A Homotopy Classification οf Symplectic Immersions

An application of the $h$-principle to $C^1$-isometric immersions in contact manifolds

“…Gray showed in [6] that a small perturbation of a contact structure is isomorphic to the original one by a small diffeomorphism to the underlying manifold. Then Moser proved a similar result for symplectic manifolds and his method was streamlined by Weinstein and presented in a form suitable for symplectic immersions on p. 337 of [7], while the contact case was worked out in [3]). …”

“…Both proofs of microextension in [7] and [3] follow the same route, essentially the one indicated in b) and briefly explained above by involving "continuity" of contact fields on parameters. So if we start from scratch the b)-approach is better as one avoids references to [7] or [3] but then one has to look into the matter more closely.…”

“…All this is done in a full generality in [7], but unfortunately tracking down our specific example there appears an unsurmontable task. On the other hand, fortunately the contact case can be handled directly without the use of Nash-Gromov implicit function theorem as indicated on p. 339 of [7] and elaborated in [3] for strict contact immersions. In fact, an appropriate extension scheme is indicated in §2 of [1], where contact extensions from V to V x R are achieved with suitable contact vector fields in W D V. Such a field starts at V and its orbit gives V x R c W with the contact structure equal the one coming from the projection V x R -> V. Moreover, the fields in [1] were chosen in such a way as to respect the (/i, a;)-regularity of the immersion theorem implying the surjectivity of the restriction homomorphism Qcr -► $ cr for rankT > 2{rankS + 1).…”

“…The microextension property for the sheaf $ = $ sc is proven in [3] and so we could repeat the above with the sheaf $ sc D $ scr instead of &con D ^scrj thus deriving our microextension proof from [3] instead…”