Related papers: Projective Structure and Holonomy in 4-dimensional…
A real projective orbifold has a radial end if a neighborhood of the end is foliated by projective geodesics that develop into geodesics ending at a common point. It has a totally geodesic end if the end can be completed to have the totally…
We revisit the subject exploring maps from the space of 4-spinors to 3+1 space-time that commute with the Lorentz transformation. All known mappings have a natural embedding in a higher five dimensional spacetime, and can be succinctly…
Almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics are considered. A linear connection $D$ is introduced such that the structure of these manifolds is parallel with respect to D. Of special interest is the class of the…
We show how an observer could measure the non-local holonomy variables that parametrise the flat Lorentzian 3d manifolds arising as spacetimes in (2+1)-gravity. We consider an observer who emits lightrays that return to him at a later time…
We give explicit bijective correspondences between three families of objects: certain pairs of quaternions, which we regard as spinors; certain flags in (1+4)-dimensional Minkowski space; and horospheres in 4-dimensional hyperbolic space…
This paper combines two classical theories, namely metric projective differential geometry and superintegrability. We study superintegrable systems on 2-dimensional geometries that share the same geodesics, viewed as unparametrized curves.…
The relationship between convex geometry and algebraic geometry has deep historical roots, tracing back to classical works in enumerative geometry. In this paper, we continue this theme by studying two interconnected problems regarding…
Let $\calM=\Gamma\bs \calH^{(n)}$, where $\calH^{(n)}$ is a product of $n+1$ hyperbolic planes and $\Gamma\subset\PSL(2,\bbR)^{n+1}$ is an irreducible cocompact lattice. We consider closed geodesics on $\calM$ that propagate locally only in…
Space-Time in general relativity is a dynamical entity because it is subject to the Einstein field equations. The space-time metric provides different geometrical structures: conformal, volume, projective and linear connection. A deep…
Usually, the topology of a 4-manifolds $M$ is restricted to admit a global hyperbolic structure $\Sigma\times\mathbb{R}$. The result was obtained by using two conditions: existence of a Lorentz structure and causality (no time-like closed…
We generalise the notions of supersymmetry and superspace by allowing generators and coordinates transforming according to more general Lorentz representations than the spinorial and vectorial ones of standard lore. This yields novel…
There are considered 4-dimensional pseudo-Riemannian spaces with inner products of signature (3,1) and (2,2). The objects of investigation are space-like and time-like hyperspheres in the respective cases. These hypersurfaces are equipped…
We offer new insight into the folding kinematics of degree-4 rigid origami vertices by drawing an analogy to spacetime in special relativity. Specifically, folded states of the vertex, described by pairs of fold angles in terms of cotangent…
In this paper, we investigate the connection between the M-projective curvature tensor and other tensors. Also, we obtain the divergence of M-projective curvature tensor. A symmetry of spacetime known as M-projective collineation has been…
We give a summary of recent results on the explicit local form of the second-order symmetric Lorentzian manifolds in arbitrary dimension, and its global version. These spacetimes turn out to be essentially a specific subclass of plane…
Our aim is to support the choice of two remarkable connections with torsion in a 3-Sasakian manifold, proving that, in contrast to the Levi-Civita connection, the holonomy group in the homogeneous cases reduces to a proper subgroup of the…
We classify two-symmetric Lorentzian manifolds using methods of the theory of holonomy groups. These manifolds are exhausted by a special type of pp-waves and, like the symmetric Cahen-Wallach spaces, they have commutative holonomy.
Some standard definitions and results concerning foliations of dimension one and codimension one are introduced. A proper time foliation of Minkowski space is defined and contrasted with the foliation that is defined by the time coordinate.…
We prove two results on geometric consequences of the representation of restricted holonomy group of a Hermitian connection. The first result concerns when such a Hermitian manifold is K\"ahler in terms of the torsion and the irreducibility…
We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields $\K$, and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems.…