Volume 6, Issue 1

The classification of affine line bundles on a compact complex space is a difficult problem. We study the affine analogue of the Picard functor and the representability problem for this functor. Let be a compact complex space with . We introduce the affine Picard functor which assigns to a complex space the set of families of linearly -framed affine line bundles on parameterized by . Our main result states that the functor is representable if and only if the map is constant. If this is the case, the space which represents this functor is a linear space over whose underlying set is , where is a Poincaré line bundle normalized at . The main idea idea of the proof is to compare the representability of to the representability of a functor considered by Bingener related to the deformation theory of -cohomology classes. Our arguments show in particular that, for = 1, the converse of Bingener’s representability criterion holds

We study some properties of curvature tensors of Norden and, more generally, metallic pseudo-Riemannian manifolds. We introduce the notion of J -sectional and J -bisectional curvature of a metallic pseudo-Riemannian manifold ( M , J , g ) and study their properties.We prove that under certain assumptions, if the manifold is locally metallic, then the Riemann curvature tensor vanishes. Using a Norden structure ( J , g ) on M , we consider a family of metallic pseudo-Riemannian structures { J a,b } a , b ∈ℝ and show that for a ≠ 0, the J -sectional and J -bisectional curvatures of M coincide with the J a , b -sectional and J a , b -bisectional curvatures, respectively. We also give examples of Norden and metallic structures on ℝ 2 n .

Let Γ ( M ) be the set of all global continuous cross sections of a continuous family M of compact complex manifolds on a compact Hausdorff space X . In this paper, we introduce a C ( X )-manifold structure on Γ ( M ). Especially, if X is contractible, then Γ ( M ) is a finite-dimensional C ( X )-manifold. Here, C ( X ) denotes the Banach algebra of all complex-valued continuous functions on X .

We describe the natural geometry of Hilbert schemes of curves in ℙ 3 and, in some cases, in ℙ n , n ≥ 4.

We study special Lagrangian fibrations of SU(3)-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group G , we decompose such SU(3)-structures into triples of solder 1-forms, connection 1-forms and equivariant 3 × 3 positive-definite symmetric matrix-valued functions on principal G -bundles over 3-manifolds. As applications, we describe regular parts of G 2 -manifolds that admit Lagrangian-type 3-dimensional group actions by constrained dynamical systems on the spaces of the triples in the cases of G = T 3 and SO(3).

We define a Kähler-like almost Hermitian metric. We will prove that on a compact Kähler-like almost Hermitian manifold ( M 2 n , J , g ), if it admits a positive ∂ ̄∂-closed ( n − 2, n − 2)-form, then g is a quasi-Kähler metric.

This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics.

This survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.

These are notes based on a mini-course at the conference RIEMain in Contact, held in Cagliari, Sardinia, in June 2018. The main theme is the connection between Reeb dynamics and topology. Topics discussed include traps for Reeb flows, plugs for Hamiltonian flows, the Weinstein conjecture, Reeb flows with finite numbers of periodic orbits, and global surfaces of section for Reeb flows. The emphasis is on methods of construction, e.g. contact cuts and lifting group actions in Boothby–Wang bundles, that might be useful for other applications in contact topology.

In this note we show families of homogeneous Sasakian manifolds G / H which are nonformal. The non-formality condition is expressed in terms of characters of a maximal torus in G .

For a contact manifold, we study a strongly pseudo-convex CR space form with constant holomorphic sectional curvature for the Tanaka-Webster connection. We prove that a strongly pseudo-convex CR space form M is weakly locally pseudo-Hermitian symmetric if and only if (i) dim M = 3, (ii) M is a Sasakian space form, or (iii) M is locally isometric to the unit tangent sphere bundle T 1 (𝔿 n +1 ) of a hyperbolic space 𝔿 n +1 of constant curvature −1.

We discuss the classifiation of simply connected, complete ( κ , µ )-spaces from the point of view of homogeneous spaces. In particular, we exhibit new models of ( κ , µ )-spaces having Boeckx invariant -1. Finally, we prove that the number (n+1)(n+2)2${{(n + 1)(n + 2)} \over 2}$ is the maximum dimension of the automorphism group of a contact metric manifold of dimension 2 n +1, n ≥ 2, whose symmetric operator h has rank at least 3 at some point; if this dimension is attained, and the dimension of the manifold is not 7, it must be a ( κ , µ )-space. The same conclusion holds also in dimension 7 provided the almost CR structure of the contact metric manifold under consideration is integrable.

We provide a new, self-contained and more conceptual proof of the result that an almost contact metric manifold of dimension greater than 5 is Sasakian if and only if it is nearly Sasakian.

A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.

It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure.We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras.

A classical theoremof Kervaire states that products of spheres are parallelizable if and only if at least one of the factors has odd dimension. Two explicit parallelizations on S m × S 2 h −1 seem to be quite natural, and have been previously studied by the first named author in [32]. The present paper is devoted to the three choices G = G 2 , Spin(7), Spin(9) of G -structures on S m × S 2 h −1 , respectively with m + 2 h − 1 = 7, 8, 16 and related with octonionic geometry.

This is partly an expository paper, where the authors’ work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-˛at metrics on nilpotent Lie groups of dimension [eight.tf] are obtained. Some related open questions are presented.

The classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups [9] states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter , is integrable for all values of the loop parameter. As a matter of fact, the same result holds for k -symmetric spaces over reductive Lie groups, [8]. In this survey we will show that to each of the five different types of real forms for a loop group of A 2 (2) there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in ℂℙ 2 , minimal Lagrangian surfaces in ℂℍ 2 , timelike minimal Lagrangian surfaces in ℂℍ 1 2 , proper definite affine spheres in ℝ 3 and proper indefinite affine spheres in ℝ 3 , respectively.

The Gauss images of isoparametric hypersufaces of the standard sphere S n +1 provide a rich class of compact minimal Lagrangian submanifolds embedded in the complex hyperquadric Q n (ℂ). This is a survey article based on our joint work [17] to study the Hamiltonian non-displaceability and related properties of such Lagrangian submanifolds.

An R -space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R -space has the canonical embedding into a Kähler C -space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R -spaces canonically embedded in Einstein-Kähler C -spaces, and provide some examples of the calculation by the formula.