Affine Connections
In this section we introduce an important additional structure on differentiable manifolds, thus making it possible to define a “covariant derivative” which transforms tensor fields into other tensor fields.
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In this section we introduce an important additional structure on differentiable manifolds, thus making it possible to define a “covariant derivative” which transforms tensor fields into other tensor fields.
13.1 Covariant Derivative of a Vector Field The following definition abstracts certain properties of the directional derivative in Rn . Definition 13.1. An affine (linear) connection or covariant differentiation on a manifold M is a mapping ∇ which assigns to every pair X, Y of C ∞ vector fields on M another C ∞ vector field ∇X Y with the following properties: (i) ∇X Y is R-bilinear in X and Y ; (ii) If f ∈ F(M ), then ∇ f X Y = f ∇X Y , ∇X (f Y ) = f ∇X Y + X(f )Y . We show that ∇ can be localized. Lemma 13.1. Let ∇ be an affine connection on M and let U be an open subset of M . If either X or Y vanishes on U , then ∇X Y also vanishes on U . Proof. Suppose Y vanishes on U . Let p ∈ U . Choose a function h ∈ F(M ) with h(p) = 0 and h = 1 on M \ U . From h Y = Y it follows that ∇X Y = ∇X (hY ) = X(h)Y + h∇X Y . This vanishes at p. The statement about X follows similarly. This lemma shows that an affine connection ∇ on M induces a connection on every open submanifold U of M . In fact, let X, Y ∈ X (U ) and p ∈ U . Then ˜ Y˜ ∈ X (M ) which coincide with X, respectively there exist vector fields X, Y on an open neighborhood V of p (see the continuation Lemma 12.1 of Sect.12.3). We then set N. Straumann, General Relativity © Springer-Verlag Berlin Heidelberg 2004
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13 Affine Connections
(∇|U )X Y |q = (∇X˜ Y˜ )|q ,
for q ∈ V .
˜ Y˜ ∈ By Lemma 13.1, the right-hand side is independent of the choice of X, X (M ). ∇|U is obviously an affine connection on U . We show that (∇X Y )p depends on X only via Xp , whence ∇v Y is welldefined for v ∈ Tp M . Lemma 13.2. Let X, Y ∈ X (M ). If X vanishes at p, then ∇X Y also vanishes at p. Proof. Let U be a coordinate neighborhood of p. On U , we have the representation X = ξ i ∂/∂xi with ξ i ∈ F(U ), where ξ i (p) = 0. Then (∇X Y )p = ξ i (p)(∇∂/∂xi Y )p = 0. Definition 13.2. Set, relative to a chart (U, x1 , . . . , xn ), ∂ ∂ k = Γ ij . ∇∂/∂xi ∂xj ∂xk
(13.1)
The n3 functions Γ kij ∈ F(U ) are called the Christoffel symbols (or connection coefficients) of the connection ∇ in the given chart. The connection coefficients are not the components of a tensor. Their transformation properties under a coordinate transformation to the chart (V, x ¯1 , . . . , x ¯n ) are obtained from the following calculation: On the one hand we have k ∂ ¯ c ∂ = Γ¯ c ∂x ∂ ∇∂/∂ x¯a = Γ (13.2) ab ab ∂x ¯b ∂x ¯c ∂x ¯c ∂xk and on the other hand the axioms imply j ∂ ∂x ∂ = ∇ ∂xai ∂ ∇∂/∂ x¯a ∂x ¯ ∂xi ∂x ¯b ∂x ¯b ∂xj j ∂x ∂ ∂ ∂xi ∂xj k ∂ Γ + = ∂x ¯a ∂ x ¯b ij ∂xk ∂xi ∂ x ¯b ∂xj ∂ 2 xj ∂ ∂xi ∂xj k ∂ Γ + . = ∂x ¯a ∂ x ¯b ij ∂xk ∂x ¯a ∂ x ¯b ∂xj Comparison with (13.2) results in
or
∂xk ¯ c ∂xi ∂xj k ∂ 2 xk = Γ + Γ ab ij ∂x ¯c ∂x ¯a ∂ x ¯b ∂x ¯a ∂ x ¯b
(13.3)
¯c k ¯c ∂xi ∂xj ∂ x ∂ 2 xk ∂ x Γ + . Γ¯ cab = ∂x ¯a ∂ x ¯b ∂xk ij ∂ x ¯a ∂ x ¯b ∂xk
(13.4)
Conversely, if for every chart there exist n3 f
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