Let
be a small increment of coordinates
in the physical spacetime. This is mapped into a coordinate increment
via a
transformation
[13, 19, 20, 21]
whose matrix elements 
are multivalued
tetrads.
The transformation can be chosen such that the length
of is measured by
the Minkowski metric 
, so that
the metric 
in the spacetime 
is given by
Parallel transport in q-spacetime is performed
with an affine connection
where 
are reciprocal multivalued tetrads.
Its antisymmetric part is
the torsion tensor [22]
and its covariant curl the curvature tensor
Note that if we were to live in x-spacetime, we would
register 
as an object of anholonomity. In the coordinate system
q, however,
is observable as a torsion.
Recall the way in which the affine
connection 
serves to define a covariant derivative of vector fields 
:
If we lower the last index of the affine connection
by a contraction, 
,
there exists the decomposition
where
is the Riemann connection, symmetric in 
,
and
is an antisymmetric tensor in 
,
called the
contortion tensor
[22],
formed from the
torsion tensor
by lowering the last index 
.
With the help of the Riemann connection,
we may define another covariant derivative
