One illuminating example of the interplay between mathematics and physics is the relation between symplectic geometry and mechanics. A symplectic manifold is characterised by a closed, non-degenerate form of degree two. In modern physics higher degree differential forms play an important role too. In this thesis, we study geometries that are either completely or partly specified in terms of a differential form.
In the first part of the thesis, three-forms play the main role. When the form is closed, we call the geometry strong. One particular class of examples comes from torsion geometry, where the three-form appears as the torsion of a metric connection. Our first main result is a classification of invariant strong Kähler with torsion structures on four-dimensional solvable Lie groups.
We then pass on to study strong geometries in general. When these come with a Lie group action which preserves the strong structure, we introduce a notion of moment map. While the basic ideas come from the theory of symplectic moment maps, the adaption to strong geometry with symmetry group requires several fundamentally new approaches. We show existence of our multi-moment maps in many circumstances, including mild topological assumptions on the underlying manifold. Such maps are also shown to exist for all groups whose second and third Lie algebra Betti numbers are zero. We show that these form a special class of solvable Lie groups and provide a structural characterisation. We give many examples of multi-moment maps for different geometries, including strong hyperKähler manifolds with torsion and strict nearly Kähler six-manifolds.
By generalising the arguments, we obtain a notion of multi-moment map for geometries with closed forms of higher degree. As in the three-form case, these maps often exist, for instance, under mild topological assumptions on the underlying manifold, or if the Lie group of symmetries has a vanishing pair of Lie algebra Betti numbers.
One intriguing application of multi-moment maps addresses the classification of Riemannian manifolds with exceptional holonomy and an isometric action of a torus. We explore the cases when the multi-moment map is a scalar function. Via a reduction procedure, the study of these exceptional holonomy spaces is related to tri-symplectic geometry in dimension four.
In the last part of the thesis, we introduce a Calabi-Yau problem for hyper-Kähler manifolds with torsion, and we take the first steps towards a solution via the continuity method.