By Lawrence Conlon (auth.)

ISBN-10: 0817641343

ISBN-13: 9780817641344

ISBN-10: 081764766X

ISBN-13: 9780817647667

ISBN-10: 3764341343

ISBN-13: 9783764341343

The fundamentals of differentiable manifolds, international calculus, differential geometry, and similar issues represent a middle of knowledge crucial for the 1st or moment yr graduate pupil getting ready for complex classes and seminars in differential topology and geometry. *Differentiable Manifolds *is a textual content designed to hide this fabric in a cautious and sufficiently certain demeanour, presupposing just a solid origin mostly topology, calculus, and smooth algebra. This moment variation incorporates a major volume of recent fabric, which, as well as lecture room use, will make it an invaluable reference textual content. themes that may be passed over effectively in a primary direction are truly marked, making this variation more straightforward to exploit for this kind of path, in addition to for personal examine through non-specialists wishing to survey the field.

The issues of linearization, (re) integration, and international as opposed to neighborhood calculus are emphasised all through. extra gains comprise a therapy of the weather of multivariable calculus, formulated to evolve conveniently to the worldwide context, an exploration of package conception, and another (optional) improvement of Lie concept than is familiar in textbooks at this point. New to the second one version is an in depth remedy of overlaying areas and the elemental group.

Students, lecturers and execs in arithmetic and mathematical physics should still locate this a so much stimulating and precious text.

"This is a delicately written and wide-ranging textbook appropriate more often than not for graduate classes, even though a few complicated undergraduate classes may gain advantage from the early chapters. the subject material is differential topology and geometry, that's, the learn of curves, surfaces and manifolds the place the idea of differentiability provides the instruments of differentiable and vital calculus to these of topology. inside of this zone, the booklet is surprisingly comprehensive…. the fashion is apparent and exact, and this makes the booklet an outstanding reference textual content. there are numerous solid exercises."—**The Mathematical Gazette** (Review of the second one Edition)

"This textbook, most likely the easiest creation to differential geometry to be released due to the fact Eisenhart's, significantly advantages from the author's wisdom of what to prevent, anything newbie is probably going to overlook. The presentation is delicate, the alternative of themes optimum, and the booklet will be profitably used for self teaching."—**The Bulletin of Mathematical Books** (Review of the 1st Edition)

"The writer has rather well succeeded in writing an attractive, stimulating and delightful interpreting e-book [with] an clever equilibrium among rigor and informal."—**Zentralblatt Math** (Review of the 1st Edition)

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**Additional resources for Differentiable Manifolds**

**Sample text**

This enables us to define an induced map f, : 7cl(X, xo) --* 7rl(Y, y0) f,([~]) = If o ~]. We use the notation f ~xo g for homotopy rood the singleton {x0}. This is a homotopy through basepoint-preserving maps. The following is a very routine exercise, but important. 37. Prove that f . is well defined and is a group homomorphism. Show that, whenever (x, xo) ~ (z, yo) Z (z, zo), then ( f o g ) , = f, og,, and t h a t id, = id (where we use "id" for identity maps on any suitable domain). Finally, if f ~xo g, prove t h a t f , = g,.

22. Consider the map p : S 1 x[0,1]---*D 2 defined by viewing S 1 C D 2 C C and writing p(z,t) = (1 - t)z. This is one-to-one on S 1 x [0, 1) and collapses S 1 x {1} to the single point 0 C C. Arguing as in the previous example, we see that the quotient space (S a x [0, 1 ] ) / ( S 1 x {0}) is canonically homeomorphic to D 2. Intuitively, we have collapsed the top of the cylinder S 1 x [0, 1] to a point, obtaining a cone that can then be flattened to a disk. Generally, if A C_ X, one can define the equivalence relation ~ A by writing x ~ A Y if and only if either x = y or x, y E A.

6. More generally, the projection p : R ~-~ Rn/Z ~ = T n is a covering map with covering group isomorphic to the integer lattice Z n. 7. 26, is a covering map. The group of covering transformations is generated by the antipodal interchange map, hence is Z2. 8. If y, X' f f ~y ~X is a commutative diagram of continuous maps, where p~ and p are covering maps, we say t h a t f i s a lift of f to the covering spaces. In the case that X ~ = X and f = i d x , such a lift is called a homomorphism of covering spaces.

### Differentiable Manifolds by Lawrence Conlon (auth.)

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