lecture32

# lecture32 - Lecture 32 6.4 Tangent Spaces of Manifolds We...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 n-dimensional 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 one-to-one. → 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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

lecture32 - Lecture 32 6.4 Tangent Spaces of Manifolds We...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online