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3' There is no guarantee that this integral will is not well defined. On the other hand, since 3' is given by the integral of a local expression, the formal expression for d 3' (II) will also 9 be given by the integral of a local expression; in fact, a bit of computation will show that d:J (II) = 9 1 M t T • lI(vol) 9 where where R is the scalar curvature. Now if then the integral in the definition of II E Zo d:J (II) makes sense since the integrand has compact support. Thus, in interpreting (S) 9 we can, for each fixed II E Zo replace, in the definition of 3', integration over M by integration over a sufficiently large compact region.

The group map of P G acts on P by right multiplication. pa Let on g denote the Lie algebra of P G. which is vertical and satisfies form on P is a -1 Ra *Sp SEg = (Ada S)p g - valued linear differential form, = denote the a gives rise to a vector field, for all a E G. A connection 8, which satisfies for all Ada a EG on the right-hand side of the first equation means that we apply The (81. (8I(Sp) the vector field S. Evaluating The second equation asserts that to each point of Ada in the second equation denotes the value of the one form function on P.

Similarly, it may be that fJ. will Zo which contains all the 3< is not defined on all of X, and yet (S) makes good sense where both sides are regarded as linear functions on Ta~ ZO. Let us illustrate what we have in mind for the case of general relativity. Let M be a (four dimensional) manifold. X be the space of all pseudo- + ---). Let Q = DiffO(M) be the group of all Riemannian metrics on M diffeomorphisms of of compact support. ) The action of one: a diffeomorphism sends the metric C(l g the preceeding section, the "tangent space" to into the metric X at -1* g.