Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. Sub-Riemannian Geometry enjoys major differences from the Riemannian being a generalization of the latter at the same time, e.g., geodesics are not unique and may be singular, the Hausdorff dimension is larger than the manifold topological dimension. There exists a la...

We summarize the main ideas of General Relativity and Lorentzian geometry, leading to a proof of the simplest of the celebrated Hawking-Penrose singularity theorems. The reader is assumed to be familiar with Riemannian geometry and point set topology. Introduction Historically, much of the development of Riemannian geometry has been driven by General Relativity. This theory models spacetime as ...

The Batalin-Vilkovisky data for Polyakov string theory on a manifold with (non-null) boundary is shown to induce compatible Batalin--Fradkin--Vilkovisky data, thus allowing BV-quantisation manifolds boundary. On the other hand, analogous formulation of Nambu--Goto fails satisfy needed regularity requirements. As byproduct, concise description given reduced phase spaces both models and their rel...

Spacetimes have conventionally been described by a global Lorentzian metric on a differentiable four-manifold. Herein we explore the possibility of spacetimes defined by a connection, which is locally but not globally Levi-Civita. The general method of obtaining such connections is presented for the nondegenerate case followed by an example that modifies the Robertson-Walker spacetimes for flat...

We exhibit an example of a homogeneous Lorentzian manifold G/H whose homogeneous geodesics form just the light-cone and a hyperplane in the tangent space. For all light-like homogeneous geodesics, the natural parameter of the orbit is not the affine parameter of the geodesic.

Using the theory of measurable categories developped in [Yet03], we provide a notion of representations of 2-groups more well-suited to physically and geometrically interesting examples than that using 2-VECT (cf. [KV94]). Using this theory we sketch a 2-categorical approach to the state-sum model for Lorentzian quantum gravity proposed in [CY03], and suggest state-integral constructions for 4-...

A new general procedure to construct realistic spacetimes is introduced. It is based on the null congruence on a time-oriented Lorentzian manifold associated to a certain timelike vector field. As an application, new examples of stably causal Petrov type D spacetimes which obey the timelike convergence condition and which admit an isometric spacelike circle action are obtained.