Some Details and Supplements
In this chapter we give detailed proofs of some of the theorems formulated in previous chapters. In addition, we clarify the notion of vector fields along maps and their induced covariant derivatives, because this is used at various places in the book. Fo
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Some Details and Supplements
In this section we give detailed proofs of some of the theorems formulated in previous sections. In addition, we clarify the notion of vector fields along maps and their induced covariant derivatives, because this is used at various places in the book. For a convenient formulation we introduce the tangent bundle of a manifold, the prototype of a vector bundle. Applications to variations of curves will illustrate the usefulness of the concepts.
16.1 Proofs of Some Theorems Proof of Theorem 13.6 (a) First we show the following: Let DX , DY be the derivations of F(M), F(N ) respectively, belonging to the vector fields X and Y . Then X and Y are φrelated in the sense of Definition 13.6 if and only if φ ∗ ◦ DY = DX ◦ φ ∗ . To see this we note DX ◦ φ ∗ (f )|p = DX (f ◦ φ)|p = Xp (f ◦ φ) = (Tp φ · Xp )f and φ ∗ ◦ DY (f )|p = (DY f )|φ(p) = Yφ(p) f. The left-hand sides of these two equations are obviously equal if and only if X and Y are φ-related. (b) With a straightforward calculation one establishes the following: Suppose that derivations Di and Di , i = 1, 2, are related as φ ∗ ◦ Di = Di ◦ φ ∗ , then φ ∗ ◦ [D1 , D2 ] = [D1 , D2 ] ◦ φ ∗ . N. Straumann, General Relativity, Graduate Texts in Physics, 665 DOI 10.1007/978-94-007-5410-2_16, © Springer Science+Business Media Dordrecht 2013
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Some Details and Supplements
(c) Using the notation of Theorem 13.6, the result (b) implies that φ ∗ ◦ [DY1 , DY2 ] = [DX1 , DX2 ] ◦ φ ∗ D[Y1 ,Y2 ]
D[X1 ,X2 ]
and hence by (a) that [X1 , X2 ] and [Y1 , Y2 ] are φ-related.
Proof of Theorem 13.8 Only the last statement LX Y = [X, Y ] is not obvious. To show this the following remark is useful. Let f : (−ε, ε) × M −→ R be smooth (C ∞ ) and f (0, p) = 0 for all p ∈ M, then there is a C ∞ function g : (−ε, ε) × M −→ R with ∂f (0, p) = g(0, p). ∂t
f (t, p) = tg(t, p),
(16.1)
A function with these properties is
1
g(t, p) =
f (st, p) ds,
0
where the prime denotes the derivative of f with respect to the first argument. Now let f ∈ F(M) and φt be the flow of X, |t| < ε. From what has just been said we know that there is a family of C ∞ functions gt on M such that f ◦ φt = f + tgt ,
g0 = Xf.
(16.2)
The last proposition of Theorem 13.8 is equivalent to 1 [X, Y ]p = lim [Yp − Tp φt · Yq ], t→0 t
q = φ−t (p)
or 1 [X, Y ] = lim [Y − φt∗ Y ]. t→0 t For calculating the right-hand side we determine its action on f . We have, using Eq. (16.2), (φt∗ Y )p (f ) = (Tq φt · Yq )(f ) = Yq (f ◦ φt ) = Yφ−t (p) (f + tgt ), so 1 Yp − (φt∗ Y )p (f ) t→0 t
1 = lim (Yf )(p) − (Yf ) φ−t (p) − lim (Y gt ) φ−t (p) t→0 t t→0
= LX (Yf ) (p) − (Y g0 )(p) = X(Yf ) − Y (Xf ) (p). lim
This proves LX Y = [X, Y ].
16.1
Proofs of Some Theorems
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Proof of Theorem 13.9 Both statements will immediately follow from the Lemma 16.1 Two derivations of the algebra of tensor fields T (M), which preserve the type and commute with all contractions, are equal if they agree on F(M) and X (M). Before we prove this, we show that the Lemma
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