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Unformatted text preview: Math 6455 Nov 1, 2006 1 Differential Geometry I Fall 2006, Georgia Tech Lecture Notes 14 Connections Suppose that we have a vector field X on a Riemannian manifold M . How can we measure how much X is changing at a point p ∈ M in the direction Y p ∈ T p M ? The main problem here is that there exists no canonical way to compare a vector in some tangent space of a manifold to a vector in another tangent space. Hence we need to impose a new kind of structure on a manifold. To gain some insight, we first study the case where M = R n . 0.1 Differentiation of vector fields in R n Since each tangent space T p R n is canonically isomorphic to R n , any vector field on R n may be identified as a mapping X : R n → R n . Then for any Y p ∈ T p R n we define the covariant derivative of X with respect to Y p as ∇ Y p X := ( Y p ( X 1 ) ,...,Y p ( X n ) ) . Recall that Y p ( X i ) is the directional derivative of X i at p in the direction of Y , i.e., if γ : (- , ) → M is any smooth curve with γ (0) = p and γ (0) = Y , then Y p ( X i ) = ( X i ◦ γ ) (0) = grad X i ( p ) ,Y . The last equality is an easy consequence of the chain rule. Now suppose that Y : R n → R n is a vector field on R n , p Y 7-→ Y p , then we may define a new vec- tor field on R n by ( ∇ Y X ) p := ∇ Y p X. Then the operation ( X,Y ) ∇ 7-→ ∇ X Y may be thought of as a mapping ∇ : X ( R n ) × X ( R n ) → X ( R n ), where X denotes the space of vector fields on R n . Next note that if X ∈ X ( R n ) is any vector field and f : M → R is a function, then we may define a new vector field fX ∈ ( R n ) by setting ( fX ) p := f ( p ) X p (do not confuse fX , which is a vector field , with Xf which is a function defined by Xf ( p ) := X p ( f )). Now we observe that the covariant differentiation of vector fields on R n satisfies the following properties: 1. ∇ Y ( X 1 + X 2 ) = ∇ Y X 1 + ∇ Y X 2 1 Last revised: November 14, 2006 1 2. ∇ Y ( fX ) = ( Y f ) ∇ Y X + f ∇ Y X 3. ∇ Y 1 + Y 2 X = ∇ Y 1 X + ∇ Y 2 X 4. ∇ fY X = f ∇ Y X It is an easy exercise to check the above properties. Another good exercise to write down the pointwise versions of the above expressions. For instance note that item (2) implies that ∇ Y p ( fX ) = ( Y p f ) ∇ Y p X + f ( p ) ∇ Y p X, for all p ∈ M . 0.2 Definition of connection and Christoffel symbols Motivated by the Euclidean case, we define a connection ∇ on a manifold M as any mapping ∇ : X ( M ) × X ( M ) → X ( M ) which satisfies the four properties mentioned above. We say that ∇ is smooth if whenever X and Y are smooth vector fields on M , then ∇ Y X is a smooth vector field as well. Note that any manifold admits the trivial connection ∇ ≡ 0. In the next sections we study some nontrivial examples....
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- Spring '08
- Geometry, Metric space, Manifold, γ, Parallel transport, Affine connection