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Unformatted text preview: Lecture 32 6.4 Tangent Spaces of Manifolds We generalize our earlier discussion of tangent spaces to tangent spaces of manifolds. First we review our earlier treatment of tangent spaces. Let p ∈ R n . We define T p R n = { ( p, v ) : v ∈ R n } . (6.31) Of course, we associate T p R n ∼ R n by the map ( p, v ) → v . = If U is open in R n , V is open in R k , and f : ( U, p ) → ( V, q ) (meaning that f maps U → V and p → p 1 ) is a C ∞ map, then we have the map df p : T p R n T q R k . ∼ → Via the identifications T p R n = R n and T p R k ∼ R k , the map df p is just the map = Df ( p ) : R n R k . Because these two maps can be identified, we can use the chain → rule for C ∞ maps. Specifically, if f : ( U, p ) → ( V, q ) and g : ( V, q ) → ( R , w ), then d ( g ◦ f ) p = ( dg ) q ◦ ( df ) p , (6.32) because ( Dg )( q )( Df ( p )) = ( Dg ◦ f )( p ). You might be wondering: Why did we make everything more complicated by using df instead of Df ? The answer is because we are going to generalize from Euclidean space to manifolds. Remember, a set X ⊆ R N is an ndimensional manifold if for every p ∈ X , there exists a neighborhood V of p in R N , an open set U in R n , and a diffeomorphism φ : U → V ∩ X . The map φ : U → V ∩ X is called a parameterization of X at p . Let us think of φ as a map φ : U → R N with Im φ ⊆ X . Claim. Let φ − 1 ( p ) = q . Then the map ( dφ ) q : T q R n T p R N is onetoone. → Reminder of proof: The map φ − 1 : V ∩ X → U is a C ∞ map. So, shrinking V if necessary, we can assume that this map...
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 Fall '04
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 Topology, Vector Space, Manifold, Euclidean space, Topological space, Tangent space

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