# 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)