The geometry of recursion operators

Details The geometry of recursion operators The geometry of recursion operators

2007-03-12

arxiv.org/abs/math/0703348v2

Commun. Math. Phys. 280 (2008), 737--749

Mathematics

Symplectic Geometry

cosmetic changes only; to appear in Comm. Math. Phys

Scientific paper

10.1007/s00220-008-0477-6

We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser's theorem for simultaneous isotopies of two families of symplectic forms. We also consider the geometric structures defined by pairs and triples of symplectic forms for which the squares of the intertwining endomorphisms are plus or minus the identity. For pairs of forms we recover the notions of symplectic pairs and of holomorphic symplectic structures. For triples we recover the notion of a hypersymplectic structure, and we also find three new structures that have not been considered before. One of these is the symplectic formulation of hyper-Kaehler geometry, which turns out to be a strict generalization of the usual definition in terms of differential or Kaehler geometry.

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