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A general theory of vector-valued modular functions, holomorphic in the upper half-plane, is presented for finite dimensional representations of the modular group. This also provides a description of vector-valued modular forms of arbitrary…
The book is designed for a semester-long course in Foundations of Geometry and meant to be rigorous, conservative, elementary and minimalist. List of topics: Euclidean geometry: The Axioms / Half-planes / Congruent triangles / Perpendicular…
Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and also on the choice of the contravariant components, it was shown that a wide variety of third,…
This review presents an overview of various kinds of models -- physical, abstract, mathematical, visual -- that can be used to present the concepts and applications of Einstein's general theory of relativity at the level of undergraduate…
A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions,…
The main result is the identification of the orthogonal complement of the subalgebra of conformal vector field inside the algebra of all vector fields of a compact flat 2-manifold. As a fundamental tool, the complete Hodge decomposition for…
In this paper, we present a series of techniques to describe General Relativity using Geometric Algebra (GA). We emphasize the physical interpretation of quantities and provide a step-by-step guide for performing calculations. In doing so,…
We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors. We demonstrate the effectiveness of the approach by proving of a number of integral identities with vector integrands.
This paper presents a brief but comprehensive introduction to certain mathematical techniques in General Relativity. Familiar mathematical procedures are investigated taking into account the complications of introducing a non trivial…
We consider generalized inverses of linear operators on arbitrary vector spaces and study the question when their product in reverse order is again a generalized inverse. This problem is equivalent to the question when the product of two…
In this work, we make new developments in generic cotangent bundle geometries, depending on all phase-space variables. In particular, we will focus on the so-called generalized Hamilton spaces, discussing how the main ingredients of this…
A brief introduction to geometric valuation theory is given. The focus is on classification results for valuations on convex bodies and on function spaces.
We study geometric structures arising from Hermitian forms on linear spaces over real algebras beyond the division ones. Our focus is on the dual numbers, the split-complex numbers, and the split-quaternions. The corresponding geometric…
We first review the definition of the angle between subspaces and how it is computed using matrix algebra. Then we introduce the Grassmann and Clifford algebra description of subspaces. The geometric product of two subspaces yields the full…
Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This…
We discuss domestic affairs of metric spaces, keeping away from any extra structure. Topics include universal spaces, injective spaces, Hausdorff and Gromov--Hausdorff convergences, and ultralimits.
These are expanded lecture notes of a mini-course whose objectives were to introduce the basic concepts, constructions and techniques of noncommutative geometry, as well as their uses as a framework for modelling quantum spacetime. Key…
The generalized projection-tensor geometry introduced in an earlier paper is extended. A compact notation for families of projected objects is introduced and used to summarize the results of the previous paper and obtain fully projected…
Consider the smooth sections of the tangent bundle of a reductive homogeneous space. This is a vector space over the field of real numbers. The canonical connection acts as a linear binary operator on this vector space, making it an…
Starting from a short review of spaces of generalized sections of vector bundles, we give a concise systematic description, in precise geometric terms, of Leray densities, principal value densities, propagators and elementary solutions of…