Infinitesimal Affine Automorphisms of Symplectic Connections

Conditions are given under which an infinitesimal automorphism of a torsion-free connection preserving a symplectic form is necessarily a symplectic vector field. An example is given of a compact symplectic nilmanifold admitting a flat symplectic connection and an infinitesimal automorphism that is not symplectic. On a symplectic manifold (M,Ω), a symplectic connection is a torsion-free affine connection ∇ that preserves the symplectic form Ω. This note addresses the question of when an infinitesimal affine automorphism of a symplectic connection must be an infinitesimal symplectomorphism. Recall that an infinitesimal affine automorphism of an affine connection ∇ is a vector field X such that the Lie derivative LX∇ vanishes. The automorphism group of the standard flat affine connection∇ on n-dimensional affine space is the full group of affine transformations. On the other hand, the automorphism group of a ∇-parallel metric or symplectic form is the proper subgroup of isometric affine transformations or symplectic affine transformations. In particular, the full group of affine transformations preserves the standard flat connection, but only its subgroup comprising symplectic affine transformation preserves a parallel Darboux symplectic form. Similar examples on solvable symplectic Lie groups abound. This shows that some condition is necessary to conclude that an infinitesimal automorphism of a connection preserving some tensor necessarily preserves the tensor. On the other hand, for the Levi-Civita connection D of a Riemannian metric g on a compact manifold, Theorem 4 of K. Yano’s [5] shows that an infinitesimal automorphism of D is a g-Killing field, and consequently that if a connected Lie group acts on a compact Riemannian manifold by automorphisms of the Levi-Civita connection, it acts by isometries. Related more general results are recorded in chapter VI of [3]. Yano’s theorem means that, if M is compact, the quotient of the affine automorphism group of D by the isometry group of g is discrete. For pseudo-Riemannian manifolds the analogous claims are false. In [1], the compact Lorentzian 3-manifolds admitting an affine automorphism that is not an isometry are classified. Many of the examples admit a one-parameter group of affine automorphisms that are not isometric. One expects the symplectic situation to more closely resemble the Lorentzian case than the Riemannian setting because the existence of isotropic subspaces means that holonomy need not act completely irreducibly. Here there are obtained results for infinitesimal affine automorphisms of symplectic connections analogous to those available for metric connections: Theorem. Let (M,Ω) be a 2n-dimensional symplectic manifold and let ∇ be a torsion-free affine connection preserving Ω. Suppose the compactly supported vector field X is an infinitesimal automorphism of ∇. Then X is symplectic, meaning LXΩ = 0, in any of the following situations: (1) M is noncompact. (2) dimM = 2. (3) M is compact with nonzero Euler characteristic. (4) M is compact and the linear holonomy of ∇ is irreducible. 1