Prostory a geometrie

The book explores the relationship between geometry and arithmetic and traces the development of geometry from its illustrative nature to abstractness and generality. It presents the logically complete Tarski’s axioms of elementary Euclidean geometry as well as the logically complete axioms of Veblen and Young’s projective geometry. It shows how spherical geometry and trigonometry inspired the discovery of non-Euclidean geometries. It also discusses the differential geometry of curves and surfaces embedded in a three-dimensional Euclidean space and shows how the notions of metric tensor, curvature, geodesics, and parallel transport of vector fields are related to it, becoming the basis of the abstract Riemannian differential geometry of smooth manifolds. Finally, the book demonstrates how Riemannian differential geometry became the mathematical basis of Albert Einstein’s general theory of relativity.