Let $M$ be a $2n$-dimensional symplectic manifold. A question: are there special terms for isotropic submanifolds of $M$ of dimensions $<n$ (i.e., isotropic submanifolds that are not Lagrangian) and for coisotropic submanifolds of $M$ of dimensions $>n$ (i.e., coisotropic submanifolds that are not Lagrangian)? Of course, the terms “strictly isotropic” and “strictly coisotropic” come to mind, but they are already used to mean other things, see e.g. R.Baer, Linear Algebra and Projective Geometry, second ed., Dover Publ., Mineola, NY, 2005; M.S.Borman, F.Zapolsky, Quasimorphisms on contactomorphism groups and contact rigidity, Geom. Topol. 19 (2015) 365-411.
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$\begingroup$ I think I've seen "non-maximally isotropic" somewhere. $\endgroup$– AlexArvanitakisCommented Apr 23, 2020 at 19:53
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3 Answers
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I've seen "subcritical(ly)" used in the isotropic case, in the context of an h-principle for "subcritical isotropic immersions/embeddings" (see e.g. the book "Introduction to the h-principle" by Eliashberg and Mishachev). So if you said "subcritically isotropic" or "supercritically coisotropic," some subset of the symplectic population would know what you were talking about, and my vote would be that this become standard terminology.
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I must be getting conservative in my old age: I'd advise against coining new terminology unless it makes your paper substantially more readable. Definitely "subcritical" (and possibly "supercritical") would be understood by today's symplectic community as KSackel says, but in 50 years' time the community may no longer exist, and it will make your paper much easier to understand if you say "coisotropic of dimension > n" rather than introducing a new adjective which may or may not survive.
Of course, I reserve the right to ignore my own advice (in the past, present and future).
answered Apr 24, 2020 at 18:43
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1$\begingroup$ As a counterpoint to this: "non-Lagrangian" would work (e.g. non-Lagrangian coisotropic). $\endgroup$ Commented Apr 24, 2020 at 18:46
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Note that in the context of KAM theory, given a Hamiltonian system with $n$ degrees of freedom, isotropic invariant tori of dimensions $<n$ are often said to be lower dimensional, and coisotropic invariant tori of dimensions $>n$ are sometimes said to be higher dimensional.