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Let p be an n-form. If = CS= A i jq. for 1 Ii I n, then AVlr . . , V,) = (det 4pW1,. . , W . The proof is a standard combinatorial argument. TENSOR DERIVATIONS Previous sections have dealt with tensor algebra; we now consider some tensor calculus. 11. Definition. A tensor derivation 9 on a smooth manifold M is a set of R-linear functions 93 = 9;: q M ) +q ( M ) (r 2 0, s 2 0) such that for any tensors A and B : 9 ( A @ B) = 9 A @ B + A @ 9 B , (2) 9 ( C A ) = C ( 9 A )for any contraction C. (1) Thus 3 is R-linear, preserves tensor type, obeys the usual Leibnizian product rule, and commutes with all contractions.

But by (c), dnl 7;P,4)Mis an isomorphism. Thus 7;,,$4 n T P , 4 ) N= 0. The result then follows by linear algebra, since by (c) the sum of the dimensions of rn these two subspaces is dim(M x N ) . To relate the calculus of M x N to that of its factors the crucial notion is that of lifting, as follows. Some Special Manifolds 25 Iff E g ( M ) the lft off to M x N is f = f n E g ( M x N ) . If x E Tp(M)and q E N then the l f t I of x to ( p , q ) is the unique vector in qp,q)(M)such that d n ( I ) = x.

B) By considering a subset of R2 shaped like a figure 8, show that an immersed submanifold need not have the induced topology and that Corollaries 29 and 30 both fail for immersed submanifolds. 16. Let $ be the flow of V E X(M). (a) If $,(p) = p for a sequence of t-values approaching zero, then V, = 0 (hence $,(p) = p for all t). (b) If an integral curve a : [0, co) -+ M of V is extendible, with endpoint q, then % = 0. 17. (a) Let a E 1 c R. Iff E % ( I ) and f(a) = 0 show that there exists a function g E S(I)such that f ( s ) = (s - a)g(s) for all s E 1.

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