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

This publication, written through a number of the grasp expositors of contemporary arithmetic, is an creation to fashionable differential geometry with emphasis on concrete examples and ideas, and it's also specified to a physics viewers. each one subject is prompted with examples that support the reader have fun with the necessities of the topic, yet rigor isn't really sacrificed within the e-book.

within the first bankruptcy the reader will get a flavor of differentiable manifolds and Lie teams, the later gving upward thrust to a dialogue of Lie algebras by means of contemplating, as traditional, the tangent house on the id of the Lie workforce. Projective area is proven to be a manifold and the transition features explicitly written down. The authors provide a neat instance of a Lie team that isn't a matrix workforce. a slightly speedy creation to advanced manifolds and Riemann surfaces is given, might be too quickly for the reader requiring extra info. Homogeneous and symmetric areas also are mentioned, and the authors plunge correct into the idea of vector bundles on manifolds. therefore 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 element.

The authors think of extra basic questions in gentle manifolds in bankruptcy three, with walls of cohesion used to end up the lifestyles of Riemannian metrics and connections on manifolds. additionally they end up Stokes formulation, and turn out the life of a tender embedding of any compact manifold into Euclidean house of measurement 2n + 1. houses of gentle maps, equivalent to the facility to approximate a continual mapping through a gentle mapping, also are mentioned. an explanation of Sard's theorem is given, hence permitting the examine of singularities of a mapping. The reader does get a flavor of Morse conception the following additionally, in addition to transversality, and hence a glance at a few easy notions of differential topology. a fascinating dialogue is given on tips to receive Morse capabilities on delicate manifolds through the use of focal issues.

Notions of homotopy are brought in bankruptcy three, besides extra strategies from differential topology, reminiscent of the measure of a map. a truly attention-grabbing dialogue is given at the relation among the Whitney variety of a airplane closed curve and the measure of the Gauss map. This ends up in an evidence of the $64000 Gauss-Bonnet theorem. measure thought is additionally utilized to vector fields after which to an program for differential equations, specifically the Poincare-Bendixson theorem. The index conception of vector fields can be 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 displaying how orientation should be transported alongside a direction, therefore giving a non-traditional characterization as to while a hooked up manifold is orientable, particularly if this shipping round any closed course preserves the orientation type. extra homotopy idea, through the elemental team, is additionally mentioned, with a number of examples being computed and the relationship of the elemental workforce with orientability. it truly is proven that the elemental staff of a non-orientable manifold is homomorphic onto the cyclic crew of order 2. Fiber bundles with discrete fiber, often referred to as overlaying areas, also are mentioned, in addition to their connections to the speculation 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 truly exact dialogue of the motion of the discrete team at the Lobachevskian airplane is given.

Absolute and relative homotopy teams are brought in bankruptcy five, and plenty of examples are given in their calculation. the assumption of a masking homotopy ends up in a dialogue of fiber areas. the main attention-grabbing dialogue during this bankruptcy is the only on Whitehead multiplication, as this can be often no longer coated in introductory books comparable to this one, and because it has develop into vital 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 is dependent upon utilizing framed general bundles.

the idea of soft fiber bundles is taken into account within the subsequent bankruptcy. The physicist reader may still pay shut recognition to this bankruptcy is it offers 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 concept of fiber bundles is especially dense yet well worth the time analyzing. curiously, the authors contain a dialogue of the Picard-Lefschetz formulation, for instance of a category of "fiber bundles with singularities". these drawn to the geometry of gauge box theories will relish 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 aspect mechanics. the idea of foliations is additionally mentioned, even if the dialogue is just too short to be of a lot use. The authors additionally examine variational difficulties, and given its significance in physics, they proceed the remedy within the final bankruptcy of the e-book, giving a number of examples more often than not relativity, and in gauge conception through a attention of the vacuum options of the Yang-Mills equation. The physicist reader will savor this dialogue of the classical conception of gauge fields, because it is nice training for additional interpreting on instantons and the eventual quantization of gauge fields.