As indicated earlier, Riemann's geometry played a fundamental
role in the mathematical formulation of relativity theory. Basically,
Riemannian geometry is concerned with the properties of a coordinate
space (x
, . . . , x
) in which there is a nondegenerate quadratic differential form
called the element of arc (see 311).
This geometry reduces to Euclidean geometry if the element of
arc takes the special form ds
= (dx
) + . . .
+ (dx
). The two-dimensional
case had been considered before Riemann by C.F. Gauss as the
intrinsic geometry on a surface in ordinary Euclidean space.
In pure mathematics, the differential form ds
is generally supposed to be positive definite, an assumption
that is essential to many of the important consequences. In
applications to general relativity, however, in which the Riemannian
space is the physical universe, ds
is supposed to be hyperbolic; i.e., reducible to a sum
of squares minus the square of a linear differential form.
in equation (311) allows
the definition of the arc length of a curve x
= x
(t) as an integral (see 312).
Geodesics are curves that are shortest between
any two of their points, sufficiently near to each other. An
n-dimensional domain has a volume given by the integral
{sqroot |g|}
dx
. . . dx
, in which g is the determinant of g{sub ik}.
The coordinates x
are local coordinates in the sense that they can be subjected
to a general differentiable transformation x'
= f{sup i}(x
, . . . , x
), in which i = 1, . . . , n, with nonvanishing
Jacobian determinant (see above Differential
geometry: Manifolds and tensor bundles). For ds
to be invariant
under such changes of local coordinates, the quantities g{sub
ik} should transform according to certain equations (see
313) and are, therefore,
the components of a symmetric covariant tensor field of order
two. The elements g{sup jl} of the inverse matrix
of the matrix (g{sub ik}), which are related to
the elements g{sub ik} by equations (see 314),
are the components of a symmetric contravariant tensor field
of order two. The fundamental tensors g{sub ik}
and g{sup jl} can be used to derive new tensors
from those given. Thus, if A{sup ij}
is a given tensor field then A
{sub jk} (see 315)
is a new tensor field. The two fields are said to be associated
to each other, and the process is described as that of raising
or lowering indices. The length of a vector A
is defined (see 316). The
angle 
between
two vectors A
, B
, of lengths different from zero, is defined by a formula (see
317).
From the fundamental (or metric) tensor g{sub ik},
Elwin Bruno
Christoffel, a German mathematician, constructed
(in 1869) a quantity expressed in terms of partial derivatives
(see 318). These are not
the components of a tensor, but can be used to define the covariant
derivative of a tensor field, giving a tensor field of one more
covariant order. In the cases of contravariant and covariant
vector fields and that of a mixed tensor field, their covariant
derivatives are defined symbolically (see 319).
Similar expressions are valid for the covariant derivatives
of more general tensor fields. The covariant derivative of g{sub
ik} vanishes identically. Conversely, 
can be
characterized as the set of quantities that are symmetric in
i, k and for which the covariant derivative of the fundamental
tensor is zero. In geometrical language,
defines
a parallel displacement of vectors along curves: the vector
field A
is parallel along a curve x
= x
(t) if a condition holds (see 320).
This is the Levi-Civita parallelism described previously.
It is an affine connection in the general terminology of connections,
but is related in a special way to the Riemannian metric. Under
the parallelism of Levi-Civita, the scalar product of two vectors
remains unchanged. A geodesic is either a curve of zero length
or one along which the unit tangent vector is parallel. The
former possibility does not appear in the case of a positive
definite metric.
The notion of curvature arises in studying the parallel displacement
of a vector around a closed curve. When the closed curve is
an infinitesimal parallelogram, the new position of the vector
can be obtained from the initial position by an infinitesimal
rotation. Analytically, this association is described by the
Riemann-Christoffel curvature tensor (see
321). The components of the curvature tensor have symmetry
properties (see 322). As
a result, there are n
(n
- 1)/12 independent components; e.g., six for a three-space,
20 for a four-space, and one for a two-space. In the latter
case the quotient K = -R{sub 1212}/g is
a scalar, called the
Gaussian curvature. If
a vector is given a parallel displacement around a closed curve
on a surface, it returns with its direction changed by an amount
equal to Kd
, the Gaussian curvature of the surface integrated over
the surface area bounded by the closed curve. As a consequence,
the excess (over 
)
of the angle sum in any geodesic triangle is
Kd, a theorem
proposed by Gauss.
In the case of three or more dimensions, the curvature properties
can no longer be expressed by a single magnitude, but require
for their description the whole curvature tensor. The concept
of Gaussian curvature is replaced by that of
Riemannian curvature,
or sectional curvature. This is, at any point p and any
plane element (two-dimensional) through p, the Gaussian
curvature of the geodesic surface through p and tangent
to the plane element. If the latter is spanned by the vectors
A, B{sup
j}, the Riemannian curvature is given by a formula (see
323). In general, K will
depend both on the point and on the plane element through it.
In other words, with regard to curvature, the space may be nonhomogeneous
as well as anisotropic. For n 
3, however, Schur's theorem (named after Friedrich Heinrich
Schur, a 19th-20th-century Polish-German mathematician)
says that if K is everywhere isotropic (i.e., independent
of the choice of the plane element at every point), then it
is constant throughout. Riemannian spaces, for which the Riemannian
curvature is constant for all points and all plane elements
through them, are said to be of constant curvature. They include
the Euclidean and non-Euclidean spaces. They are characterized
by analytic conditions (see 324)
with constant K. The curvature tensor gives, by contraction,
the symmetric tensor (see 325)
called the
Ricci tensor (named for Gregorio Ricci-Curbastro).
A further contraction gives the scalar invariant (see 326).
= 
+
+ 
-
(
, . . . ,
are linear differential forms). The components g{sub
ik} of the fundamental tensor define the gravitational
potential. Thus, light rays are null curves defined by the differential
equation ds
= 0, whereas the trajectories of particles, without the action
of external forces, are the geodesics. In this way, the geometry
of the space is not a priori given, but is determined by matter.
An essential restriction on the metric is given by
Einstein's field equation
R{sub ik} = -H(T{sub ik}
- 1/2 g{sub ik}T), in which T{sub
ik} is the energy-momentum tensor, T is the contraction
of T{sub ik} relative to the fundamental tensor,
and H is a constant. Many attempts have been made, some
by Einstein himself in his late years, to develop a
unified field theory
that would include both the gravitational and the electromagnetic
fields. This leads to a generalization of Riemannian geometry
to more general geometric objects, such as the Weyl geometry,
the Kaluza five-dimensional metric, and the projective relativity
of Oswald Veblen. Results are inconclusive.
