By Vaisman

ISBN-10: 0824770633

ISBN-13: 9780824770631

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**Extra resources for A First Course in Differential Geometry**

**Example text**

A homeomorphism. 3). Tn are same n g12 n by e l l i p t i c find a subsequence, solution since B n is a g a i n a c i r c u l a r d o m a i n as n to (gij). T h e m a p s Yn s a t i s f y respect the c o r r e s p o n d i n g the a system , where with as C a e s t i m a t e s we can that T is a w e a k tisfy : Bq + G H I'2 as w e l l and therefore and Tn T n is c o n f o r m a l satisfy the metrics to uniform (gnj)i c a n be c h o s e n (gij). 1) C I'~ e s t i m a t e s , we to c o n v e r g e infer that and consequently with the the limit T is a d i f f e o m o r p h i s m .

3 a n d of B. e. 1, with represent of t h o s e which to t h e only a theorem variations a -priori about We of ~ come (cf. 1 solutions theory together of in- on E . 1) via uniqueness on and o f v i e w of K o d a i r a - S p e n c e r to d e a l theory under ~ , divided differentials of Thm. of of the to T e i c h m H l l e r can be proved have struc- respect [EE] hhe point thus that the invariant quadratic dependance They employ are with of E a r l e - E e l l s [ESz] ), w h i c h structures back orthogonal it is s h o w n of c o n f o r m a l is t h e q u o t i e n t structures of h o l o m o r p h i c continuous estimates.

E. 9 6 C2'~(G) y is a h o m e o m o r p h i s m between T is n o t a h o m e o m o r p h i s m . Then in t h e n o t a - B and T is n o t exist two points z I , z 2 , z I ~ z2 with a shortest segment Yn joining T n is a h o m e o m o r p h i s m , Yn: = T n I (yn) Yn ' then Tn(Z 1) a n d is a c u r v e S. injec- T(z I) = Tn(Z2). joining z I and z2 . If P n ' @ is a p o i n t subsequence the of Tn c o n v e r g e Thus, a whole on (pn,~) ~B(Zl,6) N converging uniformly continuum to for n ÷ ~ we can to s o m e p o i n t T , we is m a p p e d see t h a t onto the P6 o n T(p6) find a ~B(z1,6).

### A First Course in Differential Geometry by Vaisman

by Mark

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