Moduli space of symplectic connections of Ricci type on $T^{2n}$; a formal approach

Details Moduli space of symplectic connections of Ricci type on $T^{2n}$; a formal approach Moduli space of symplectic connections of Ricci type on $T^{2n}$; a formal approach

2002-01-18

arxiv.org/abs/math/0201167v2

Mathematics

Symplectic Geometry

LaTeX, 19 pages. Removed claim to handle all symplectic structures on the torus

Scientific paper

We consider analytic curves $\nabla^t$ of symplectic connections of Ricci type on the torus $T^{2n}$ with $\nabla^0$ the standard connection. We show, by a recursion argument, that if $\nabla^t$ is a formal curve of such connections then there exists a formal curve of symplectomorphisms $\psi_t$ such that $\psi_t\cdot\nabla^t$ is a formal curve of flat invariant symplectic connections and so $\nabla^t$ is flat for all $t$. Applying this result to the Taylor series of the analytic curve, it means that analytic curves of symplectic connections of Ricci type starting at $\nabla^0$ are also flat. The group $G$ of symplectomorphisms of the torus $(T^{2n},\omega)$ acts on the space $\E$ of symplectic connections which are of Ricci type. As a preliminary to studying the moduli space $\E/G$ we study the moduli of formal curves of connections under the action of formal curves of symplectomorphisms.

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