122 XVI DIFFERENTIAL MANIFOLDS

of T(B) (resp. T(F)) over U (resp. W). Identifying T(X) with T(B) x T(F),
we have a frame of T(X) over U x W, obtained by taking the vector fields
(b'9 0^(7^0, 0) and (4', z')^(0, Z/z'))- For each x' = (&', z') e U x W,
the vectors (0, Z/z')) form a basis of V^, which proves our assertion (cf.
Section 16.19, Problem 4).

(16.17.2) With the hypotheses and notation of (16.17.1), let E'b = Eb/E'b for
all b e B, and let E" be the disjoint union of the sets E£ as b runs through B.
Let n": E" -> B be the mapping which sends each element of E'b to 6, and let
p : E -» E" be the mapping whose restriction to Eb is the canonical mapping
Ed -* El, for each b e B. Then there exists on E" a unique structure of a vector
bundle over B with projection it", such that/? is a morphism of vector bundles.
For if U is an open set in B satisfying condition (2) of (16.17.1), and if Fy is
the vector subspace of Ey spanned by $m+i(y),.. •, sM(y), for each y e U, then
it is clear that the union F of the spaces Fy f or y e U is a subbundle of n ~~1 (U),
and the restriction q ofp to F must be an isomorphism of F onto n"~l(\J\ in
view of (16.12.2.1). From this follows the uniqueness of the bundle structure
on E". On the other hand, if we put $£(3;) = p(sk(y)) f°r m + 1 =s k =* n and all
y e U, it follows from above that ($JJ,+1,..., sj) must be & frame of E" over
U. The existence of a vector bundle structure on E" possessing these frames
is then verified by the same method as in (16.16.1), and we leave the details
to the reader.

The vector bundle E" thus constructed is denoted by E/E' and is called the
quotient of E by the vector subbundle E', It is clear that the canonical mor-
phism p: E -+E" is a submersion (16.12.2.1).

(16.17.3) With the notation and hypotheses of (16.17.1) and (16.17.2), there
exists a morphism r: E'' -*E such that p o r = 1E». Consider a locally finite
denumerable open covering (Ua) of B such that each Ua satisfies condition
(2) of (16.17.1). Let Ea = TiT1^), E£ = Ti""1^), and let p,: Ea -> EJ be the
restriction of p; then it is immediately clear that there exists a morphism
ra: EJ -»Ea such that pK o ra = 1ESJ. Let (/a) be a partition of unity subordinate
to the covering (Ua) and consisting of C°°-functions (16.4.1). For each index
a, let r^: E" -» E be the morphism whose restriction to E'6' is/a(i)(ra | E^) when
b e Ua, and 0 when b $ Ua. Then the morphism r = J] r^ has the required
property. a

It follows immediately that the morphism j + r:E'® E" ~* E is an iso-
morphism; F = r(E/') is a subbundle of E, such that E; and Fb are supple-
mentary subspaces of Eb for each b e B. Every vector subbundle of E with
this property is called a supplement of E' in E.) such that Un, P(C) © co£(Un, n_p(C)) is trivializable as a real-analytic bundle.