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# HW01 - Diﬀerential Geometry 1 Homework 01 1(a Prove...

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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, |w|2 − |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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