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The reasons of the crisis in the contemporary (Riemannian) geometry are discussed. The conventional method of the generalized geometries construction, based on a use of the topology, leads to a overdetermination of the Riemannian geometry.…

Geometry and topology have generated impacts far beyond their pure mathematical primitive, providing a solid foundation for many applicable tools. Typically, real-world data are represented as vectors, forming a linear subspace for a given…

A formulation of nucleon-nucleon scattering is developed in which the S-matrix, rather than an effective-field theory (EFT) action, is the fundamental object. Spacetime plays no role in this description: the S-matrix is a trajectory that…

Paths in an appropriate geometry are usually used as trajectories of test particles in geometric theories of gravity. It is shown that non-symmetric geometries possess some interesting quantum features. Without carrying out any quantization…

Excitations of a relativistic geometry are used to represent the theory of quantum electrodynamics. The connection excitations and the frame excitations reduce, respectively, to the electromagnetic field operator and electron field…

We explain how quantum gravity can be defined by quantizing spacetime itself. A pinpoint is that the gravitational constant G = L_P^2 whose physical dimension is of (length)^2 in natural unit introduces a symplectic structure of spacetime…

The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief…

A complete canonical formulation of general covariance makes it possible to construct new modified theories of gravity that are not of higher-curvature form, as shown here in a spherically symmetric setting. The usual uniqueness theorems…

The geometrical nature of gravity emerges from the universality dictated by the equivalence principle. In the usual formulation of General Relativity, the geometrisation of the gravitational interaction is performed in terms of the…

The Euclidean interpretation of special relativity which has been suggested by the author is a formulation of special relativity in ordinary 4D Euclidean space-time geometry. The natural and geometrically intuitive generalization of this…

Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding…

We consider non minimal coupling between matters and gravity in modified theories of gravity. In contrary to the current common sense, we report that quantum mechanics can effectively emerge when the space-time geometry is sufficiently…

Classical methods of differential geometry are used to construct equations of motion for particles in quantum, electrodynamic and gravitational fields. For a five dimensional geometrical system, the equivalence principle can be extended.…

We show that by employing the standard projected curvature as a measure of spatial curvature, we can make a certain generalization of optical geometry (Abramowicz and Lasota 1997, Class. Quantum Grav. 14 (1997) A23). This generalization…

In this paper, we propose to study a new geometric optimization problem called "geometric prototype" in Euclidean space. Given a set of patterns, where each pattern is represented by a (weighted or unweighted) point set, the geometric…

We describe a theory amalgamating quantum theory and general relativity through the identification of a continuous 4-dimensional spacetime arena constructed from the substructures of a generalised multi-dimensional form for proper time. In…

Quantum particles confined to surfaces in higher dimensional spaces are acted upon by forces that exist only as a result of the surface geometry and the quantum mechanical nature of the system. The dynamics are particularly rich when…

The gravitation equations of the general relativity, written for Riemannian space-time geometry, are extended to the case of arbitrary (non-Riemannian) space-time geometry. The obtained equations are written in terms of the world function…

This article explores the overall geometric manner in which human beings make sense of the world around them by means of their physical theories; in particular, in what are nowadays called pregeometric pictures of Nature. In these, the…

In the recently introduced gauge theory of translations, dubbed Coincident General Relativity, gravity is described with neither torsion nor curvature in the spacetime affine geometry. The action of the theory enjoys an enhanced symmetry…