It is assumed to be a Hausdorff and second countable topological space.
Being paracompact, a world manifold admits a partition of unity by smooth functions.
Paracompactness is an essential characteristic of a world manifold.
are trivial, i.e., there exists a global section (a frame field) of
It is essential that the tangent and associated bundles over a world manifold admit a bundle atlas of finite number of trivialization charts.
These transformations are gauge symmetries of gravitation theory on a world manifold.
Thus, a world manifold always admits a Riemannian metric which makes
In accordance with the geometric Equivalence Principle, a world manifold possesses a Lorentzian structure, i.e., a structure group of a frame bundle
It is treated as a gravitational field in General Relativity and as a classical Higgs field in gauge gravitation theory.
Therefore, a world manifold is assumed to satisfy a certain topological condition.
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.
Thus, there is the commutative diagram of the reduction of structure groups of a frame bundle
These transition functions preserve a time-like component
of local frame fields which, therefore, is globally defined.
Accordingly, the dual time-like covector field
also is globally defined, and it yields a spatial distribution
is a one-dimensional fibre bundle spanned by a time-like vector field
(ii) Given the above-mentioned diagram of reduction of structure groups, let
obeying the relation Conversely, let a world manifold
yields the pseudo-Riemannian metric It follows that a world manifold
admits a pseudo-Riemannian metric if and only if there exists a nowhere vanishing vector (or covector) field on
-compatible Riemannian metrics and the corresponding distance functions are different for different spatial distributions
It follows that physical observers associated with these different spatial distributions perceive a world manifold
The well-known relativistic changes of sizes of moving bodies exemplify this phenomenon.
If a space-time satisfies the strong causality condition, such topologies coincide with a familiar manifold topology of a world manifold.
A space-time structure is called integrable if a spatial distribution
In this case, its integral manifolds constitute a spatial foliation of a world manifold whose leaves are spatial three-dimensional subspaces.
This condition is equivalent to the stable causality of Stephen Hawking.
is a fibre bundle, it is trivial, i.e., a world manifold