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In this paper, we discuss some problems of elementary plane differential geometry and kinematics. Although the results are not new, the consistent use of complex-valued functions (plane curves) of a real variable (parameter) allows to…
The mathematical treatment and graphical representation of Special Relativity (SR) are well established, yet carry deep implications that remain hard to visualize. This paper presents a new graphical interpretation of the geometry of SR…
We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties…
We discuss the most elementary properties of the hyperbolic trigonometry and show how they can be exploited to get a simple, albeit interesting, geometrical interpretation of the special relativity. It yields indeed a straightforword…
Physical geometry studies mutual disposition of geometrical objects and points in space, or space-time, which is described by the distance function d, or by the world function \sigma =d^{2}/2. One suggests a new general method of the…
In this paper, we extend the method developed in [17, 18] to curves in the Minkowski plane. The method proposes a way to study deformations of plane curves taking into consideration their geometry as well as their singularities. We deal in…
These notes give an introduction to the quantization procedure called geometric quantization. It gives a definition of the mathematical background for its understanding and introductions to classical and quantum mechanics, to differentiable…
In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of…
We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…
In this review, we give a pedagogical introduction to a systematic framework for constructing and analyzing supersymmetric field theories on curved spacetime manifolds. The framework is based on the use of off-shell supergravity background…
We advocate that the dual picture of spacetime noncommutativity , i.e. the existence of a curved momentum space, could be a way out to solve some of the open conceptual problems in the field, such as the basis dependence of observables. In…
To unify general relativity and quantum theory is hard in part because they are formulated in two very different mathematical languages, differential geometry and functional analysis. A natural candidate for bridging this language gap, at…
Considering Riemannian submersions, we find necessary and sufficient conditions for when sub-Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvatures. We…
This survey article is an introduction to Diophantine Geometry at a basic undergraduate level. It focuses on Diophantine Equations and the qualitative description of their solutions rather than detailed proofs.
One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…
This is an introduction to the subject of the differential topology of the space of smooth loops in a finite dimensional manifold. It began as the background notes to a series of seminars given at NTNU and subsequently at Sheffield. I am…
The kinematical part of general theory of deformational structures on smooth manifolds is developed. We introduce general concept of d-objects deformation, then within the set of all such deformations we develop some special algebra and…
This paper is the second part of a series that develops the mathematical framework necessary for studying field theories on ``T-Minkowski'' noncommutative spacetimes. These spacetimes constitute a class of noncommutative geometries,…
These lecture notes provide an introduction to the theory and application of symmetry methods for ordinary differential equations, building on minimal prerequisites. Their primary purpose is to enable a quick and self-contained approach for…
Differential calculus on discrete spaces is studied in the manner of non-commutative geometry by representing the differential calculus by an operator algebra on a suitable Krein space. The discrete analogue of a (pseudo-)Riemannian metric…