has
proposed such a theory.” ([116], pp. 367-368)
and
“His geometry also is included in the one now proposed and it may be that the latter, because of its greater generality and adaptability will serve better as the basis for the mathematical formulation of the results of physical experiments.” ([116], p. 369)
The spreading of knowledge about properties of differential geometric objects like connection and curvature took time, however, even in Leningrad. Seven years after Schouten’s classification of connections, Fréedericksz of Leningrad - known better for his contributions to the physics of liquid crystals - put forward a classification of his own by using both the connection and the curvature tensor [138].