Glanon groupoids - Mathematisches Institut - GWDG

Lemma 3.28. Let J : P Γ → P Γ be a multiplicative map. Then
is a vector bundle morphism if and only if
A(J ) : P A → P A
is a vector bundle morphism.
J : P Γ → P Γ
Since the map N J : P Γ × Γ P Γ → P Γ is then also a Lie groupoid morphism by Theorem
3.26, we can also consider
A(N J ) : P A × A P A → P A
defined by
A(P Γ ) × A A(P Γ )
A(N J )
A(P Γ )
Σ 2 Γ
P A × A P A .
Σ Γ
P A
A(N J )
The main result of this section is the following.
Theorem 3.29. Let J : P Γ → P Γ be a vector bundle homomorphism and a Lie groupoid
morphism. Then
(a)
(b)
The map A(J ) is 〈· , ·〉 A -orthogonal if and only if J is 〈· , ·〉 Γ -orthogonal and,
in that case,
A(N J ) = N A(J ) . (20)
For the proof, we need a couple of lemmas.
Definition 3.30. Let M be a smooth manifold and ι : N ↩→ M a submanifold of M.
(a)
(b)
(c)
A section e N =: X N + α N is ι-related to e M = X M + α M if ι ∗ X N = X M | N and
α N = ι ∗ α M . We write then e N ∼ ι e M .
Two vector bundle morphisms J N : P N → P N and J M : P M → P M are said to
be ι-related if for each section e N of P N , there exists a section e M ∈ P M such
that e N ∼ ι e M and J N (e N ) ∼ ι J M (e M ).
Two vector bundle morphisms N N : P N × N P N → P N and N M : P M × M P M →
P M are ι-related if for each pair of sections e N , f N ∈ Γ(P N ), there exist sections
e M , f M ∈ Γ(P M ) such that e N ∼ ι e M , f N ∼ ι f M and N N (e N , f N ) ∼ ι
N M (e M , f M ).
Lemma 3.31. Let M be a manifold and N ⊆ M a submanifold. If e N , f N ∈ Γ(P N ) are
ι-related to e M , f M ∈ Γ(P M ), then e N , f N ∼ ι e M , f M , for the Dorfman and the Courant
bracket.
Proof. This is an easy computation, see also [33].
Lemma 3.32. If M is a manifold, N ⊆ M a submanifold and J N : P N → P N and
J M : P M → P M two ι-related vector bundle morphisms. Then the generalized Nijenhuis
tensors N JN and N JM are ι-related.
□
Proof. This follows immediately from Lemma 3.31 and the definition.
□
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