Download A Comprehensive Introduction To Differential Geometry by Michael Spivak PDF

By Michael Spivak

Booklet by means of Michael Spivak, Spivak, Michael

Show description

Read or Download A Comprehensive Introduction To Differential Geometry PDF

Similar differential geometry books

Convex analysis: theory and applications

This ebook is an advent to convex research and a few of its functions. It begins with easy concept, that's defined in the framework of finite-dimensional areas. the single necessities are uncomplicated research and easy geometry. the second one bankruptcy provides a few purposes of convex research, together with difficulties of linear programming, geometry, and approximation.

Discrete Groups, Expanding Graphs and Invariant Measures

The booklet provides the suggestions to 2 difficulties: the 1st is the development of increasing graphs – graphs that are of primary value for communique networks and machine technology; the second one is the Ruziewicz challenge about the finitely additive invariant measures on spheres. either difficulties have been partly solved utilizing the Kazhdan estate (T) from illustration conception of semi-simple Lie teams.

Extra info for A Comprehensive Introduction To Differential Geometry

Example text

A , ",k + t (a), . . , ", m (a» . I\1oreover, given any coordinate system )" the appropriate coordinate system on N can be obtained merely by permuting the component functions of y. (2) If J has rank k in a neighborhood of p, then there are coordinate systems (x, U) and (y, V) such that y o J o x- t (a t , . . , a n ) = (a t , . . , ak , O, . . , O) . Remark: The special case M = � n , N = �m i s equivalent t o the general theo­ rem, which gives only local results. If y is the identity of �m , part (I) says that by first performing a diffeomorphism on �n, and then permuting the coordi­ nates in �m, we can insure that J keeps the first k components of a point fixed.

An) (a', ... ,an,O, ... ,O). Even if we clo not perform ¢-' first, the map f still takes ]Rn into the subset ¢ -f- 0 <� = --f -". �)-�(1/r:(7f(IT{n»:12 f(]Rn) which 1/1 takes to ]Rn {OJ ]Rm-the points of ]Rn just get moved to the wrong place in ]Rn {OJ. This can be conected by another map on ]Rm. Define A by x C x Then A f(a', ... ,an) = A 1/1 f o¢-'(b', ... ,bn) for (b I , ... ,bn) '" ¢(a) '" A(b', ... ,bn,O, ... (CP-I (bI , ... ,bn),O, ... ,O) (a I , ... ,an,O, ... so A 1/1 is the desired If we are given a coordinate system x on ]Rn other than the identify, we just define A(bI , ...

A) Show that pI is homeomorphic to Sl . (b) Since we can consider sn-l C S n, and since antipodal points in sn-I are still antipodal when considered as points in sn, we can consider pn-I C pn in an obvious way. Show that pn - pn-l is homeomorphic to interior Dn = {x E �n: d(x,O) < J}. 16. A classical theorem of topology states that every compact surface other than S 2 is obtained by gluing together a certain number of tori and projective spaces, and that all compact surfaces-with-boundary are obtained from these by cutting out a finite number of discs.

Download PDF sample

Rated 4.88 of 5 – based on 32 votes