By Norbert Straumann

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In addition, 1'/ is a volume element. 19) corresponding to g. We shall now use 1] to associate with each form WE Ap(M) another form *W E An-p(M). We consider a positive local coordinate system (Xl, ... , Xl and write W in the form W =, 1 p. . Wh ... i. dXl' /\ ... /\ dXl•. 20) (Wi, ... i. is totally antisymmetric). 19) we have 1'/i,... i. = vrgr eh... 22) where + 1 if i], ... , in is an even permutation of (1, ... , n) { ei, ... i. = - b ~~h::~;i~:. in is an odd permutation of (1, ... , n) 38 4.

3 Derivations and Antiderivations Let Mbe an n-dimensional differentiable manifold and n A(M) = EEl Ap(M) p=O be the graded 1R-algebra of exterior differential forms on M. 2: A mapping 0: A (M) ---t A (M) is called a derivation (antiderivation) of degree k E 7L, provided (i) 0 is 1R-linear (ii) O(Ap(M») c Ap+kCM) for p = 0, 1, ... , n (iii) O(rx A /J) = Orx A P + rx A 0P, rx, PEA (M) (Leibniz rule) [O(rxA/J)=OrxAP+(-I)prxAOP, rxEAp(M), PEA(M) ("anti-Leibniz rule")]. 3: If 0 and 0' are antiderivations of degree k and k' on A (M), then 0 0 0' + 0' 00 is a derivation of degree k + k'.

Y on an open neighborhood of p (see the continuation lemma of Sect. 3). We then set (V I U)x Y = (Vi' Y) I U. 1, the right hand side is independent of the choice of i, V I U is obviously an affine connection on U. Y. 48 5. 2: Let X, Y E ~(M). If X vanishes at p, then Vx Y also vanishes at p. Proof: Let U be a coordinate neighborhood of p. On U, we have the representation X=~iO/OXi, where ~i(p) = ~iEY(U), O. Then (Vx Y)p = e(P)(Va/axi Y)p = O. We define, re1ati ve to a chart (U, . k X l , ... , X n), k VO/axi(O/OxJ)=rijO/Ox .