UNIFIED FIELD THEORY 339are necessary and sufficient conditions for a Riemann space to be everywhere flat
(pseudo-Euclidean).Now comes the generalization. Forget Eqs. 17.20 and 17.21 and the second
group of statements. Retain the first group. This leads not to one new geometry
but to a new class of geometries, or, as one also says, a new class of connections.
Let us note a few general features.
a) There is no longer a metric. There are only connections. Equation 17.25,
now imposed rather than derived from the transformation properties of gm, is
sufficient to establish that AK, and R^ are tensors. Thus we still have a tensor
calculus.
b) A general connection is defined by the 128 quantities F^ and f^,. If these are
given in one frame, then they are given in all frames provided we add the rule
that even if Fj, =£ fj, then fj, still transforms according to Eq. 17.25.
c) In the first group, we retained one reference to g^, in Eq. 17.24. The reason
for doing so is that in these generalizations one often introduces a fundamental
tensor g^, but not via the invariant line element. Hence this fundamental tensor
no longer deserves the name metrical tensor. A fundamental tensor g^ is never-
theless of importance for associating with any contravariant vector A* a covariant
vector Af by the rule Af = g^A" and likewise for higher-rank tensors. The g^
does not in general obey Eq. 17.31, nor need it be symmetric (if it is not, then, of
course, g^ A' =£ g^A').
d) Since Eq. 17.28 does not necessarily hold, the order of the ju,j> indices in Eq.
17.23 is important and should be maintained. For unsymmetric Fj,, the replace-
ment of Fj, in Eq. 17.23 by F^ also defines a connection, but a different one.
e) Even if Fj, is symmetric in /i and v, it does not follow that R^ is symmetric:
we may use Eq. 17.27 but not Eq. 17.30. This remark is of importance for the
Weyl and Eddington theories discussed in what follows.
f) For any symmetric connection, the Bianchi identities
(17.37)die vtiiiu.
g) R^1 ^, is still a tensor, but R^, = 0 does not in general imply flatness; see the
theory of distant parallelism discussed in the next section.
h) We can always contract the curvature tensor to the Ricci tensor, but, in the
absence of a fundamental tensor, we cannot obtain the curvature scalar from the
Ricci tensor.
i) the contracted Bianchi identities Eq. 17.35 are in general not valid, nor even
defined. These last two observations already make clear to the physicist that the
use of general connections means asking for trouble.The theory of connections took off in 1916, starting with a paper by the math-
ematician Gerhard Hessenberg [H2]. These new developments were entirely a