Narrow Results

We characterize the special Lagrangian submanifolds of a generalized Calabi–Yau manifold, with vanishing Maslov class. Then, we carefully describe several examples, including a non-Kähler generalized Calabi–Yau manifold foliated by special Lagrangian submanifolds.

We prove that the group of contact diffeomorphisms is closed in the group of all diffeomorphisms with respect to the C⁰-topology. By Gromov's alternative, it suffices to exhibit a diffeomorphism that cannot be approximated uniformly by contact diffeomorphisms. Our construction uses Eliashberg's shape invariant.

We present a novel $C^0$-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by Sikorav and Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of $C^0$-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from $J$-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a $C^0$-characterization of contact embeddings and diffeomorphisms in terms of coisotropic (or pre-Lagrangian) embeddings, which in turn leads to a proof of $C^0$-rigidity of contact embeddings and diffeomorphisms. We give a detailed treatment of the shape invariants of symplectic and contact manifolds, and demonstrate that shape is often a natural language in symplectic and contact topology. We consider homeomorphisms that preserve shape, and propose a hierarchy of notions of Lagrangian topological submanifold. Moreover, we discuss shape-related necessary and sufficient conditions for symplectic and contact embeddings, and define a symplectic capacity from the shape.

We briefly introduce the concept of Euler–Lagrange cohomology groups on a symplectic manifold (ℳ²ⁿ, ω) and systematically present the general form of volume-preserving equations on the manifold from the cohomological point of view. It is shown that for every volume-preserving flow generated by these equations there is an important two-form that plays the analog role with the Hamiltonian in the Hamilton mechanics. In addition, the ordinary canonical equations with Hamiltonian H are included as a special case with the two-form . The other volume preserving systems on (ℳ²ⁿ, ω) are studied. The relations between our approach and Feng–Shang's volume-preserving systems as well as the Nambu mechanics are also explored.

This paper begins the development of a theory of what we will call symplectic alternating algebras. They have arisen in the study of 2-Engel groups but seem also to be of interest in their own right. The main part of the paper deals with the challenging classification of some algebras of this kind which arise in the context of 2-Engel groups and give some new information about these groups. The main result is that there are 31 such algebras with dimension 6 over the field of three elements.

In this paper and its sequel we continue our study of nilpotent symplectic alternating algebras. In particular we give a full classification of such algebras of dimension 10 over any field. It is known that symplectic alternating algebras over GF(3) correspond to a special rich class ${𝒞}$ of 2-Engel 3-groups of exponent 27 and under this correspondence we will see that the nilpotent algebras correspond to a subclass of ${𝒞}$ that are those groups in ${𝒞}$ that have an extra group theoretical property that we refer to as being powerfully nilpotent and can be described also in the context of $p$-groups where $p$ is an arbitrary prime.

In this paper, we finish our classification of nilpotent symplectic alternating algebras of dimension 10 over any field $𝔽$.

We continue developing the theory of nilpotent symplectic alternating algebras. The algebras of upper bound nilpotent class, that we call maximal algebras, have been introduced and well studied. In this paper, we continue with the external case problem of algebras of minimal nilpotent class. We show the existence of a subclass of algebras over a field $F$ that is of certain lower bound class that depends on the dimension only. This suggests a minimal bound for the class of nilpotent algebras of dimension $2n\ge 8$ of rank $2$ over any field.

Previously, we constructed an infinite family of knotted symplectic tori representing a fixed homology class in the symplectic four-manifold E(n)_K, which is obtained by Fintushel–Stern knot surgery using a nontrivial fibered knot K in S³, and distinguished the (smooth) isotopy classes of these tori by indirectly computing the Seiberg–Witten invariants of their complements. In this note, we compute the fundamental groups of the complements of these knotted tori and show that for each nontrivial fibered knot K these groups constitute an infinite collection of nonisomorphic groups. We also review some other constructions of symplectic tori in 4-manifolds and show that the fundamental groups of the complements do not distinguish homologous tori in those cases.

