  A global vector field $v$ on a "spacetime" differentiable manifold
$\mathrm{V}$, of dimension $N+1$, defines a congruence of world lines: the
maximal integral curves of $v$, or orbits. The associated global space
$\mathrm{N}\_v$ is the set of these orbits. A "$v$-adapted" chart on
$\mathrm{V}$ is one for which the $\mathbb{R}^N$ vector ${\bf x}\equiv (x^j)\
(j=1,...,N)$ of the "spatial" coordinates remains constant on any orbit $l$. We
consider non-vanishing vector fields $v$ that have non-periodic orbits, each of
which is a closed set. We prove transversality theorems relevant to such vector
fields. Due to these results, it can be considered plausible that, for such a
vector field, there exists in the neighborhood of any point $X\in \mathrm{V}$ a
chart $\chi $ that is $v$-adapted and "nice", i.e., such that the mapping
$\bar{\chi }: l\mapsto {\bf x}$ is injective --- unless $v$ has some
"pathological" character. This leads us to define a notion of "normal" vector
field. For any such vector field, the mappings $\bar{\chi }$ build an atlas of
charts, thus providing $\mathrm{N}\_v$ with a canonical structure of
differentiable manifold (when the topology defined on $\mathrm{N}\_v$ is
Hausdorff, for which we give a sufficient condition met in important physical
situations). Previously, a local space manifold $\mathrm{M}\_\mathrm{F}$ had
been associated with any "reference frame" $\mathrm{F}$, defined as an
equivalence class of charts. We show that, if $\mathrm{F}$ is made of nice
$v$-adapted charts, $\mathrm{M}\_\mathrm{F}$ is naturally identified with an
open subset of the global space manifold $\mathrm{N}\_v$.
