differential geometry w notes from teacher_Part_6

differential geometry w notes from teacher_Part_6 - 1.2...

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Unformatted text preview: 1.2. MANIFOLDS 11 – The set of oriented lines through the origin of R n + 1 is S n . – Topologically R P n is the sphere S n in R n + 1 with the antipodal points identified, which is the unit ball in R n with the antipodal points on the boundary (which is a unit sphere S n- 1 ) identified. – R P n is covered by ( n + 1) sets U j = { L ∈ R P n | L with x j , } , j = 1 , . . . , n + 1 . – The local coordinates in U j are v 1 = x 1 x j , ··· , v n = x n x j . – The ( n + 1)-tuple ( x 1 , . . . , x n + 1 ) identified with ( λ x 1 , . . . , λ x n + 1 ), λ , 0, are called homogeneous coordinates of a point in R P n . 1.2.3 Rigorous Definition of a Manifold • Let M be a set (without topology) and { U α } α ∈ A be a collection of subsets that is a cover of M , that is, [ α ∈ A U α = M . • Let ϕ α : U α → R n , α ∈ A , be injective maps such that ϕ α ( U α ) are open subsets in R n ....
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_6 - 1.2...

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