By Alexander J. W.

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D);eJ (d(X. '/ a... " = 1Ja... de 8s[s ••• 8 t t 8"olXo; u 1 = n- 1Ja ... de X";". 37) twice. 8 holds for any vector field X; this is Gauss' theorem. Note that the orientation on ~ for which this theorem is valid is that given by the normal form 1) such that (n, X) is positive if X is a vector which points out of~. If the metric g is such that gOOnanb is negative, the vector gabnb will point into ~. 9 Fibre bundles Some of the geometrical properties of a manifold JI can be most easily examined by constructing a manifold called a fibre bundle, which is locally a direct product of JI and a suitable space.

Where no confusion can arise, we will denote the bundle simply by G. In general, the inverse image 17-1(P) of a point p e JI need not be homeomorphic to 17-1 (q) for another point qeJl. 91 is some manifold and the projection 17 is defined bYl7(P, v) = p for allpeJl, ved. 91 as the real line Rl, one constructs the cylinder 0 2 as a product bundle over 8 1 • A bundle which is locally a product bundle is called a fibre bundle. ,{w) is a diffeomorphism ljr: l7-l(~) -7 ~ x F. Since JI is paracompact, we can choose a locally finite covering of JI by such open sets ~...

Obtained by joining smoothly part of the curve y sin (l/x) to the curve FIGURE = ((y,O); - C() < y < I}. A'). Thus we can define a map 9* ofT~(p)toT~(9(P» for any r. , (¢>-l )*ljB, ¢>*Xl .... , ¢>*x,>! ¢(p) for any XieTp , ljieT*p. This map of tensors of type (r. g. the contraction of ¢>* T is equal to ¢>* (the contraction of T). 5) by placing extra structure on the manifold. 4] 25 EXTERIOR DIFFERENTIATION The exterior differentiation operator d maps r-form fields linearly to (r+ 1)-form fields.