By B. A. Dubrovin, A. T. Fomenko, S. P. Novikov

ISBN-10: 0387908722

ISBN-13: 9780387908724

ISBN-10: 0387972714

ISBN-13: 9780387972718

ISBN-10: 3540972714

ISBN-13: 9783540972716

This publication, written by means of a few of the grasp expositors of contemporary arithmetic, is an advent to fashionable differential geometry with emphasis on concrete examples and ideas, and it's also precise to a physics viewers. every one subject is prompted with examples that support the reader relish the necessities of the topic, yet rigor isn't really sacrificed within the e-book.

within the first bankruptcy the reader will get a style of differentiable manifolds and Lie teams, the later gving upward thrust to a dialogue of Lie algebras by way of contemplating, as traditional, the tangent area on the identification of the Lie staff. Projective house is proven to be a manifold and the transition capabilities explicitly written down. The authors supply a neat instance of a Lie staff that's not a matrix crew. a slightly fast advent to advanced manifolds and Riemann surfaces is given, probably too quickly for the reader requiring extra info. Homogeneous and symmetric areas also are mentioned, and the authors plunge correct into the speculation of vector bundles on manifolds. hence there's a lot packed into this bankruptcy, and the authors must have thought of spreading out the dialogue extra, because it leaves the reader in need of for extra aspect.

The authors ponder extra primary questions in gentle manifolds in bankruptcy three, with walls of team spirit used to turn out the lifestyles of Riemannian metrics and connections on manifolds. additionally they end up Stokes formulation, and turn out the life of a soft embedding of any compact manifold into Euclidean area of size 2n + 1. houses of soft maps, similar to the facility to approximate a continuing mapping by means of a soft mapping, also are mentioned. an evidence of Sard's theorem is given, therefore permitting the learn of singularities of a mapping. The reader does get a style of Morse thought right here additionally, besides transversality, and therefore a glance at a few user-friendly notions of differential topology. an enticing dialogue is given on tips on how to receive Morse services on gentle manifolds by utilizing focal issues.

Notions of homotopy are brought in bankruptcy three, in addition to extra recommendations from differential topology, corresponding to the measure of a map. a really attention-grabbing dialogue is given at the relation among the Whitney variety of a aircraft closed curve and the measure of the Gauss map. This ends up in an explanation of the $64000 Gauss-Bonnet theorem. measure conception is additionally utilized to vector fields after which to an software for differential equations, particularly the Poincare-Bendixson theorem. The index idea of vector fields is additionally proven to guide to the Hopf consequence at the Euler attribute of a closed orientable floor and to the Brouwer fixed-point theorem.

bankruptcy four considers the orientability of manifolds, with the authors exhibiting how orientation will be transported alongside a direction, hence giving a non-traditional characterization as to whilst a hooked up manifold is orientable, particularly if this delivery round any closed direction preserves the orientation classification. extra homotopy conception, through the elemental staff, is usually mentioned, with a number of examples being computed and the relationship of the elemental team with orientability. it truly is proven that the elemental workforce of a non-orientable manifold is homomorphic onto the cyclic staff of order 2. Fiber bundles with discrete fiber, often referred to as masking areas, also are mentioned, in addition to their connections to the idea of Riemann surfaces through branched coverings. The authors exhibit the application of masking maps within the calculation of the basic team, and use this connection to introduce homology teams. a really specified dialogue of the motion of the discrete staff at the Lobachevskian aircraft is given.

Absolute and relative homotopy teams are brought in bankruptcy five, and plenty of examples are given in their calculation. the belief of a masking homotopy ends up in a dialogue of fiber areas. the main fascinating dialogue during this bankruptcy is the single on Whitehead multiplication, as this is often frequently now not lined in introductory books reminiscent of this one, and because it has turn into very important in physics purposes. The authors do take a stab on the challenge of computing homotopy teams of spheres, and the dialogue is a piece unorthodox because it will depend on utilizing framed common bundles.

the idea of tender fiber bundles is taken into account within the subsequent bankruptcy. The physicist reader may still pay shut realization to this bankruptcy is it supplies many insights into the homotopy concept of fiber bundles that can't be present in the standard books at the topic. The dialogue of the class thought of fiber bundles is especially dense yet well worth the time examining. curiously, the authors comprise a dialogue of the Picard-Lefschetz formulation, as an instance of a category of "fiber bundles with singularities". these attracted to the geometry of gauge box theories will get pleasure from the dialogue at the differential geometry of fiber bundles.

Dynamical platforms are brought in bankruptcy 7, first as outlined over manifolds, after which within the context of symplectic manifolds through Hamaltonian mechanics. Liouville's theorem is confirmed, and some examples are given from relativistic element mechanics. the idea of foliations can also be mentioned, even if the dialogue is just too short to be of a lot use. The authors additionally think about variational difficulties, and given its significance in physics, they proceed the remedy within the final bankruptcy of the booklet, giving numerous examples as a rule relativity, and in gauge thought through a attention of the vacuum recommendations of the Yang-Mills equation. The physicist reader will savour this dialogue of the classical concept of gauge fields, because it is nice coaching for additional interpreting on instantons and the eventual quantization of gauge fields.