Title:
|
Completeness and its stability of manifolds with connection
|
Singularities ~n General Relativity are due to incompleteness of
space-time. This thesis examines the relationships between some of
the different notions of incompleteness of a manifold with
connection, together with the stability of geodesic completeness and
incompleteness.
Some p(parallelisation)-completions of Rand S1 are compared
with the b(bundle)-completions, and the applicability of the
p-boundary construction to General Relativity is discussed.
Relationsips are shown between g(geodesic), b(bundle) and
b.a.(bounded acceleration) incompleteness. Further acceleration
is
dependent notions of incompleteness are defined, and it shown that
A
they are all equivalent to the existing notions of completeness for a
Riemannian manifold.
The Whitney C
K
topologies provide a way of topologising the space
of metrics on a manifold, in order to consider stability of geodesic
completeness or incompleteness. It is shown how the spaces of
connections and sprays may also be topologised, and the continuity of
some important mappings is demonstrated. It turns out that for R
both geodesic completeness and incompleteness are stable with respect
to perturbation of the spray. Incompleteness of st ~s also stable,
but the complete sprays are closed. For S1~S1 and R~S1 it is
shown that null and time like geodesic completeness and incompleteness
all fail to be stable with respect to the space of Lorentz metrics.
Given a connection/spray on a pair of manifolds, one can construct a
connection/spray on their product, and this is used to show how
instability of completeness/incompleteness may arise if the product
~s compact. It is also shown how to induce sprays on the retraction
of a manifold.
|