Symplectic connection

An affine connection on a smooth manifold $ M $ of dimension $ 2n $ with a non-degenerate $ 2 $- form $ \Phi $ that is covariantly constant with respect to it. If the affine connection on $ M $ is given by the local connection forms

$$ \omega ^ {i} = \ \Gamma _ {k} ^ {i} dx ^ {k} ,\ \ \mathop{\rm det} \| \Gamma _ {k} ^ {i} \| \neq 0, $$

$$ \omega _ {j} ^ {i} = \Gamma _ {jk} ^ {i} \omega ^ {k} $$

and

$$ \Phi = \ a _ {ij} \omega ^ {i} \wedge \omega ^ {j} ,\ \ a _ {ij} = - a _ {ji} , $$

then the condition that $ \Phi $ be covariantly constant can be expressed in the form

$$ da _ {ij} = \ a _ {kj} \omega _ {i} ^ {k} + a _ {ik} \omega _ {j} ^ {k} . $$

The $ 2 $- form $ \Phi $ defines a symplectic (or almost-Hamiltonian) structure on $ M $ that converts every tangent space $ T _ {x} ( M) $ into a symplectic space with the skew-symmetric scalar product $ \Phi ( X, Y) $. A symplectic connection can also be defined as an affine connection on $ M $ which preserves this product under parallel transfer of vectors. In every $ T _ {x} ( M) $ one can choose a frame such that

$$ \Phi = 2 \sum _ {\alpha = 1 } ^ { n } \omega ^ \alpha \wedge \omega ^ {n + \alpha } . $$

The set of all such frames forms a principal fibre bundle over $ M $, whose structure group is the symplectic group. A symplectic connection is just a connection in this principal fibre bundle. There are manifolds $ M $ of even dimension on which there is no non-degenerate globally defined $ 2 $- form $ \Phi $ and, consequently, no symplectic connection. However, if $ \Phi $ exists, a symplectic connection with respect to which $ \Phi $ is covariantly constant is not uniquely determined.

References

[1]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)

References

[a1]	R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin/Cummings (1978)