116 XVI DIFFERENTIAL MANIFOLDS

They therefore define (16.133) a fibration (E' ® E", B, jx), and it is clear
that E' ® E" is thus endowed with a vector bundle structure satisfying the
condition on the sections stated above. Conversely, suppose that E' ® E" is
endowed with a vector bundle structure satisfying this condition. With the
notation used above, let (O be a basis of F; and «,.) a basis of F£ . Then the
sections bt-xp&b, e^) (resp. b*-*(p'i (A, <,.)) form a frame (s^) (resp. ($£,)) of
E' (resp. E") over Ua, and the condition on the sections shows that the
$ii ® </ form SL frame for E' ® E" over Ua . Let <pa : Ua x (F; ® F£) -* n~ \Ua)
be the diffeomorphism corresponding to this frame. It is then immediately
verified that for any two indices a, j8 the transition diffeomorphisms

(U. n U,) x (F; ® F^) -> (U. n U,) x (FJ ® FJ)

corresponding to the <pa are precisely the mappings (1 6.1 6.1..2). This establishes
the uniqueness of the vector bundle structure on E' ® E".

It follows moreover that if U is any open set in B such that Tr""1^) and
n'^CU) aretrivializable, then F(U, E' ® E") is an $ (U; R) module isomorphic

If F', F" are two vector bundles over B and if u' : E' -> F' and u" : E" -> F"
are two B-morphisms, then one shows in the same way that there exists a
unique B-morphism u' ® u" : E' ® E" -> F' ® F" such that, if $' and s" are
sections of E', E", respectively, over an open set U in B, and if w(j , w(J and
(u' ® w")u are the restrictions of u', u" and u' ® u" to Ti'^U), Tc''""1^), and
H~l(U), respectively, then we have

(U' ® W'% ° («' ® SW) = («{, o $') ® (lift o $").

The restriction of u1 ® w" to a fiber (E; ® E")b = E; ® E'J is the tensor product
M^ ® wj of the linear mappings u'b : E'b -> F^ and w^ : Eb ->• F'b' . If i/ and w" are
B-isomorphisms, then so is u' ® u" '.

The proofs of the corresponding assertions for E; ® E" are analogous
and simpler. For each open U c B such that Tt'"1^) and Tc"""1^) are
trivializable, F(U, E' © E") is an <f (U ; R)-module isomorphic to

r(u,E')er(u,E").

There are canonical injective B-morphisms/ : E' -* E' ® E",/ : E" -+ E' © E"
and canonical surjective B-morphisms p' : E' © E" -» E', p" : E' © Ew ~> E"
such that, with the same notation as above,

Furthermore, if F is a vector bundle over B and if u' : E' -+ F and u" : E" -» F
are B-morphisms, then there exists a unique B-morphism w : E' © E" -+ Fctions of the fibrations