By Michael Spivak

Booklet by means of Michael Spivak, Spivak, Michael

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**Extra info for A Comprehensive Introduction To Differential Geometry**

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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.