In this paper, we examine the phase space structure of a noncanonical formulation of four-dimensional gravity referred to as the Instanton representation of Plebanski gravity (IRPG). The typical Hamiltonian (symplectic) approach leads to an obstruction to the definition of a symplectic structure on the full phase space of the IRPG. We circumvent this obstruction, using the Lagrange equations of motion, to find the appropriate generalization of the Poisson bracket. It is shown that the IRPG does not support a Poisson bracket except on the vector constraint surface. Yet there exists a fundamental bilinear operation on its phase space which produces the correct equations of motion and induces the correct transformation properties of the basic fields. This bilinear operation is known as the almost-Poisson bracket, which fails to satisfy the Jacobi identity and in this case also the condition of antisymmetry. We place these results into the overall context of nonsymplectic systems.

Let M be a closed oriented 4-manifold, with Riemannian metric g, and a -structure induced by an almost-complex structure ω. Each connection A on the determinant line bundle induces a unique connection ∇^A, and Dirac operator on spinor fields. Let be the natural squaring map, taking self-dual spinors to self-dual 2-forms.

In this paper, we characterize the self-dual 2-forms that are images of self-dual spinor fields through σ. They are those α for which (off zeros) c₁(α)=c₁(ω), where c₁(α) is a suitably defined Chern class. We also obtain the formula: .

Using these, we establish a bijective correspondence between: {Kähler forms α compatible with a metric scalar-multiple of g, and with c₁(α)=c₁(ω)} and {gauge classes of pairs (φ,A), with ∇^Aφ=0}, as well as a bijective correspondence between: {Symplectic forms α compatible with a metric conformal to g, and with c₁(α)=c₁(ω)} and {gauge classes of pairs (φ,A), with , and <∇^Aφ,iφ>_ℝ=0, and φ nowhere-zero}.

For each member of an infinite family of homology classes in the K3 surface E(2), we construct infinitely many non-isotopic symplectic tori representing this homology class. This family has an infinite subset of primitive classes. We also explain how these tori can be non-isotopically embedded as homologous symplectic submanifolds in many other symplectic 4-manifolds including the elliptic surfaces E(n) for n > 2.

Based on Hamilton's principle, a new accurate solution methodology is developed to study the torsional bifurcation buckling of functionally graded cylindrical shells in a thermal environment. The effective properties of functionally graded materials (FGMs) are assumed to be functions of the ambient temperature as well as the thickness coordinate of the shell. By applying Donnell's shell theory, the lower-order Hamiltonian canonical equations are established, from which the eigenvalues and eigenvectors are solved as the critical loads and buckling modes of the shell of concern, respectively. The effects of various aspects, including the combined in-plane and transverse boundary conditions, dimensionless geometric parameters, FGM parameters and changing thermal surroundings, are discussed in detail. The results reveal that the in-plane axial edge supports do have a certain influence on the buckling loads. On the other hand, the transverse boundary conditions only affect extremely short shells. With increasing thermal loads, the material volume fraction has a different influence on the critical stresses. It is concluded that the optimized FGM mixtures to withstand thermal torsional buckling are Si₃N₄/SUS304 and Al₂O₃/SUS304 among the materials studied in this paper.

An accurate buckling response analysis for functionally graded graphene platelet (GPL) reinforced piezoelectric cylindrical nanoshells subject to thermo-electro-mechanical loadings is presented by a rigorous symplectic expansion approach. Three types of GPL reinforced patterns are considered, and the modified Halpin–Tsai model is employed to determine their effective material properties. By using Eringen’s nonlocal stress theory and Reissner’s shell theory, new governing equations are established in the Hamiltonian form. Exact solutions are expanded into symplectic series and three possible forms are derived. A comparison with the existing study is presented to validate the solution and very good agreement is observed. The effects of material and geometrical properties of GPLs, electric voltage and temperature rise on critical buckling stresses are investigated and discussed in detail.

We consider generalized Calabi–Yau manifolds and we give a formula for the Maslov class of a Lagrangian submanifold of a generalized Calabi–Yau manifold. In particular, we characterize the Lagrangian submanifolds with vanishing Maslov class. In the 6-dimensional case, we refine our definition. Finally, we construct some examples.

Each and every observational information we obtain from the sky regarding the brightnesses, distances or image distortions resides on the deviation of a null geodesic bundle. In this talk, we present the symplectic evolution of this bundle on a reduced phase space. The resulting formalism is analogous to the one in paraxial Newtonian optics. It allows one to identify any spacetime as an optical device and distinguish its thin lens, pure magnifier and rotator components. We will discuss the fact that the distance reciprocity in relativity results from the symplectic evolution of this null bundle. Other potential applications like wavization and its importance for both electromagnetic and gravitational waves will also be summarized.