Let be a symplectic manifold. It said to be exact if for some one-form on .

(1) If is exact, then there is a canonical isomorphism between the v.f. and 1-forms. In particular, there exists a v.f. such that . Then we have , and , and .

(2) Suppose there exists a vector field on such that its Lie-derivative (notice the difference with ). Then Cartan’s formula says that , where . So is exact, and .

A hypersurface is of restricted contact type if is a contact form on . For example, let be a smooth function, be its Hamiltonian vector field, be a regular level set of . Then is of restricted contact type if on .

Given a time-periodic Hamiltonian on , its Hamiltonian v.f. is said to be Reeb-like if the one-form defines a contact form on the -dimensional manifold . In this case one can compute the kernel :

for all .

This is equivalent to , .

So . Then being a contact form on is equivalent to that

.

Let be a Riemannian metric on , be the Gauss curvature. Then Gauss-Bonnet Formula gives that . So if , then and hence is a flat metric. Hopf generalized this argument and proved that if has no conjugate point, then is flat.

The starting point is the geodesic flow on and the induced Ricatti equation: along a geodesic . For each , let be the solution with (the geodesic variation focuses backward at time ). By the assumption of non-conjugate point, is defined for all . Consider the limit as . This function describes the geodesic curvature of the unstable horocycle at $(x,v)$, and satisfies the following cocycle condition: . Note that the limit function $u$ is measurable, but may not be continuous.

Then we integrate over , and note that since the geodesic flow preserves the measure; .

Therefore, , and hence : the metric is flat.

Riemannian manifold , symplectic manifold , (almost) complex manifold , holomorphic tangent space . A Hermitian metric on a complex vector bundle over a smooth manifold is a smoothly varying positive-definite Hermitian form on each fiber . A (almost) Hermitian manifold is a (almost) complex manifold with a Hermitian metric on its holomorphic tangent space . Every (almost) complex manifold admits a Hermitian metric.

The real part of defines a Riemannian metric on , while the (minus) imaginary part of defines a (nondegenerate) 2-form on , the fundamental form. Any one of the three forms , , and uniquely determine the other two: , and .

If the fundamental class of a Hermitian manifold is closed, then it is called a Kahler manifold. In this case the form is called a Kahler form. A Kahler form is a symplectic form, and so Kahler manifolds are naturally symplectic.

An almost Hermitian manifold with closed is naturally called an almost Kahler manifold. Any symplectic manifold admits a compatible almost complex structure making it into an almost Kahler manifold. In holomorphic local coordinates , any Kahler form can be written as , where is a real function, the so-called Kahler potential.

As a compact manifold, admits no symplectic structure since any closed 2-form must be exact (). Note that has a well-known almost complex structure , which comes from the octonions, when the 6-sphere is viewed as the unit sphere in the 7-space of imaginary octonions. This, however, is not integrable (by the non-associativity of the octonions). An open question is whether admits a complex structure.

Kodaira and Thurston constructed a manifold that admits a symplectic form and a complex structure , which fails to be Kahler. In particular, and are not compatible. The construction is quite simple. Consider the direct product , where is the 3D Heisenberg group, be the integer lattice in , and .

Let be the natural basis of the Lie algebra , and be the natural basis of its dual . Then only non-zero bracket is , and the only non-zero differential is .

Let . Clearly the corresponding left-invariant 2-form on is nondegenerate and closed. Note that the more natural 2-form is not closed.

Let be the almost complex structure with and . Then the tensor vanishes. For example:

;

.

Therefore, the corresponding left-invariant almost complex structure on is integrable, and is a complex manifold. One can check that is not compatible with : (although compatible with ).

Moreover, the first homology group and then the first Betti number . The odd-degree Betti numbers of any Kahler manifold are ever. So is not Kahler, and there is no compatible symplectic–complex structure on .