Related papers: Gravity, Cartan geometry, and idealized waywisers
An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels,…
The classical world structures borne by spacetimes endowed with torsionful affinities are reviewed. Subsequently, the definition and symmetry properties of a typical pair of Witten curvature spinors for such spacetimes are exhibited along…
In this paper we introduce a generalization of Hamiltonian mechanics that replaces configuration spaces, conventionally regarded simply as smooth manifolds, with line bundles over smooth manifolds. Classical observables are then identified…
On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of…
A generalized model of space-time is given, taking into consideration the anisotropic structure of fields which are depended on the position and the direction (velocity).In this framework a generalized FRW-metric the Raychaudhouri and…
This paper defines the spacetime geometry attached with observor as vacuum geometry (it defines the idea physical measurement geometry) and the spacetime geometry attached with matter as spacetime geometry. The initial spacetime geometry…
Two pseudo-Riemannian metrics are called projectively equivalent if their unparametrized geodesics coincide. The degree of mobility of a metric is the dimension of the space of metrics that are projectively equivalent to it. We give a…
By a real alphabeta-geometry we mean a four-dimensional manifold M equipped with a neutral metric h such that (M,h) admits both an integrable distribution of alpha-planes and an integrable distribution of beta-planes. We obtain a local…
Manifolds occur naturally as configuration spaces of robotic systems. They provide global descriptions of local coordinate systems that are common tools in expressing positions of robots. The purpose of this survey is threefold. Firstly, we…
In a former paper we proposed a model for the quantization of gravity by working in a bundle $E$ where we realized the Hamilton constraint as the Wheeler-DeWitt equation. However, the corresponding operator only acts in the fibers and not…
We describe noncommutative geometric aspects of twisted deformations, in particular of the spheres in Connes and Landi [8] and in Connes and Dubois Violette [7], by using the differential and integral calculus on these spaces that is…
Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces $(M=G/H,g)$ whose geodesics are orbits of one-parameter subgroups of $G$. The corresponding metric $g$ is called a geodesic orbit metric. We study the…
Realistic metric spaces (such as road/transportation networks) tend to be much more algorithmically tractable than general metrics. In an attempt to formalize this intuition, Abraham et~al.\ (SODA 2010, JACM 2016) introduced the notion of…
We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a…
Two sets of spatially diffeomorphism invariant operators are constructed in the loop representation formulation of quantum gravity. This is done by coupling general relativity to an anti- symmetric tensor gauge field and using that field to…
In the general relativity theory the basic ingredient to describe gravity is the geometry, which interacts with all forms of matter and energy, and as such, the metric could be interpreted as a true physical quantity. However the metric is…
The communications and interrelations between different locations on the Earth's surface have far-reaching implications for both social and natural systems. Effective spatial analytics ideally require a spatial representation, where…
Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3…
The aim of this note is to describe the geometry of $\mathbb{C}^2$ equipped with a K\"{a}hler metric defined by Warren. It is shown that with that metric $\mathbb{C}^2$ is a flat manifold. Explicit formulae for geodesics and volume of…
Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case…