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Unformatted text preview: Differential Geometry 1 Homework 01 1. (a) Prove explicitly that the real projective line P 1 R = P ( R 2 ) is homeo morphic to the unit circle S 1 ⊂ R 2 . (b) Prove explicitly that the complex projective line P 1 C = P ( C 2 ) is home omorphic to the unit sphere S 2 ⊂ R 3 . Suggestion : Identify R 4 = C 2 and R 3 = C × R ; consider the map ρ given by ρ ( z,w ) = (2 zw,  w  2  z  2 ) . 2. Let Σ denote the image of the map F : S 2 → R 3 : ( x,y,z ) 7→ ( yz,zx,xy ) . (a) Show that Σ is obtained by removing from the set { ( X,Y,Z ) : Y 2 Z 2 + Z 2 X 2 + X 2 Y 2 = XY Z } ⊂ R 3 all points on each coordinate axis at distance greater than 1 / 2 from the origin. (b) Show explicitly that the space Σ ∩ { ( X,Y,Z ) : XY Z 6 = 0 } is locally Euclidean. Solutions (1a) View P 1 R as the quotient of S 1 via antipodal identifications: thus, π : S 1 → P 1 R : a 7→ [ a ] = [ a ]. Regarding S 1 as the unit circle in the complex plane, the squaring map s : S 1 → S 1 : a 7→...
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This note was uploaded on 07/08/2011 for the course MTG 6256 taught by Professor Robinson during the Spring '09 term at University of Florida.
 Spring '09
 Robinson

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