It is possible to map the 
-space locally into a q-space
via an infinitesimal transformation
with coefficient functions
which are
not integrable in the sense of Eq. (22), i.e.,
Such a mapping will be called nonholonomic.
There exists no
single-valued function 
for which 
.
Nevertheless, we shall write (25) in analogy to
(22)
as
since this equation involves only the differential 
.
This violation of
mathematical conventions will not cause any
problems.
From Eq. (25) we see that
the image space of a
nonholonomic mapping
carries torsion. The
connection
has a nonzero antisymmetric part, called the torsion tensor [17]:
In contrast to 
,
the antisymmetric part 
is a proper tensor under holonomic coordinate transformations.
The contracted tensor
transforms like a vector, whereas the contracted connection
does not.
Even though 
is not a tensor,
we shall freely lower and raise its indices using contractions with
the metric or the inverse metric, respectively: 
, 
, 
. The same thing will
be done with
.
In the presence of torsion, the connection
is no longer equal to the Christoffel symbol.
In fact,
by rewriting 
trivially as
and using
,
we find the decomposition
where the combination of torsion tensors
is called the contortion tensor. It is antisymmetric in the last two indices so that
In Einstein's theory of gravitation, torsion is assumed to be absent,
i.e.,
the integrability condition for
is not violated as in (26).
The main effect of matter in Einstein's theory
of gravitation
manifests itself in the violation of the
integrability condition for the derivative of
the coordinate transformation , namely,
A transformation for which itself is integrable, while the
first derivatives 
are not,
carries a flat-space region into a purely curved one.
The quantity which
records the nonintegrability is the Cartan curvature tensor
Working out the derivatives using (19) we see that
can be written as a covariant curl
of the connection,
In the last term we have used a matrix notation for the connection.
The tensor components
are viewed as matrix elements
, so that we can use the matrix
commutator
Einstein's original theory of gravity assumes the absence of torsion. The space properties are completely specified by the Riemann curvature tensor formed from the Riemann connection (the Christoffel symbol)
The relation between the two curvature tensors is
In the last term, the 
's are viewed as matrices
. The symbols 
denote the
covariant derivatives
formed with the Christoffel symbol.
Covariant derivatives act like ordinary derivatives
if they are applied to a scalar field.
When applied to a vector field,
they act as follows:
The effect upon a tensor
field is the generalization of this;
every index receives a corresponding
additive 
contribution.
In the presence of torsion, there exists another covariant derivative
formed with the affine connection
rather than the Christoffel symbol which acts
upon a vector field as
The two derivatives (38) and (39) are equally covariant under holonomic coordinate transformations. Thus, in conventional differential geometry it is not clear which of them should play a more fundamental role in physics. They do differ, however, in their transformation behavior under nonholonomic transformations, and there (39) is definitely the preferred object for reasons of simplicity.
From either of the two curvature tensors, and
, one
can form the once-contracted tensors of rank 2, the Ricci tensor
and the curvature scalar
The celebrated Einstein equation for the gravitational field postulates that the tensor
the so-called Einstein tensor, is proportional to
symmetric energy-mo-mentum tensor of all matter fields.
This postulate was made only for spaces with no torsion, in which case
and 
are
both symmetric. As mentioned in the Introduction, it is not yet
clear how Einstein's
field equations should be generalized in the presence of torsion
since the experimental consequences are
as yet too small to be
observed.
In this paper, we are not concerned with
the generation of curvature and torsion but only with
their consequences upon the motion of point particles.
