Connections in Poisson geometry. I. Holonomy and invariants

By Rui Loja Fernandes. 2000

155 citations recorded by the source corpus.

We discuss contravariant connections on Poisson manifolds. For vector bun dles, the corresponding operational notion of a contravariant derivative had been introduced by I. Vaisman. We show that these connections play an important role in the study of global properties of Poisson manifolds and we use them to define Poisson holonomy and new invariants of Poisson manifolds.

Subfields: Algebra and Number Theory, Mathematical Physics, Pathology and Forensic Medicine

View Paper: https://www.semanticscholar.org/paper/d0c19c46433fdeceea54379b443001c489bfdb65

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In Algebra and Number Theory

In Mathematical Physics

In Pathology and Forensic Medicine

Citations in the corpus

Lie Algebroids, Holonomy and Characteristic Classes (2000, methodology): “For contravariant connections this was already discussed in [10, 29].”
Integrability of Lie brackets (2001, methodology): “This is explained in detail in [11, 12], and here we recall only the results we need.”
Bimodule deformations, picard groups and contravariant connections (2002, background): “Example 4.2 ([31, 14 ]) Let (M, π) be a Poisson manifold, L a line bundle over M and ∇ any (ordinary) connection on L. For f ∈ C ∞(M), let Xf denote its Hamiltonian vector field.”
Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras (2002, background): “For any pseudo-metric <, > on T P , we define a contravariant connection in the sens of Fernandes [2] and Vaisman [4] by 2 < Dαβ, γ > = #π(α).”
Integrability of Poisson Brackets (2002, background): “If (M, {·, ·}) is a Poisson manifold, then the symplectic foliation with the induced symplectic forms on the leaves gives a smooth foliation with a smooth family of symplectic forms.”
Lectures on integrability of Lie brackets (2006, unknown): “A geometric approach to the theory of connections on Lie algebroids was given by Fernandes in [24, 25].”