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.

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.

This thesis studies the construction of noncompact Einstein manifolds of cohomogeneity one on some vector bundles. Cohomogeneity one vector bundle whose isotropy representation of the principal orbit G/K has two inequivalent irreducible summands has been studied in [Böh99][Win17]. However, the method applied does not cover all cases. This thesis provides an alternative construction with a weaker assumption of G/K admits at least one invariant Einstein metric. Some new Einstein metrics of Taub-NUT type are also constructed. This thesis also provides construction of cohomogeneity one Einstein metrics for cases where G/K is a Wallach space. Specifically, two continuous families of complete smooth Einstein metrics are constructed on vector bundles over CP2, HP2 and OP2 with respective principal orbits the Wallach spaces SU(3)/T2, Sp(3)/(Sp(1)Sp(1)Sp(1)) and F4/Spin(8). The first family is a 1-parameter family of Ricci-flat metrics. All the Ricci- flat metrics constructed have asymptotically conical limits given by the metric cone over a suitable multiple of the normal Einstein metric. All the Ricci-flat metrics constructed have generic holonomy except that the complete metric with G2 holonomy discovered in [BS89][GPP90] lies in the interior of the 1-parameter family on manifold in the first case. The second family is a 2-parameter family of Poincaré–Einstein metrics. / Thesis / Doctor of Philosophy (PhD)

In this thesis we study Spin(7)-manifolds, that is Riemannian 8-manifolds with torsion-free Spin(7)-structures, and Cayley submanifolds of such manifolds. We use a construction of compact Spin(7)-manifolds from Calabi–Yau 4-orbifolds with antiholomorphic involutions, due to Joyce, to find new examples of compact Spin(7)-manifolds. We search the class of well-formed quasismooth hypersurfaces in weighted projective spaces for suitable Calabi–Yau 4-orbifolds. We consider antiholomorphic involutions induced by the restriction of an involution of the ambient weighted projective space and we classify anti-holomorphic involutions of weighted projective spaces. We consider the moduli problem for Cayley submanifolds of Spin(7)-manifolds and show that there is a fine moduli space of unobstructed Cayley submanifolds. This result improves on the work of McLean in that we consider the global issues of how to patch together the local result of McLean. We also use the work of Kriegl and Michor on ‘convenient manifolds’ to show that this moduli space carries a universal family of Cayley submanifolds. Using the analysis necessary for the study of the moduli problem of Cayleys we find examples of compact Cayley submanifolds in any compact Spin(7)-manifold arising, using Joyce’s construction, from a suitable Calabi–Yau 4-orbifold with antiholomorphic involution. For the analysis to work, we need to show that a given Cayley submanifold is unobstructed. To show that particular examples of Cayley submanifolds are unobstructed, we relate the obstructions of complex surfaces in Calabi–Yau 4-folds as complex submanifolds to the obstructions as Cayley submanifolds.

This thesis is mainly concerned with the study of hypersymplectic structures in gauge theory. These structures arise via applications of the hypersymplectic quotient construction to the action of the gauge group on certain spaces of connections and Higgs fields. Motivated by Kobayashi-Hitchin correspondences in the case of hyperkähler moduli spaces, we first study the relationship between hypersymplectic, complex and paracomplex quotients in the spirit of Kirwan's work relating Kähler quotients to GIT quotients. We then study dimensional reductions of the ASD equations on $mathbb R^{2,2}$. We discuss a version of twistor theory for hypersymplectic manifolds, which we use to put the ASD equations into Lax form. Next, we study Schmid's equations from the viewpoint of hypersymplectic quotients and examine the local product structure of the moduli space. Then we turn towards the integrability aspects of this system. We deduce various properties of the spectral curve associated to a solution and provide explicit solutions with cyclic symmetry. Hitchin's harmonic map equations are the split signature analogue of the self-duality equations on a Riemann surface, in which case it is known that there is a smooth hyperkähler moduli space. In the case at hand, we cannot expect to obtain a globally well-behaved moduli space. However, we are able to construct a smooth open set of solutions with small Higgs field, on which we then analyse the hypersymplectic geometry. In particular, we exhibit the local product structures and the family of complex structures. This is done by interpreting the equations as describing certain geodesics on the moduli space of unitary connections. Using this picture we relate the degeneracy locus to geodesics with conjugate endpoints. Finally, we present a split signature version of the ADHM construction for so-called split signature instantons on $S^2 imes S^2$, which can be given an interpretation as a hypersymplectic quotient.

In dieser Arbeit studieren wir Lorentz-Mannigfaltigkeiten mit spezieller Holonomie, d.h. ihre Holonomiedarstellung wirkt schwach-irreduzibel aber nicht irreduzibel. Aufgrund der schwachen Irreduzibilität lässt die Darstellung einen ausgearteten Unterraum invariant und damit also auch eine lichtartige Linie. Geometrisch hat dies zur Folge, dass wir zwei parallele Unterbündel (die Linie und ihr orthogonales Komplement) des Tangentialbündels erhalten. Diese Arbeit nutzt diese und weitere Objekte um zu beweisen, dass kompakte Lorentzmannigfaltigkeiten mit Abelscher Holonomie geodätisch vollständig sind. Zudem werden Lorentzmannigfaltigkeiten mit spezieller Holonomie und nicht-negativer Ricci-Krümung auf den Blättern der Blätterung, induziert durch das orthogonale Komplement der parellelen Linie, und maximaler erster Bettizahl untersucht. Schließlich werden vollständige Ricci-flache Lorentzmannigfaltigkeiten mit vorgegebener voller Holonomie konstruiert. / In the present thesis we study dimensional Lorentzian manifolds with special holonomy, i.e. such that their holonomy representation acts indecomposably but non-irreducibly. Being indecomposable, their holonomy group leaves invariant a degenerate subspace and thus a light-like line. Geometrically, this means that, since being holonomy invariant, this line gives rise to parallel subbundles of the tangent bundle. The thesis uses these and other objects to prove that Lorentian manifolds with Abelian holonomy are geodesically complete. Moreover, we study Lorentzian manifolds with special holonomy and non-negative Ricci curvature on the leaves of the foliation induced by the orthogonal complement of the parallel light-like line whose first Betti number is maximal. Finally, we provide examples of geodesically complete and Ricci-flat Lorentzian manifolds with special holonomy and prescribed full holonomy group.