We classify fibrations of abstract $3$-regular GKM graphs over $2$-regular ones, and show that all fibrations satisfying the known necessary conditions for realizability are, in fact, realized as the projectivization of equivariant complex rank-$2$ vector bundles over quasitoric $4$-manifolds or $S^4$. We investigate the existence of invariant (stable) almost complex, symplectic, and Kähler structures on the total space. In this way, we obtain infinitely many Kähler manifolds with Hamiltonian non-Kähler actions in dimension $6$ with prescribed one-skeleton, in particular with a prescribed number of isolated fixed points.

This paper studies a class of singular fibrations, called self-crossing boundary fibrations, which play an important role in semitoric and generalized complex geometry. These singular fibrations can be conveniently described using the language of Lie algebroids. We will show how these fibrations arise from (nonfree) torus actions, and how to use them to construct and better understand self-crossing stable generalized complex four-manifolds. We moreover show that these fibrations are compatible with taking connected sums, and use this to prove a singularity trade result between two types of singularities occurring in these types of fibrations (a so-called nodal trade).

Our main result is the $\mathbb {\mathcal {C}}^{0}$-rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic disks in cotangent bundles with boundary on the zero section, whose boundaries represent any integral homology class. We discuss further applications of these punctured disks in symplectic geometry.

We prove that every symplectic manifold is a coadjoint orbit of the group of automorphisms of its integration bundle, acting linearly on its space of momenta, for any group of periods of the symplectic form. This result generalizes the Kirilov–Kostant–Souriau theorem when the symplectic manifold is homogeneous under the action of a Lie group and the symplectic form is integral.

In this note, we study homology classes in the mirror quintic Calabi–Yau threefold that can be realized by special Lagrangian submanifolds. We have used Picard–Lefschetz theory to establish the monodromy action and to study the orbit of Lagrangian vanishing cycles. For many prime numbers $p,$ we can compute the orbit modulo p. We conjecture that the orbit in homology with coefficients in $\mathbb {Z}$ can be determined by these orbits with coefficients in $\mathbb {Z}_p$ .

Given a vector field on a manifold $M$, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well behaved under transgression to spaces of maps into $M$. We focus on the case of multisymplectic manifolds and Hamiltonian vector fields. Our main result is that in the presence of a Lie group of symmetries admitting a homotopy co-momentum map, one obtains a whole family of globally conserved quantities. This extends a classical result in symplectic geometry. We carry this out in a general setting, considering several variants of the notion of globally conserved quantity.

In this paper, we introduce the notion of complex product structures on hom-Lie algebras and show that a hom-Lie algebra carrying a complex product structure is a double hom-Lie algebra and it is also endowed with a hom-left symmetric product. Moreover, we show that a complex product structure on a hom-Lie algebra determines uniquely a left symmetric product such that the complex and the product structures are invariant with respect to it. Finally, we introduce the notion of hyper-para-Kähler hom-Lie algebras and we present an example of hyper-para-Kähler hom-Lie algebras.

Any singular level of a completely integrable system (c.i.s.) with non-degenerate singularities has a singular affine structure. We shall show how to construct a simple c.i.s. around the level, having the above affine structure. The cotangent bundle of the desingularized level is used to perform the construction, and the c.i.s. obtained looks like the simplest one associated to the affine structure. This method of construction is used to provide several examples of c.i.s. with different kinds of non-degenerate singularities.

We give a characterization of contact manifolds in terms of symplectic Lie–Rinehart–Jacobi algebras. We also give a sufficient condition for a Jacobi manifold to be a contact manifold.

Symplectic induction was first introduced by Weinstein as the symplectic analogue of induced representations, and was further developed by Guillemin and Sternberg. This paper deals with the case where the symplectic manifold in question is a semisimple coadjoint orbit of a Lie group. In this case, the construction is generalized by adding a smooth mapping, in order to obtain various symplectic forms. In particular, when the orbit is elliptic, a study of the complex geometry shows that quantization commutes with induction.

