By Kapovich M.

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We shall now extend this to a map of F‘(M’) to F‘(M) for all r, that is, to a map sending a differential form o’on M’ buck to a differential form +*(o’) on M . Recall that the linear map 4*:M , + M i ( , ) induces a dual map 4* from covectors on M i ( , ) back to covectors on M , , that is, 4*:Mi:,’, + MF‘. Regard o’as a cross sectior, p’ -+ w’(p’) E MA*‘ of T * r ( M ’ ) Then . 4*(u‘)is by definition the cross section p + 4*(0’(4(p))); that is, --f -+ ” “ q5*(u’>(ul, . . , u,) = o’(4*(u1), . .

Combining this with the remark about the integral curves of X and X ’ , we see that: The one-parameter group generated by Y permutes the integral of X with a change of parametrization if [ Y , X I = g X for some function g E F(D). Suppose now that [ Y , X I = 0. The coordinate system may be chosen so that Y = d/dx,. 3, we see that this coordinate system may be easily found whpn the explicit equations of the one-parameter group determined by Y are known. If we write X = djdx,, we must have “ ” asi a ax, axi’ O=[X,Y]=--- hence the Bi are functions Bi(x,, .

Recall that the linear map 4*:M , + M i ( , ) induces a dual map 4* from covectors on M i ( , ) back to covectors on M , , that is, 4*:Mi:,’, + MF‘. Regard o’as a cross sectior, p’ -+ w’(p’) E MA*‘ of T * r ( M ’ ) Then . 4*(u‘)is by definition the cross section p + 4*(0’(4(p))); that is, --f -+ ” “ q5*(u’>(ul, . . , u,) = o’(4*(u1), . . , 4*(ur)) for u l , . . , u, E ” T ( M ) . 10) 20 Part 1. Calculus on Manifolds Now we have the following very nice property of the exterior derivative operation : t$*(df) = dt$":(f) for f~ F(M').

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