Affine Connections
In this central chapter 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. This allows us to introduce many import
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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.
15.1 Covariant Derivative of a Vector Field The following definition abstracts certain properties of the directional derivative in Rn . Definition 15.1 An affine (linear) connection or 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. Such a mapping ∇ is called a covariant derivative. We show that ∇ can be localized. Lemma 15.1 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, as in Sect. 14.3, a function h ∈ F(M) with h(p) = 0 and h = 1 on M \ U . From hY = Y it follows that ∇X Y = ∇X (hY ) = X(h)Y + h∇X Y . This vanishes at p. The statement about X follows similarly. N. Straumann, General Relativity, Graduate Texts in Physics, 631 DOI 10.1007/978-94-007-5410-2_15, © Springer Science+Business Media Dordrecht 2013
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Affine Connections
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 there ˜ Y˜ ∈ X (M) which coincide with X, respectively Y on an open exist vector fields X, neighborhood V of p (see the continuation Lemma 14.1 of Sect. 14.3). We then set (∇|U )X Y |q = (∇X˜ Y˜ )|q ,
for q ∈ V .
˜ Y˜ ∈ X (M). By Lemma 15.1, the right-hand side is independent of the choice of X, ∇|U is obviously an affine connection on U . We show that (∇X Y )p depends on X only via Xp , whence ∇v Y is well-defined for v ∈ Tp M. Lemma 15.2 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 ∂/∂x i with ξ i ∈ F(U ), where ξ i (p) = 0. Then, using the previous Lemma, (∇X Y )p = ξ i (p)(∇∂/∂x i Y )p = 0. Definition 15.2 Set, relative to a chart (U, x 1 , . . . , x n ), ∂ ∂ k = Γ ij . ∇∂/∂x i ∂x j ∂x k
(15.1)
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 ∂ ∂x k ∂ ∂ ¯ cab ¯ cab ∇∂/∂ x¯ a = Γ (15.2) = Γ ∂ x¯ b ∂ x¯ c ∂ x¯ c ∂x k and on the other hand the axioms imply j ∂ ∂x ∂ ∇∂/∂ x¯ a = ∇ i a i (∂x /∂ x¯ )(∂/∂x ) ∂ x¯ b ∂ x¯ b ∂x j ∂x i ∂x j k ∂ ∂ ∂x j ∂ = a Γ + ∂ x¯ ∂ x¯ b ij ∂x k ∂x i ∂ x¯ b ∂x j =
∂ 2xj ∂ ∂x i ∂x j k ∂ Γ ij k + a b j . a b ∂ x¯ ∂ x¯ ∂x ∂ x¯ ∂ x¯ ∂x
Comparison with (15.2) results in ∂x k c ∂x i
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