A world manifold is a four-dimensional orientable real smooth manifold. It is assumed to be a Hausdorff and second countable topological space. Consequently, it is a locally compact space which is a union of a countable number of compact subsets, a separable space, a paracompact and completely regular space. Being paracompact, a world manifold admits a partition of unity by smooth functions. It should be emphasized that paracompactness is an essential characteristic of a world manifold. It is necessary and sufficient in order that a world manifold admits a Riemannian metric and necessary for the existence of a pseudo-Riemannian metric. A world manifold is assumed to be connected and, consequently, it is arcwise connected.
The tangent bundle
T
X
of a world manifold
X
and the associated principal frame bundle
F
X
of linear tangent frames in
T
X
possess a general linear group structure group
G
L
+
(
4
,
R
)
. A world manifold
X
is said to be parallelizable if the tangent bundle
T
X
and, accordingly, the frame bundle
F
X
are trivial, i.e., there exists a global section (a frame field) of
F
X
. It is essential that the tangent and associated bundles over a world manifold admit a bundle atlas of finite number of trivialization charts.
Tangent and frame bundles over a world manifold are natural bundles characterized by general covariant transformations. These transformations are gauge symmetries of gravitation theory on a world manifold.
By virtue of the well-known theorem on structure group reduction, a structure group
G
L
+
(
4
,
R
)
of a frame bundle
F
X
over a world manifold
X
is always reducible to its maximal compact subgroup
S
O
(
4
)
. The corresponding global section of the quotient bundle
F
X
/
S
O
(
4
)
is a Riemannian metric
g
R
on
X
. Thus, a world manifold always admits a Riemannian metric which makes
X
a metric topological space.
In accordance with the geometric Equivalence Principle, a world manifold possesses a Lorentzian structure, i.e., a structure group of a frame bundle
F
X
must be reduced to a Lorentz group
S
O
(
1
,
3
)
. The corresponding global section of the quotient bundle
F
X
/
S
O
(
1
,
3
)
is a pseudo-Riemannian metric
g
of signature
(
+
,
−
−
−
)
on
X
. It is treated as a gravitational field in General Relativity and as a classical Higgs field in gauge gravitation theory.
A Lorentzian structure need not exist. Therefore a world manifold is assumed to satisfy a certain topological condition. It is either a noncompact topological space or a compact space with a zero Euler characteristic. Usually, one also requires that a world manifold admits a spinor structure in order to describe Dirac fermion fields in gravitation theory. There is the additional topological obstruction to the existence of this structure. In particular, a noncompact world manifold must be parallelizable.
If a structure group of a frame bundle
F
X
is reducible to a Lorentz group, the latter is always reducible to its maximal compact subgroup
S
O
(
3
)
. Thus, there is the commutative diagram
G
L
(
4
,
R
)
→
S
O
(
4
)
↓
↓
S
O
(
1
,
3
)
→
S
O
(
3
)
of the reduction of structure groups of a frame bundle
F
X
in gravitation theory. This reduction diagram results in the following.
(i) In gravitation theory on a world manifold
X
, one can always choose an atlas of a frame bundle
F
X
(characterized by local frame fields
{
h
λ
}
) with
S
O
(
3
)
-valued transition functions. These transition functions preserve a time-like component
h
0
=
h
0
μ
∂
μ
of local frame fields which, therefore, is globally defined. It is a nowhere vanishing vector field on
X
. Accordingly, the dual time-like covector field
h
0
=
h
λ
0
d
x
λ
also is globally defined, and it yields a spatial distribution
F
⊂
T
X
on
X
such that
h
0
⌋
F
=
0
. Then the tangent bundle
T
X
of a world manifold
X
admits a space-time decomposition
T
X
=
F
⊕
T
0
X
, where
T
0
X
is a one-dimensional fibre bundle spanned by a time-like vector field
h
0
. This decomposition, is called the
g
-compatible space-time structure. It makes a world manifold the space-time.
(ii) Given the above mentioned diagram of reduction of structure groups, let
g
and
g
R
be the corresponding pseudo-Riemannian and Riemannian metrics on
X
. They form a triple
(
g
,
g
R
,
h
0
)
obeying the relation
g
=
2
h
0
⊗
h
0
−
g
R
.
Conversely, let a world manifold
X
admit a nowhere vanishing one-form
σ
(or, equivalently, a nowhere vanishing vector field). Then any Riemannian metric
g
R
on
X
yields the pseudo-Riemannian metric
g
=
2
g
R
(
σ
,
σ
)
σ
⊗
σ
−
g
R
.
It follows that a world manifold
X
admits a pseudo-Riemannian metric if and only if there exists a nowhere vanishing vector (or covector) field on
X
.
Let us note that a
g
-compatible Riemannian metric
g
R
in a triple
(
g
,
g
R
,
h
0
)
defines a
g
-compatible distance function on a world manifold
X
. Such a function brings
X
into a metric space whose locally Euclidean topology is equivalent to a manifold topology on
X
. Given a gravitational field
g
, the
g
-compatible Riemannian metrics and the corresponding distance functions are different for different spatial distributions
F
and
F
′
. It follows that physical observers associated with these different spatial distributions perceive a world manifold
X
as different Riemannian spaces. The well-known relativistic changes of sizes of moving bodies exemplify this phenomenon.
However, one attempts to derive a world topology directly from a space-time structure (a path topology, an Alexandrov topology). If a space-time satisfies the strong causality condition, such topologies coincide with a familiar manifold topology of a world manifold. In a general case, they however are rather extraordinary.
A space-time structure is called integrable if a spatial distribution
F
is involutive. In this case, its integral manifolds constitute a spatial foliation of a world manifold whose leaves are spatial three-dimensional subspaces. A spatial foliation is called causal if no curve transversal to its leaves intersects each leave more than once. This condition is equivalent to the stable causality of Stephen Hawking. A space-time foliation is causal if and only if it is a foliation of level surfaces of some smooth real function on
X
whose differential nowhere vanishes. Such a foliation is a fibred manifold
X
→
R
. However, this is not the case of a compact world manifold which can not be a fibred manifold over
R
.
The stable causality does not provide the simplest causal structure. If a fibred manifold
X
→
R
is a fibre bundle, it is trivial, i.e., a world manifold
X
is a globally hyperbolic manifold
X
=
R
×
M
. Since any oriented three-dimensional manifold is parallelizable, a globally hyperbolic world manifold is parallelizable.
