Related papers: Affine and convex spaces: blending the analytic an…
A common approach to metric-affine, local Poincar\'e, special-relativistic and Galilei spacetime geometry is developed. Starting from an affine composite bundle, we introduce local reference frames and their evolution along worldlines and…
Notes prepared for the introductory general relativity course PHYSICS 748 at The University of Auckland. They are designed to introduce general relativity to upper-year undergraduate students directly using the modern language of…
We study here systems of symmetries on $|1|$--graded parabolic geometries. We are interested in smooth systems of symmetries and we discuss non--flat homogeneous $|1|$--graded geometries. We show the existence of an invariant admissible…
In this work, we propose a convenient framework for infinite-dimensional analysis (including both real and complex analysis in infinite dimensions), in which differentiation (in some weak sense) and integration operations can be easily…
Geometrical properties of holonomic and non holonomic varieties defined by the Pfaff equations connected with a first order systems of differential equations are studied. The Riemann extensions of affine connected spaces for investigation…
This is a slightly updated version of lectures notes for a course on analytic geometry taught in the winter term 2019/20 at the University of Bonn. The material presented is part of joint work with Dustin Clausen. This is intended as a…
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,…
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,…
We make an attempt to develop "noncommutative algebraic geometry" in which noncommutative affine schemes are in one-to-one correspondence with associative algebras. In the first part we discuss various aspects of smoothness in affine…
These informal notes deal with some basic properties of metric spaces, especially concerning lengths of curves.
The basic principles of Affine Quantum Gravity are presented in a pedagogical style with a limited number of equations.
Learning to use math in physics involves combining (blending) our everyday experiences and the conceptual ideas of physics with symbolic mathematical representations. Graphs are one of the best ways to learn to build the blend. They are a…
Congruences, or $2$-parameter families of lines in $3$-space are of interest in many situations, in particular in geometric optics. In this paper we consider elements of their geometry which are invariant under affine changes of…
This brief note, written for non-specialists, aims at drawing an introductive overview of the multiverse issue.
We give a survey of the theory of affine spheres, emphasizing the convex cases and relationsships to Monge-Ampere equations and geometric structures on manifolds.
Affine correspondences have received significant attention due to their benefits in tasks like image matching and pose estimation. Existing methods for extracting affine correspondences still have many limitations in terms of performance;…
An affine vector space partition of $\operatorname{AG}(n,q)$ is a set of proper affine subspaces that partitions the set of points. Here we determine minimum sizes and enumerate equivalence classes of affine vector space partitions for…
We give an overview of the affine surface area, its properties and its history.
Intrinsic volumes are fundamental geometric invariants generalizing volume, surface area, and mean width for convex bodies. We establish a unified Laplace-Grassmannian representation for intrinsic and dual volumes of convex polynomial…
The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine. In this note we prove…