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Informally a smooth manifold is a space which locally looks like an open subset of Rn . It need not be a subset of RN globally. Example 0.0.1. Let X be the configuration space of two distinct points in R2 i.e. the set of two point subsets of R2 . It’s equal to (R2 × R2 \∆R2 )/ ∼ where (x, y) ∼ (y, x). This space is a manifold but it’s not canonically embedded into any RN . Definition 0.0.2. A generalized smooth manifold of dimension n is a set M together with a collection of subsets {Uα }α∈A and maps ψα : Vα → Uα where Vα is an open subset of Rn such that the following properties are satisfied (1) ∪α Uα = M (2) ψα : Vα → Uα is a bijection for every α (3) For any α, β the set Uαβ = ψα−1 (Uα ∩ Uβ ) is open in Rn (4) For any α, β the map ψβ−1 ◦ ϕα : Uαβ → Uβα is smooth. The collection {Uα }α∈A together with the maps ψα : Vα → Uα is called a smooth atlas on M . The maps ψα : Vα → Uα are called local parameterizations. The inverse maps ϕα = ψα−1 : Uα → Vα are called local charts or local coordinates. Two atlases on M are said to define the same smooth structure if their union is still a smooth atlas on M . Examples • M = Rn , A = {1}, U1 = V1 = Rn , ψ1 = id. • M = R, A = {1}, U1 = V1 = R, ψ1 : R → R is given by ϕ(x) = x3 . • M = Γf - the graph of f where f : Rn → Rk . ( here Γf = {(x, y) ∈ Rn+k | y = f (x)}). A = {1}, U1 = Rn , V1 = M, ψ(x) = (x, f (x))} • S n = {(x0 , x1 , . . . , xn )| Σi x2i = 1}. For each i set Ui+ = {(x0 , x1 , . . . , xn ) ∈ S n | xi > 0} and Ui− = {(x0 , x1 , . . . , xn ) ∈ S n | xi < 0}. 2 < 1} and let ψ ± : V ± → U ± be Let Vi± = {(u0 , . . . , un−1 )| Σi up i i i i given by ψi (u) = (u0 , . . . , ui−1 , ± 1 − |u|2 , ui , . . . , un−1 ). The inverse map is just the projection ϕ± i (u0 , . . . , un ) = (u0 , . . . , ûi , . . . un ) This collection is a smooth atlas on S n (verify!). If j < i then p ψj−1 ◦ψj (u0 , . . . , un−1 ) = (u0 , . . . , uj−1 , uj+1 , . . . , ui−1 , ± 1 − |u|2 , ui , . . . , un−1 ) • RPn is the set of lines through 0 in Rn+1 . It can be equivalently described as the set of equivalence classes of points in Rn+1 \{0} modulo the equivalence relation (x0 , . . . , xn ) ∼ (x00 , . . . , x0n ) iff (x00 , . . . , x0n ) = t(x0 , . . . , xn ) for some t 6= 0. The equivalence class of (x0 , . . . , xn ) will be denoted by [x0 : x1 : . . . : xn ] Let Ui ⊂ RPn be the set of lines of the form R(x0 , . . . , xn ) with xi 6= 0 for any nonzero point. any such line intersects the hyperplane 1 2 {xi = 1} in a single point. This gives us a bijective map ϕi : Ui → Rn given by ϕi ([x0 : x1 : . . . : xn ]) = ( xx0i , . . . , xxi−1 , xxi+1 , . . . xxni ) with i i the inverse given by ψi (u0 , . . . , un−1 ) = [u0 : . . . : ui−1 : 1 : ui : . . . : un ]. For j < i we have ϕj ◦ ψi (u0 , . . . , un1 ) = ϕj ([u0 : . . . , ui−1 : 1 : ui : u u . . . : un−1 ]) = ( uu0j , . . . , uj−1 , uj+1 , . . . , uui−1 , u1j , uui−1 , uuji , . . . , un−1 uj ). j j j j This map is smooth on the open set {uj 6= 0} = ψi−1 (Uj ). A similar formula holds in the case j > i. This gives a smooth atlas on RPn . Example 0.0.3. Different smooth structures on R. The first structure is given by U1 = V1 = R with ψ : R → R given by the identity map ψ1 (x) = x. the second smooth structure is given by Ũ1 = Ṽ1 = R with p̃si : R → R given by ψ̃1 (x) = x3 . These smooth structure are distinct because these two atlases together √ do not form a smooth atlas since ψ̃1−1 ◦ ψ1 (x) = 3 x is not smooth at zero. Definition 0.0.4. let X be a generalized smooth n-dimensional manifold with an atlas {ψi : Vα → Uα }α∈A . A subset U of X is called open if ψα−1 (U ) is an open subset of Rn for any α. It’s easy to see that open sets satisfy the following properties • X and ∅ are open. • Uα is open for any α. • Union of any collection of open sets is open • Intersection of finitely many open sets is open. Definition 0.0.5. A generalized smooth manifold M n is called Hausdorff for any two distinct points p1 , p2 ∈ X there exist open sets W1 , W2 ⊂ X such that p1 ∈ W1 , p2 ∈ W2 and W1 ∩ W2 = ∅. Definition 0.0.6. A generalized smooth manifold M n is called a smooth manifold if it is Hausdorff and it admits a countable atlas {ψi : Vα → Uα }α∈A . Example 0.0.7. Let I1 = (−1, 1) × {1}, I2 = (−1, 1) × {2}. Let X be the space obtained from I1 ∪ I2 by identifying points (x, 1) with (x, 2) for all x 6= 0. Let π : I1 ∪ I2 → X be the natural projection map and let ψi : (−1, 1) → X be given by ψi (x) = π(x, i). This gives a smooth atlas on X with the transition map ψ2−1 ◦ψ1 equal to the identity map of (−1, 1)\{0} to itself. Thus X is a generalized 1−dimensional manifold. However it is not Hausdorff (why?) and hence is not a smooth manifold. Example 0.0.8. Rn , S n , RPn are Hausdorff and admit countable atlases. hence they are smooth manifolds.