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19 THE COVARIANT EXTERIOR DIFFERENTIAL 337

for all a e A and y e F, and

'</>J (*), L?O))

for all x e E and y e F.

(b) Show that for each pair (p, g) of integers ]>0 there exists a unique mapping

LS : E®" ® F®« -+• E'®^p <g) F'««
such that

for a e A and zeE®^p(x)F®«, where pj is the canonical extension of /oj and p? to the
tensor product);

L{«(« ® y) = Lj(n)
for w e E®^p ® F®« and v e E®^r ® F®⁸; and

'(/o«^p(«), LJ(»*))

for w e E®^p ® F®« and u* e E®« ® F®^p, where <E> and €>' are the canonical extensions of
the bilinear forms to the tensor product. (Follow the proof of (17.14.6).)

2. Let E be a vector bundle over M, and let Diff^E) denote the «f(M)-module of differen-
tial operators of order <J1 from E to E. Show that every <^(M)-linear mapping X\-*P_Xof ^J(M) into DifT₁(E) such that P_x - (aY) * (0* - <r)Y + o(P_x • Y) for all a e ^(M)
and all C°° -sections Y of E, is of the form X\- > V^ relative to a unique connection in E.
(Show first that if X vanishes in an open set U, then P_x\ U « 0.)

3. Generalize the result of (17.18.1) to linear connections in an arbitrary vector bundle E
over M. Consider in particular a C°°-section G* of the dual E* of E, and associate with
it the scalar function on E given by u_xH>H(u_x) = <(u_x> <•*(#)>. Show that

4. With the notation of Section 16.19, Problem lls, how that a linear connection in E is a
mapping C : E X_BT(B)-^T(E) such that /uo C= 1_EXBT(B), and such that C is a
bundle morphism of E x _B T(B) into T(E) both as bundles over E and as bundles over
T(B).

5. With the notation of (17.18.4), show that V(cjU) « cj(VU) for any contraction cj.

19. THE COVARIANT EXTERIOR DIFFERENTIAL

(17.19.1) The formula (17.15,3.6) enables us to calculate at each point x e M
the value (d(a)(x), H^A k_x> of the exterior differential of a l-form co by
considering two C°° vector fields X, Y on M, such that X(x) = h_x and
Y(x) = k^, and calculating the value of the right-hand side of (17.15.3.6) at



	19 THE COVARIANT EXTERIOR DIFFERENTIAL 337 for all a e A and y e F, and

	'</>J (), L?O)) for all x e E and y e* F. (b) Show that for each pair (p, g) of integers ]>0 there exists a unique mapping LS : E®" ® F®« -+• E'®^p <g) F'«« such that

	for a e A and zeE®^p(x)F®«, where pj is the canonical extension of /oj and p? to the tensor product);

	L{«(« ® y) = Lj(n) for w e E®^p ® F®« and v e E®^r ® F®⁸; and '(/o«^p(«), LJ(»*))

	for w e E®^p ® F®« and u* e E®« ® F®^p, where <E> and €>' are the canonical extensions of the bilinear forms to the tensor product. (Follow the proof of (17.14.6).) 2. Let E be a vector bundle over M, and let Diff^E) denote the «f(M)-module of differen- tial operators of order <J1 from E to E. Show that every <^(M)-linear mapping X\-P_Xof ^J(M) into DifT₁(E) such that P_x - (aY)* * (0* - <r)Y + o(P_x • Y) for all a e ^(M) and all C°° -sections Y of E, is of the form X\- > V^ relative to a unique connection in E. (Show first that if X vanishes in an open set U, then P_x\ U « 0.) 3. Generalize the result of (17.18.1) to linear connections in an arbitrary vector bundle E over M. Consider in particular a C°°-section G* of the dual E* of E, and associate with it the scalar function on E given by u_xH>H(u_x) = <(u_x> <•*(#)>. Show that

	4. With the notation of Section 16.19, Problem lls, how that a linear connection in E is a mapping C : E X_BT(B)-^T(E) such that /uo C= 1_EXBT(B), and such that C is a bundle morphism of E x _B T(B) into T(E) both as bundles over E and as bundles over T(B). 5. With the notation of (17.18.4), show that V(cjU) « cj(VU) for any contraction cj.

	19. THE COVARIANT EXTERIOR DIFFERENTIAL (17.19.1) The formula (17.15,3.6) enables us to calculate at each point x e M the value (d(a)(x), H^A k_x> of the exterior differential of a l-form co by considering two C°° vector fields X, Y on M, such that X(x) = h_x and Y(x) = k^, and calculating the value of the right-hand side of (17.15.3.6) at