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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 (R2 ) is homeomorphic to the unit circle S 1 ⊂ R2 .
(b) Prove explicitly that the complex projective line P 1 C = P (C2 ) is homeomorphic to the unit sphere S 2 ⊂ R3 .
Suggestion: Identify R4 = C2 and R3 = C × R; consider the map ρ given by
ρ(z, w) = (2zw, w2 − z 2 ).
2. Let Σ denote the image of the map
F : S 2 → R3 : (x, y, z ) → (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 } ⊂ R3
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 = 0}
is locally Euclidean. 1 ...
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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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