Lemma 5

Tag 0ADD, permanent. Cite it as [Bourbaki, Tag 0ADD].

Lie Groups and Lie Algebras, chapter III, § 4. Passage from Lie algebras to Lie groups

Let $X$ be a manifold of class $C^r$ , $F$ and $F'$ vector bundles of class $C^r$ with base space $X$ and $\phi$ a morphism of $F$ into $F'$ . For all $x \in X$ , let $S_x$ be the set of

(a, \phi(a)) \in F_x \oplus F'_x

for

a \in F_x

. Then the union

S

of the

S_x

is a vector subbundle of

F \oplus F'

Let $\theta$ and $\theta'$ be the mappings of $F \oplus F'$ into itself defined as follows: if $(u, v) \in F_x \oplus F'_x$ , then

\theta(u, v) = (u, v + \phi(u)), \qquad \theta'(u, v) = (u, v - \phi(u)).

By Differentiable and Analytic Manifolds, R, 7.7.1,

\theta

and

\theta'

are morphisms of

F \oplus F'

into itself. Clearly

\theta \circ \theta' = \theta' \circ \theta = \mathrm{Id}_{F \oplus F'}

. Hence

\theta

and

\theta'

are automorphisms of

F \oplus F'

. Therefore,

S = \theta(F \oplus \{0\})

is a vector subbundle of

F \oplus F'

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Cites

Nothing.