Absolute parallelism: Spherical symmetry and singularities

Crushing singularities in spacetimes with spherical, plane and hyperbolic symmetry

It is shown that the initial singularities in spatially compact spacetimes with spherical, plane or hyperbolic symmetry admitting a compact constant mean curvature hypersurface are crushing singularities when the matter content of spacetime is described by the Vlasov equation (collisionless matter) or the wave equation (massless scalar field). In the spherically symmetric case it is further sho...

Emergence, Singularities, and Symmetry Breaking

Absolute Parallelism Geometry: Developments, Applications and Problems

Absolute parallelism geometry is frequently used for physical applications. It has two main defects, from the point of view of applications. The first is the identical vanishing of its curvature tensor. The second is that its autoparallel paths do not represent physical trajectories. The present work shows how these defects were treated in the course of development of the geometry. The new vers...

Metrics with conic singularities and spherical polygons

A spherical n-gon is a bordered surface homeomorphic to a closed disk, with n distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these polygons and enumerate them in the case that two angles at the corners are ...

Coning, Symmetry and Spherical Frameworks

In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning [28], (b) the further transfer of these results to spherical space via associated rigidity matrices [19], and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetryderived orbit rigidity...