By P. M. H. Wilson
This self-contained 2007 textbook provides an exposition of the well known classical two-dimensional geometries, comparable to Euclidean, round, hyperbolic, and the in the neighborhood Euclidean torus, and introduces the elemental options of Euler numbers for topological triangulations, and Riemannian metrics. The cautious dialogue of those classical examples presents scholars with an advent to the extra basic concept of curved areas constructed later within the ebook, as represented by means of embedded surfaces in Euclidean 3-space, and their generalization to summary surfaces built with Riemannian metrics. subject matters working all through contain these of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the hyperlink to topology supplied by means of the Gauss-Bonnet theorem. quite a few diagrams support deliver the major issues to existence and beneficial examples and routines are integrated to assist realizing. in the course of the emphasis is put on specific proofs, making this article perfect for any scholar with a uncomplicated historical past in research and algebra.
Read Online or Download Curved Spaces: From Classical Geometries to Elementary Differential Geometry PDF
Similar Algebraic Geometry books
Fractal styles have emerged in lots of contexts, yet what precisely is a development? How can one make certain the constructions mendacity inside of gadgets and the relationships among them? This e-book proposes new notions of coherent geometric constitution to supply a clean method of this regularly occurring box. It develops a brand new thought of self-similarity known as "BPI" or "big items of itself," which makes the sector a lot more straightforward for individuals to go into.
The idea of elliptic curves is exceptional by means of its lengthy background and by means of the range of the tools which were utilized in its learn. This e-book treats the mathematics strategy in its sleek formula, by utilizing simple algebraic quantity concept and algebraic geometry. Following a quick dialogue of the mandatory algebro-geometric effects, the ebook proceeds with an exposition of the geometry and the formal workforce of elliptic curves, elliptic curves over finite fields, the complicated numbers, neighborhood fields, and international fields.
This booklet introduces the speculation of modular types, from which all rational elliptic curves come up, with a watch towards the Modularity Theorem. dialogue covers elliptic curves as advanced tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner idea; Hecke eigenforms and their mathematics houses; the Jacobians of modular curves and the Abelian forms linked to Hecke eigenforms.
Extra resources for Curved Spaces: From Classical Geometries to Elementary Differential Geometry