Download e-book for kindle: A Comprehensive Introduction to Differential Geometry by Michael Spivak

By Michael Spivak

ISBN-10: 0914098713

ISBN-13: 9780914098713

Ebook by means of Michael Spivak, Spivak, Michael

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Example text

Po is not constant. Use the results of the last exercise to prove that there is no proper map­ ping from the unit polydisk to the unit ball in e2 • Let G1 c en and G2 c em be domains and f : G 1 --+ G2 a biholomorphic map . Show that m = n. e. , an �-linear map satisfying the product rule at zo E G . f z0 . 1. Let 2. 3. 4. 5. = = 36 I. Holomorphic Functions 6. 1 , Jm ) : G ---+ e m a holomorphic mapping, and M : { (z, w ) E G x em : w f (z) }. Prove the following: If g : G X em ---+ e is a holomorphic function with 9 IM = 0, then for every point (z 0, wo ) E M there is a neighborhood U and an m-tuple of holomorphic functions in U such that ( a1 , , cy • • .

Let G C en be a domain and f : G --+ D R (O ) C e a function, r = { (z, w) E G x D R (O) : w = f(z)} its graph. Sow that if there is a holomorphic function F in G x D R(O ) that is completely singular at every point of r, then f is continuous. ) 6 . Show that the "Hartogs triangle" { ( z,w) E e 2 : JwJ < J z J < 1} is a weak domain of holomorphy. 2. Plur isubharr nonic Fu n ction s Subharmonic Functions . Recall some facts from complex analysis of one variable. A twice differentiable real-valued function h on a domain G C e is called harmonic if hzz (z) = 0 on G.

Give an example of a proper holomorphic map that is not biholomorphic. (b) Let G1 and G 2 be bounded. Show that a continuous map f : G1 --+ G2 is proper if and only if for every sequence (z k ) in G 1 tending to 8G1 , the sequence (f(z k )) tends to 8G2 • (c) Let G' , G" C e be bounded domains and f : G' x G" --+ G2 a proper holomorphic map onto a bounded domain G2 C e2 • Show that z >-+ fw (z, w) cannot vanish identically on G'. Let z0 E 8G' be an arbitrary point and ( Zk ) a sequence in G' tending to z0.

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A Comprehensive Introduction to Differential Geometry by Michael Spivak


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