Michael Spivak's A Comprehensive Introduction To Differential Geometry PDF

By Michael Spivak

Publication by way of Michael Spivak, Spivak, Michael

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Extra resources for A Comprehensive Introduction To Differential Geometry

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PROOF The stronger version ofSard 's Theorem , which we will never use (except once , in Problem 8-24), states' that the critical values of a C k map J: M n � N"' are a set of measure 0 if k 2: 1 + max (n - In, 0). Theorem 8 is the easy case, and the case m > n is the trivial case (Problem 20). Although Theorem 8 will be very important later on, for the present we are more interested in knowing what the image of J : M � N looks like lo�ally, in terms of the rank k of J at p E M. More exact information Can be given when J actually has rank k in a neighborhood of p.

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. n, and its image could, for example, contain only points with first coordinate o. 1i1IIor, Topolog)' From the D{fff'l"miiabtc Viewpoint or Sternberg, Lectures 011 Differential Geomel1)', DifJerelltiabLe Structures 43 fORn) JRn.

V"ith the new surface p2 at our disposal, we can create other surfaces in t he same way as the n-holed torus. \16bius strip. The closest we can come to picturing this is by drawing a cross­ cap sticking on a torus. We can also join together a pair of projective planes with holes cut out, which amounts to sewing two Mobius strips together along their boundary. Although this can be pictured as two cross-caps joined together, it has a nicer, and famous, representation. Consider the surface obtained from the square with identifications indicated below; it may also be obtained from the cylinder where I) x SI by identifying E I) X SI with is the reflection of x through a fixed diameter of the circle.

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A Comprehensive Introduction To Differential Geometry by Michael Spivak

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