This paper analyses the geometry of the Lie algebra of SO(6) by making use of its homomorphism with SU(4). We study the vector space of 4×4 traceless, Hermitian matrices from four different viewpoints and examine the connections between them. We review the strata of this space under group transformations using established techniques for su(N) algebras. We focus on orbits of special types of vectors and their interpretation as rotations of SO(6) spinors.

We recap the theory of diffeomorphisms and covariances acting on a space and their induced action on fields, and how this leads to their decomposition into Fourier modes and harmonics. We show that a careful consideration of this theory reveals some unexpected results. For a start, harmonics are distinguished by their ‘quantum numbers’, but this analysis is valid for classical fields, so that classical fields can be written as a linear sum of modes with ‘quantum numbers’. Furthermore, certain models of compactification, as described in previous works by the author, are found to exploit a loophole in a key ‘no-go’ theorem, allowing the Poincaré group and an internal O(N ) symmetry group to be embedded in a larger symmetry group. This embedding is achieved without the need for the fermionic generators of supersymmetry and it is manifest in the ‘decompactification limit’. Away from this limit – for a realistic universe – the homogeneous part of this larger symmetry group is non-linearly realised. The translation part of the original symmetry group is realised as additional rotational transformations on the compact factor space.

We present a geometric field theory in which the action and field equation are constructed from a vector field and its covariant derivative and have full general covariance in a higher-dimensional spacetime. The field equation is the simplest possible generalisation of the Poisson equation for gravity consistent with general covariance and the equivalence principle. It contains the Ricci tensor and metric acting as operators on the vector field. If the symmetrised covariant derivative is diagonalisable across a neighbourhood under real changes of coordinate basis, spacetime coincides with a product manifold. The dimensionalities of the factor spaces are determined by its eigenvalues and hence by its algebraic invariants. Tensors decompose into multiplets which have both Lorentz and internal symmetry indices. The vector field decomposes into conformal Killing vector fields for each of the factor spaces.The field equation has a `classical vacuum' solution which is a Cartesian product of factor spaces. The factor spaces are all Einstein manifolds or two-dimensional Riemannian manifolds. All have a Ricci curvature of roughly the same order of magnitude, or are Ricci-flat. A worked example is provided in six dimensions.Away from this classical vacuum, connection components in appropriate coordinates include $SO(N)$ gauge fields. The Riemann tensor includes their field strength. Unitary gauge symmetries act indirectly on tensor fields and some or all of the unitary gauge fields are found amongst the $SO(N)$ gauge fields. Symmetry restoration occurs at the zero-curvature `decompactification limit', in which all dimensions appear on the same footing.

This paper looks at how changes of coordinates on a pseudo-Riemannian manifold induce homogeneous linear transformations on its tangent spaces. We see that a pseudo-orthonormal frame in a given tangent space is the basis for a set of Riemann normal coordinates. A Lorentz subgroup of the general linear transformations preserves this pseudo-orthonormality. We borrow techniques from the methodology of non-linear realizations to analyze this group-subgroup structure. `Parallel maps' are used to relate tangent space at different points. `Parallelisms' across a finite region of the manifold may be built up from them. These are used to define Weitzenb\"{o}ck connections and Levi-Civita connections. This provides a new formulation of teleparallel gravity, in which the tetrad field is viewed as a field-valued group element relating the coordinate basis to the frame basis used in defining a parallelism. This formulation separates the metric degrees of freedom from those associated with the choice of parallelism. The group element can be combined by matrix multiplication with Lorentz transformations of frame or with other Jacobian matrices. We show how this facilitates a new understanding of inertial forces and local Lorentz transformations. The analysis is also applied to translations of the coordinates. If they are constant across spacetime, this has no effect on the tangent space bases. If the translation parameters become fields, they induce general linear transformations of the coordinate basis; however, the tetrad components can only be expressed in terms of translations on a flat spacetime.