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1's) representing the points of the primary space fails to be complete; but it can be completed to give a 4- dimensional family, effecting a unique embedding of the original 3-fold in a 4-fold with conformal structure, of which the conformal curvature is selfdual, in such a way that the induced conformal structure is the original one and such that the conformal torsion is related to the second conformal fundamental form of the hypersurface in a canonical linear fashion. In any case, the small deformations of the complex structure of the space of null geodesies correspond precisely to the small deformations of the conformal connexion. It is shown that a space of torsion-free null geodesies admits a holomorphic contact structure, and that conversely, for n ≥ 4, the admission of a contact structure forces the conformal torsion to vanish; for n=3, the contact form constructs automatically a unique metric on the ambient 4-fold in the previously constructed self-dual conformal class which solves Einstein's equations with cosmological constant 1 and blov/s up on the 3-fold, which is a general umbilic hypersurface. These results are in turn used to show that a real-analytic 3-fold with real-analytic positive definite conformal structure and a real-analytic symmetric form of conformal weight 1 can be embedded (in a locally unique fashion) in a real-analytic 4-fold with positive-definite conformal structure for which the conformal curvature is self-dual in such a way as to realize the given structures as the first and second conformal fundamental forms of the hypersurface; and it is shown that a real analytic 3-fold with positivedefinite conformal bounds a locally unique positive-definite solution of Einstein's equations with cosmological constant -1 as its umbilic conformal infinity. By contrast, these results fail when "real-analytic" is replaced by "smooth".

The holonomy group arising from a linear connection and differential homotopy is a classical subject in geometry. The notion was generalized first by Y. Muto ({10}) by considering horizontal subspaces in a fibred space which by construction is a differential manifold over a base space with another manifold as the fibre. He called this generalized group the restricted holonomy group Hl('o)((')M). Unlike the case of frame bundles the horizontal subspaces in a fibred space do not in general obey the right invariant rule. Hence it is not hard to imagine that Hl('o)((')M) is larger than linear holonomy groups. It may not even form a Lie group and for years the structure of this group was left unknown simply because the number of elements concerned is too large to handle. / One of the intentions here is to clarify and determine the structure of Hl('o)((')M) by setting certain conditions. Then by use of Palais' theorem about transformation groups, Nijenhuis' method for dealing with linear holonomy groups, and the standard technique of computing line integrals, the structure of Hl('o)((')M) is determined in Chapter One under certain conditions. Some properties concerning the isometric immersion from one fibred Riemannian space into another are also discussed in Chapter Two. / As far as I know, the work in this thesis is original, except where the text indicates the contrary: In particular, Chapter One is purely expository.

Bibliography: leaves 103-105. / xii, 105 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1997

The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudomodular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra. The quantum superalgebra Uq(osp(1|2n)) over C is considered with q a primitive Nth root of unity for all integers N &gt = 3. For such a q, a certain left ideal I of U_q(osp(1|2n)) is also a two-sided Hopf ideal, and the quotient algebra U^(N)_q(osp(1|2n)) = U_q(osp(1|2n))/I is a Z_2-graded ribbon Hopf algebra. For all n and all N &gt = 3, a finite collection of finite dimensional representations of U^(N)_q(osp(1|2n)) is defined. Each such representation of U^(N)_q(osp(1|2n)) is labelled by an integral dominant weight belonging to the truncated dominant Weyl chamber. Properties of these representations are considered: the quantum superdimension of each representation is calculated, each representation is shown to be self-dual, and more importantly, the decomposition of the tensor product of an arbitrary number of such representations is obtained for even N. It is proved that the quotient algebra U(N)^q_(osp(1|2n)), together with the set of finite dimensional representations discussed above, form a pseudo-modular Hopf algebra when N &gt = 6 is twice an odd number. Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-manifolds. This invariant is shown to be different to the topological invariants of 3-manifolds arising from quantum so(2n+1) at roots of unity.