Search Results

Here is the version of the Darboux-Weinstein theorem that you want to use: Let $(M_1,\omega_1)$ and $(M_2,\omega_2)$ be symplectic manifolds of dimension $2n$ and let $\iota_i:P\to M_i$ be smooth emb …

Well, already the $4$-sphere, which is even dimensional and orientable, does not carry a symplectic structure. It does not even support a $2$-form that is nowhere degenerate, much less a closed $2$-f …

Actually, one can always do a (local) surgery to 'resolve' transverse intersection points of Lagrangian submanifolds, and this works in all dimensions, not just dimension $2$. Here is a sketch of an …

I think that, instead of complex-valued functions $f:M\to\mathbb{C}$ you mean $J$-holomorphic functions, i.e., complex-valued functions $f:M\to\mathbb{C}$ that satisfy $f'(x)(Jv) = i\,f'(x)(v)$ for al …

I think that it's possible that the OP was interested in knowing something about the 'generality' of coisotropic submanifolds, even though, as José and Nicola point out, it is hard to give an explicit …

Another way to interpret this question is: Is there a 'heuristic' reason that all closed nondegenerate $2$-forms in $2n$ dimensions are locally equivalent? (It becomes more reasonable to ask this wh …

It's not completely clear to me what form of answer you would accept. In one sense, the answer is 'the structure on $T^\ast_2M$ is the pseudogroup structure that is induced by prolongation of the pse …

No, such an $a$ does not always exist. The lowest dimension in which an example without an $a$ could possibly exist is dimension $5$, and, low and behold, there is such an example. I will give one n …

The answer is 'no', but seeing this depends on whether your 'standard half-dimensional subspace' $\mathbb{R}^{2n}\times \lbrace0\rbrace$ is symplectic or not as a subspace of $\mathbb{R}^{4n}$. (You …

If the question is "Are there examples of compact real-analytic Riemannian manifolds that cannot be isometrically embedded as a special Lagrangian submanifold of a compact Calabi-Yau manifold?", then …

The general solution of your equations in a simply connected domain on which $r_2\not=0$ and $r_1\not=\pm1$ is $$ \beta = \frac12 + \frac1{{(r_1}^2{-}1)}\, \left(\frac{\partial a}{\partial\theta_1}+b( …

Yes, if you ignore 2-torsion: After all, if you give $TM$ a complex structure compatible with $\omega$, say $J$, then you'll have that $TM$ pulls back to $L$ to become isomorphic to $TL\oplus T^*L\si …

I believe that you will find a careful discussion of symplectic blow-up of symplectic submanifolds in D. McDuff, Examples of simply-connected symplectic non-Kählerian manifolds, J. Differential Geom. …

What you are asking about is very classical in the theory of exterior differential systems. The subspaces of $D$ that you are calling `isotropic' are what Élie Cartan called the integral elements of …

The injectivity case is well-known and follows quite easily from the statement (usually attributed to Lefschetz) that $L^k_\omega:\Omega^{n-k}(M)\to \Omega^{n+k}(M)$ is a isomorphism for $0\le k\le n$ …