This thesis is about the structure of the boundary of the universe, i.e., points where the geometric structures of spacetime cannot be continued. In partic­ular, we study the structure of the b-boundary by B. Schmidt. It has been known for some time that the b-boundary construction has several drawbacks, perhaps the most severe being that it is often not Hausdorff separated from interior points in spacetime. In other words, the topology makes it impossible to distinguish which points in spacetime are near the singularity and which points are ‘far’ from it. The non-Hausdorffness of the b-completion is closely related to the concept of fibre degeneracy of the fibre in the frame bundle over a b-boundary point, the fibre being smaller than the whole structure group in a specific sense. Fibre degeneracy is to be expected for many realistic spacetimes, as was proved by C. J. S. Clarke. His proofs contain some errors however, and the purpose of paper I is to reestablish the results of Clarke, under somewhat different conditions. It is found that under some conditions on the Riemann curvature tensor, the boundary fibre must be totally degenerate (i.e., a single point). The conditions are essentially that the components of the Riemann tensor and its first derivative, expressed in a parallel frame along a curve ending at the singularity, diverge sufficiently fast. We also demonstrate the applicability of the conditions by verifying them for a number of well known spacetimes. In paper II we take a different view of the b-boundary and the b-length func­tional, and study the Riemannian geometry of the frame bundle. We calculate the curvature Rof the frame bundle, which allows us to draw two conclusions. Firstly, if some component of the curvature of spacetime diverges along a horizontal curve ending at a singularity, R must tend to — oo. Secondly, if the frame bundle is extendible through a totally degenerate boundary fibre, the spacetime must be a conformally flat Einstein space asymptotically at the corresponding b-boundary point. We also obtain some basic results on the isometries and the geodesics of the frame bundle, in relation to the corresponding structures on spacetime. The first part of paper III is concerned with imprisoned curves. In Lorentzian geometry, the situation is qualitatively different from Riemannian geometry in that there may be incomplete endless curves totally or partially imprisoned in a compact subset of spacetime. It was shown by B. Schmidt that a totally imprisoned curve must have a null geodesic cluster curve. We generalise this result to partially im­prisoned incomplete endless curves. We also show that the conditions for the fibre degeneracy theorem in paper I does not apply to imprisoned curves. The second part of paper III is concerned with the properties of the b-length functional. The b-length concept is important in general relativity because the presence of endless curves with finite b-length is usually taken as the definition of a singular spacetime. It is also closely related to the b-boundary definition. We study the structure of b-neighbourhoods, i.e., the set of points reachable from a fixed point in spacetime on (horizontal) curves with b-length less than some fixed number e > 0. This can then be used to understand how the geometry of spacetime is encoded in the frame bundle geometry, and as a tool when studying the structure of the b-boundary. We also give a result linking the b-length of a general curve in the frame bundle with the b-length of the corresponding horizontal curve. / digitalisering@umu

In dieser Dissertation werden Cartan-Ränder von CR-Mannigfaltigkeiten und ihren Fefferman-Räumen besprochen. Der Fefferman-Raum einer strikt pseudo-konvexen CR-Mannigfaltigkeit ist als das Bündel aller reellen Strahlen im kanonischen, komplexen Linienbündel definiert. Eine andere Definition nutzt die Cartan-Geometrie und führt zu einer starken Beziehung zwischen den Cartan-Geometrien der CR-Mannigfaltigkeit und des zugehörigen Fefferman-Raumes. Allerdings wird hier die Existenz einer gewissen Wurzel des antikanonischen, komplexen Linienbündels, dessen Existenz nur lokal gesichert ist, vorausgesetzt. Für Randkonstruktionen benötigen wir jedoch eine globale Konstruktion des Fefferman-Raumes. Dennoch können lokale Resultate zum Fefferman-Raum von einer Konstruktion zur anderen übertragen werden können, da konforme Überlagerungen von beiden vorliegen. Der Cartan-Rand einer Mannigfaltigkeit wird mithilfe der zugehörigen Cartan-Geometrie konstruiert, welche eine globale Basis und damit auch eine Riemannsche Metrik auf dem Cartan-Bündel definiert, welches per Cauchy-Vervollständigung abgeschlossen wird. Division durch die Strukturgruppe ergibt den Cartan-Rand der Mannigfaltigkeit. Der Cartan-Rand ist eine Verallgemeinerung des Cauchy-Randes, da beide im Riemannschen übereinstimmen. Allgemein ist der Cartan-Rand nicht unbedingt Hausdorffsch, was nicht wirklich überrascht, sind doch Rand-Phänomene "irgendwie singulär". Wir stellen fest, dass für CR-Mannigfaltigkeit und ihre Fefferman-Räume die Projektion des Cartan-Randes des Fefferman-Raumes den Cartan-Rand der CR-Mannigfaltigkeit enthält. Schließlich betrachten wir die Heisenberg-Gruppe, eines der grundlegenden Beispiele für CR-Mannigfaltigkeiten. Sie ist flach aber - anders als der homogene Raum - nicht kompakt. Wir finden, dass der Cartan-Rand der Heisenberg-Gruppe ein einzelner Punkt und der Cartan-Rand des zugehörigen Fefferman-Raumes eine nicht-ausgeartete Faser über diesem ist. / The aim of this thesis is to discuss the Cartan boundaries of CR manifolds and their Fefferman spaces. The Fefferman space of a strictly pseudo-convex CR manifold is defined as the bundle of all real rays in the canonical complex line bundle. Another way of defining the Fefferman space of a CR manifold uses the tools of Cartan geometry and leads to a strong relationship between the Cartan geometries of a CR manifold and the corresponding Fefferman space. However here the existence of a certain root of the anticanonical complex line bundle is requested which can solely be guarantied locally. As we are interested in boundaries we need a global construction of the Fefferman space. Still we find that local results on the Fefferman space can be transferred from one construction to the other since we have conformal coverings of both. The Cartan boundary of a manifold is constructed with the help of the corresponding Cartan geometry, which defines a global frame and hence a Riemannian metric on the Cartan bundle which can be completed by Cauchy completion. Division by the structure group gives the Cartan boundary of the manifold. The Cartan boundary is a generalization of the Cauchy boundary since both coincide in the Riemannian case. In general the Cartan boundary is not necessarily Hausdorff, which is not really surprising since boundary phenomena are somehow ``singular''''. For CR manifolds and their Fefferman spaces we especially prove that the projection of the Cartan boundary of the Fefferman space contains the Cartan boundary of the CR manifold. We finally discuss the Heisenberg group, one of the basic examples of CR manifolds. It is flat but - contrary to the homogeneous space - not compact. We find that the Cartan boundary of the Heisenberg group is a single point and the Cartan boundary of the corresponding Fefferman space is a non degenerate fibre over that point.