340 THE LATER JOURNEYconsequence of the advent of general relativity, as is seen from persistent reference
to that theory in all papers on connections which appeared in the following years,
by authors like Weyl, Levi-Civita, Schouten, Struik, and especially Elie Cartan,
who introduced torsion in 1922 [C4], and whose memoir 'Sur les Varietes a Con-
nexion Affine et la theorie de la Relativite Generalisee' [C5] is one of the papers
which led to the modern theory of fiber bundles [C3]. Thus Einstein's labors had
a major impact on mathematics.
The first book on connections, Schouten's Der Ricci-Kalkiil [S4], published in
1924, lists a large number of connections distinguished (see [S4], p. 75) by the
relative properties of FJ, and Fj,, the symmetry properties of Fj,, and the prop-
erties of Q^. It will come as a relief to the reader that for all unified-field theories
to be mentioned below, Eq. 17.33 does hold. This leads to considerable simplifi-
cations since then, and only then, product rules of the kind
(17.38)hold true. Important note: the orders of indices in Eqs. 17.23 and 17.32 are
matched in such a way that Eq. 17.38 is also true for nonsymmetric connections.
Let us consider the Weyl theory of 1918 [W2] as an example of this formalism.
This theory is based on Eq. 17.33, on a symmetric (also called affine) connection,
and on a symmetric fundamental tensor g^ However, Qw does not vanish.
Instead:(17.39)(which reduces to Q^ = 0 for 4>p = 0). </>p is a 4-vector. This equation is invar-
iant under
(17.40)
(17.41)(17.42)where X is an arbitrary function of x". Equations 17.40-17.42 are compatible
since Eq. 17.39 implies that
(17.43)where F*^ is the Riemannian expression given by the right-hand side of Eq.
17.21. Weyl's group is the product of the point transformation group and the
group of X transformations specified by Eqs. 17.40 and 17.41. The xx are
unchanged by X transformations, so that the thing ds2 = g^dx^dx^^-Xds2. If we
dare to think of the thing ds as a length, then length is regauged (in the same
sense the word is used for railroad tracks), whence the expression gauge transfor-
mations, which made its entry into physics in this unphysical way. The quantities
R^1 ^ and F^ defined by