Glanon groupoids - Mathematisches Institut - GWDG

The following corollary is immediate.
Corollary 3.35. Let J : P Γ → P Γ be a vector bundle morphism and a Lie groupoid
morphism. Then
N A(J ) = 0 if and only if N J = 0.
3.8. Proof of the integration theorem. Proof of Theorem 3.11
By Lemma 3.28, the map A(J ) : P A → P A is a vector bundle morphism. Since J 2 =
− Id PΓ , we have using Lemma 3.21:
By Corollary 3.35 and Theorem 3.29, we get
(A(J )) 2 = − Id PA .
N A(J ) = 0
and
A(J ) is 〈· , ·〉 A -orthogonal.
Since
A(P Γ )
A(J )
A(P Γ )
is a Lie algebroid morphism and
T M ⊕ A ∗
T M ⊕ A ∗
A(P Γ )
Σ| A(PΓ )
P A
T M ⊕ A ∗
T M ⊕ A ∗
is a Lie algebroid isomorphism over the identity, the map
A(J ) = Σ| A(PΓ ) ◦ A(J ) ◦ ( −1
Σ| A(PΓ ))
= Σ|A(PΓ ) ◦ A(J ) ◦ Σ −1 | PA ,
P A
A(J )
P A
is a Lie algebroid morphism.
T M ⊕ A ∗
For the second part, consider the map
T M ⊕ A ∗
A J := Σ −1 | PA ◦ J A ◦ Σ| A(PΓ ) : A(P Γ ) → A(P Γ ).
Since J A : P A → P A is a Lie algebroid morphism, A J is a Lie algebroid morphism and
there is a unique Lie groupoid morphism J : P Γ → P Γ such that A J = A(J ). By Lemma
3.28, J is a morphism of vector bundles.
We get then immediately J A = A(J ). Since J 2 A = − Id A, we get A(J 2 ) = − Id A =
A(− Id PΓ ) and we can conclude by Theorem 3.29.
This concludes the proof of Theorem 3.11.
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