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differential geometry w notes from teacher_Part_6

differential geometry w notes from teacher_Part_6 - 11 1.2...

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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 0 } , 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 . The set ϕ α ( U α U β ) is an open subset in R n . The maps f αβ = ϕ α ϕ - 1 β : ϕ α ( U α U β ) R are called the transition functions (or the overlap functions ). We assume that the transition functions are smooth.
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