# Pfaff H.'s Neuere geometrie PDF

By Pfaff H.

By Pfaff H.

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Example text

If f ∈ C ∞ (P ) has compact support (or if Xf is complete), then XJ ∗ (f ) is complete. The global action mN : G(P ) ×P Q → Q arising in this way is compatible with the Poisson structure on Q in the sense that graph(mN ) is a Lagrangian submanifold of (G(P ), π ) × (Q, πQ ) × (Q, −πQ ),3 where π is the Poisson structure associated with the symplectic form ω on G(P ). 5) where pr G : G(P ) ×P S → G(P ) and pr S : G(P ) ×P S → S are the natural projections, see [25, 32]. On the other hand, if (Q, πQ ) is a Poisson manifold and mN is an action of a symplectic groupoid G on J : Q → P compatible with πQ in the sense just described, then J is automatically a Poisson map (this is just a leafwise version of [25, Thm.

34) shows that dJ π = −(ρM σ ∨ )∗ ; dualizing it (and using (π )∗ = −π , which holds by the ﬁrst part of the lemma), we obtain the moment map condition. 28 H. Bursztyn and M. 37. The bivector ﬁeld π is g-invariant. Proof. We have to show that LρM (v) (π (α)) = π (LρM (v) (α)) for v ∈ g, and 1-forms α. , dJ (LρM (v) (π (α))) = −(ρM σ ∨ )∗ LρM (v) (α), ∗ (LρM (v) (π (α)), C LρM (v) (α)) ∈ L. 26. 66). 33), and the fact that L is isotropic, we conclude that ([ρM (v), π (α)], LρM (v) (C ∗ α) − Lπ ∗ + d J σ (v), π (α) − iρM (v)∧π (α) (J ∗ (α) (J σ (v)) ∗ G φ )) ∈ L.

Bursztyn and M. 19), then it is a Dirac realization whose integration is mN . In order to prove the theorem, we need the following result. 8. Let (M, LM ) be a φ-twisted Dirac manifold and assume that LM is integrable. Let mN : G(LM ) ×M N → N be an action of G(LM ) on J : N → M, and assume that N is equipped with a J ∗ φ-twisted presymplectic form ωN . Then J is an f-Dirac map if and only if m∗N ωN = pr ∗N ωN + pr ∗G ω. 8) Proof. To simplify the notation, let G = G(LM ), and let us denote by A the corresponding Lie algebroid (which is just LM ).