Compatible almost complex structure ($J$)

An almost complex structure $J$ on a symplectic manifold $(M,\omega )$ is compatible with $\omega$ whenand $g_J(u,v)=\omega (u,Jv)$ is a positive-definite inner product. Every symplectic vector bundle admits a compatible almost complex structure.

The metric construction of a compatible almost complex structure is continuous in the auxiliary Riemannian metric, and recovers $J$ when applied to $g_J=\omega (\cdot ,J\cdot )$. Interpolating $g_J$ linearly to a fixed auxiliary metric therefore contracts the space of compatible almost complex structures. In particular it is nonempty and path connected.

A compatible triple consists of a symplectic form, a compatible almost complex structure, and the Riemannian metric $g(u,v)=\omega (u,Jv)$. Equivalently $g(Ju,v)=\omega (u,v)$.

Choose an auxiliary Riemannian metric $h$ and define the skew-adjoint bundle map $A$ by $\omega (u,v)=h(Au,v)$. Thenis a compatible almost complex structure. The positive square root is defined fiberwise by the spectral theorem for normal operators, and its smooth dependence on $A$ makes $J$ smooth.

If a compatible almost complex structure is prescribed along a closed symplectic submanifold $N\subset M$ and preserves $TN$, it extends to one on $M$. Choose a compatible structure separately on $TN$ and its symplectic normal bundle, use the associated metric along $N$, extend that metric to $M$, and apply the metric construction of a compatible almost complex structure.