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Unformatted text preview: Chapter 6
Connection and Curvature
6.1 Aﬃne Connection 6.1.1 Covariant Derivative
Deﬁnition 6.1.1 Let M be an ndimensional manifold. An aﬃne connection is an operator
: C ∞ (T M ) × C ∞ (T M ) → C ∞ (T M )
that assigns to two vector ﬁelds X and Y a new vector ﬁeld X Y, that
is linear in both variables, that is, for any a, b ∈ R and any vector ﬁelds
X, Y and Z, • + b Z) = a X Y + b
aX+bY Z = a X Z + b Y Z
X (aY XZ and satisﬁes the Leibnitz rule, that is, for any smooth function f ∈
C ∞ ( M ) and any two vector ﬁelds X and Y,
X ( f Y) = X( f )Y + f X Y
= (d f )(X)Y + f X Y • Let xµ , µ = 1, . . . , n, be local coordinates and ∂µ be the basis of vector ﬁelds.
It will be called a coordinate frame for the tangent bundle.
• A basis of vector ﬁelds ei = eµ ∂µ , i = 1, . . . , n, for the tangent bundle T M is
i
143 144 CHAPTER 6. CONNECTION AND CURVATURE
called a frame. • We will denote partial derivatives by
∂i f = f,i
and the action of frame vector ﬁelds on functions by
ei ( f ) = eµ ∂µ = fi .
i
• For any frame the commutator of the frame vector ﬁelds deﬁnes the commutation coeﬃcients
[ei , e j ] = C k i j ek .
• The commutation coeﬃcients are scalar functions, in general.
• Theorem 6.1.1 A frame ei is a coordinate frame if and only if for any
i, j = 1, . . . , n,
[ei , e j ] = 0 .
Proof :
1. Need to show that there is a local coordinate system xi such that ei =
∂i .
2. Use the dual basis of 1forms to prove that they are exact. Deﬁnition 6.1.2 Let {ei } be a frame of vector ﬁelds and σ j be the dual
frame of 1forms. The symbols ωi jk deﬁned by
• ωi k j = σ i ( e j ek ) are called the coeﬃcients of the aﬃne connection.
• We denote
i = ei . • Then
ie j = ωk ji ek ,
diﬀgeom.tex; April 12, 2006; 17:59; p. 143 ...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry, Derivative

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