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In the Λ-system, non-adiabatic holonomic quantum phases are used to construct holonomic quantum gates. An interesting approach would be to implement open path holonomies in the Λ-system. By dividing the loop into two curve segments with a unitary transformation between them, universality can be reached. In doing so the exibility of the scheme has been increased by the fact that one single full pulse is now enough for universality, and we have achieved a clearer proof of the geometric property of the Λ system. / I ett Λ-system så används icke-adiabatiska holonoma kvantfaser för att bygga holonoma kvantgrindar. I detta arbete undersöker vi om holonomier för öppna kurvor kan implementeras i Λsystemet. Genom att dela upp en loop i Λ-systemet i två sekvenser med en unitär transformation emellan så kan vi konstruera en universell holonom kvantgrind. Med detta så har vi ökat exibiliteten för systemet genom att vi nu bara behöver ta en loop för att nå universalitet, och vi har även erhållit en klarare bild över den geometriska egenskapen hos Λ-systemet.

This thesis looks into the holonomy algebras of Tractor/Cartan connections for both projective and conformal structures. Using a splitting formula and a cone construction in the Einstein case, it classifies all reductive, non-irreducible holonomy groups for conformal structures (thus fully solving the question in the definite signature case). The thesis then analyses the geometric consequences of of holonomy reduction for the projective Tractor connection. A general, Ricci-flat, cone construction pertains in the projective case, and this thesis fully classifies the irreducibly acting holonomy algebras by analysing which holonomy families admit a torsion-free Ricci-flat affine connection, and constructing cones with these properties.

In Berger's classification of Riemannian holonomy groups there are several infinite families and two exceptional cases: the groups Spin(7) and G_2. This thesis is mainly concerned with 7-dimensional manifolds with holonomy G_2. A metric with holonomy contained in G_2 can be defined in terms of a torsion-free G_2-structure, and a G_2-manifold is a 7-dimensional manifold equipped with such a structure. There are two known constructions of compact manifolds with holonomy exactly G_2. Joyce found examples by resolving singularities of quotients of flat tori. Later Kovalev found different examples by gluing pairs of exponentially asymptotically cylindrical (EAC) G_2-manifolds (not necessarily with holonomy exactly G_2) whose cylinders match. The result of this gluing construction can be regarded as a generalised connected sum of the EAC components, and has a long approximately cylindrical neck region. We consider the deformation theory of EAC G_2-manifolds and show, generalising from the compact case, that there is a smooth moduli space of torsion-free EACG_2-structures. As an application we study the deformations of the gluing construction for compact G_2-manifolds, and find that the glued torsion-free G_2-structures form an open subset of the moduli space on the compact connected sum. For a fixed pair of matching EAC G_2-manifolds the gluing construction provides a path of torsion-free G_2-structures on the connected sum with increasing neck length. Intuitively this defines a boundary point for the moduli space on the connected sum, representing a way to 'pull apart' the compact G_2-manifold into a pair of EAC components. We use the deformation theory to make this more precise. We then consider the problem whether compact G_2-manifolds constructed by Joyce's method can be deformed to the result of a gluing construction. By proving a result for resolving singularities of EAC G_2-manifolds we show that some of Joyce's examples can be pulled apart in the above sense. Some of the EAC G_2-manifolds that arise this way satisfy a necessary and sufficient topological condition for having holonomy exactly G_2. We prove also deformation results for EAC Spin(7)-manifolds, i.e. dimension 8 manifolds with holonomy contained in Spin(7). On such manifolds there is a smooth moduli space of torsion-free EAC Spin(7)-structures. Generalising a result of Wang for compact manifolds we show that for EAC G_2-manifolds and Spin(7)-manifolds the special holonomy metrics form an open subset of the set of Ricci-flat metrics.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 80-85). Available also in a digital version from Dissertation Abstracts.

In the present thesis we construct a new class of holonomy algebras in pseudo-Riemannian geometry. Starting from a smooth connected manifold M, we consider its (1;1)-tensor fields acting on the tangent spaces. We then prove that there exists a class of pseudo- Riemannian metrics g on M such that the (1;1)-tensor fields are g-self adjoint and their centralisers in the Lie algebra so(g) are holonomy algebras for the Levi-Civita connection of g. Our construction is elaborated with the aid of Manakov operators and holds for any signature of the metric g.

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.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 41-42).

Cayley submanifolds are naturally arising volume minimising submanifolds of $Spin(7)$- manifolds. In the special case that the ambient manifold is a four-dimensional Calabi--Yau manifold, a Cayley submanifold might be a complex surface, a special Lagrangian submanifold or neither. In this thesis, we study the deformation theory of Cayley submanifolds from two different perspectives.