In this article we are concerned with how to compute the cohomology ring of a symplectic quotient by a circle action using the information we have about the cohomology of the original manifold and some data at the fixed point set of the action. Our method is based on the Tolman-Weitsman theorem which gives a characterization of the kernel of the Kirwan map. First we compute a generating set for the kernel of the Kirwan map for the case of product of compact connected manifolds such that the cohomology ring of each of them is generated by a degree two class. We assume the fixed point set is isolated; however the circle action only needs to be “formally Hamiltonian”. By identifying the kernel, we obtain the cohomology ring of the symplectic quotient. Next we apply this result to some special cases and in particular to the case of products of two dimensional spheres. We show that the results of Kalkman and Hausmann-Knutson are special cases of our result.

We establish the correspondence between Euclidean differential geometry of submanifolds in $\mathbb{R}^n$ and projective differential geometry of submanifolds in $\mathbb{R}^{n +1 }$ under stereographic projection to quadrics of revolution (and to a more general class of quadrics called quasi-revolution quadrics). V. D. Sedykh found a relation between Lagrangian and Legendrian singularities by stereographic projection to a sphere in Euclidean space. We generalise Sedykh's results in several directions, for instance:

(1) such a relation holds for any stereographic projection of a hyperplane to a quasi-revolution quadric in $\mathbb{R}^n \times \mathbb{R}$, where here $\mathbb{R}^n$ denotes Euclidean space;

(2) when the quadric is a paraboloid the relation between Lagrangian and Legendrian singularities is the most natural one, and the calculations, formulas and proofs are simpler;

(3) using the classical theory of poles, polars and polar duality, we construct the natural isomorphism between the front of the Lagrange submanifold of the normal map (considered as a subvariety in $J^0(\mathbb{R}^n) = \mathbb{R}^n \times \mathbb{R}$) and the front of the Legendre submanifold of the tangential map (considered as a subvariety in the space $(\mathbb{R}^{n + 1})^\vee$ of affine n-dimensional subspaces in $\mathbb{R}^{n + 1}$).

As a consequence of our results we obtain a formula to calculate the vertices of smooth curves in $\mathbb{R}^n$. Our results may be applied to calculate and study umbilic points of surfaces in $\mathbb{R}^3$, or more generally to study the contact of submanifolds in Euclidean space $\mathbb{R}^n$ with k-spheres, for $k = 1,\ldots,n - 1$, in terms of the contact of submanifolds in $\mathbb{R}^{n + 1}$ with $(k + 1)$-planes.

We remark that it is possible to obtain additional geometric information when stereographic projection is replaced by an inversion.

Let $X$ be a closed, symplectic 4-manifold. Suppose that there is either a symplectic or an anti-symplectic involution $\sigma : X\,{\to}\, X$ with a 2-dimensional compact, oriented submanifold $\Sigma$ as a fixed point set.

If $\sigma$ is a symplectic involution then the quotient $X/\sigma$ with $b_2^+(X/\sigma)\,{\ge}\, 1$ is a symplectic 4-manifold.

If $\sigma$ is an anti-symplectic involution and $\Sigma$ has genus greater than 1 representing non-trivial homology class, we prove a vanishing theorem on Seiberg-Witten invariants of the quotient $X/\sigma$ with $b_2^+(X/\sigma)\,{ >}\,1.$

If $\Sigma$ is a torus with self-intersection number 0, we get a relation between the Seiberg-Witten invariants on $X$ and those of $X/\sigma$ with $b_2^+(X), b_2^+(X/\sigma)\,{ >}\,2$ which was obtained in [21] when the genus $g(\Sigma)\,{ >}\,1$ and $\Sigma\cdot\Sigma\,{=}\,0$.This work was supported by a Korea Research Foundation Grant (No KRF-2002-072-C00010).

This paper studies the following question: Given an ${\omega }'$-symplectic action of a Lie group on a manifold $M$ which coincides, as a smooth action, with a Hamiltonian $\omega$-action, when is this action a Hamiltonian ${\omega }'$-action? Using a result of Morse-Bott theory presented in Section 2, we show in Section 3 of this paper that such an action is in fact a Hamiltonian ${\omega }'$-action, provided that $M$ is compact and that the Lie group is compact and connected. This result was first proved by Lalonde-McDuff-Polterovich in 1999 as a consequence of a more general theory that made use of hard geometric analysis. In this paper, we prove it using classical methods only.