arXiv research

Study para-Sasakian φ-symmetric spaces using Boothby-Wang fibration.

We classify all proper-biharmonic Legendre curves in a Sasakian space form and point out some of their geometric properties. Then we provide a method for constructing anti-invariant proper-biharmonic submanifolds in Sasakian space forms. Finally, using the Boothby-Wang fibration, we determine all proper-biharmonic Hopf…

We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby-Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonic…

We consider manifolds endowed with a contact pair structure. To such a structure are naturally associated two almost complex structures. If they are both integrable, we call the structure a normal contact pair. We generalize the Morimoto's Theorem on product of almost contact manifolds to flat bundles. We construct som…

New theorem generalizes contact manifolds with symplectic properties.

We prove an analogue of the Atiyah-Bott-Berline-Vergne localization formula in the setting of equivariant basic cohomology of $K$-contact manifolds. As a consequence, we deduce analogues of Witten's nonabelian localization and the Jeffrey-Kirwan residue formula, which relate equivariant basic integrals on a contact man…

We construct symplectic and Kähler ray reduced spaces and discuss their relation with the Marsden-Weinstein (point) reduction. This Kähler reduction is well defined even when the momentum value is not totally isotropic. The compatibility of the ray reduction with the cone construction and the Boothby-Wang fibration is …

The study constructs K-contact manifolds with minimal closed Reeb orbits and provides conditions for their homeomorphism to spheres.

Established a generalized Boothby-Wang theorem in contact geometry.

Stein fillability of circle bundles over symplectic manifolds is restricted.

In the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds. The main feature of our approach is to describe the Cartan-Ehresmann connection (gauge field) for principal circle bundles over complex flag manifolds by using elements of representation theory …

Study embeds ruled surfaces into symplectic manifolds, finds Stein fillability results.

An $f$-structure on a manifold $M$ is an endomorphism field $φ$ satisfying $φ^3+φ=0$. We call an $f$-structure {\em regular} if the distribution $T=\kerφ$ is involutive and regular, in the sense of Palais. We show that when a regular $f$-structure on a compact manifold $M$ is an almost $§$-structure, as defined by Dugg…

We examine open books with powers of fibered Dehn twists as monodromy. The resulting contact manifolds can be thought of as Boothby-Wang orbibundles over symplectic orbifolds. Using the mean Euler characteristic of equivariant symplectic homology we can distinguish these contact manifolds and hence show that some fiber…

The paper studies regular contact manifolds and their products.

New method constructs degenerate Sasakian manifolds from hyperkähler bundles.

We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its minimal symplectic fillings, and moreover, fillings cannot contain symplectic surfaces of positive genus. Applying these obstructions to canonical contact struct…

The paper finds CSC Sasaki metrics on specific 7-manifolds.

Unified framework for rigidity results on $(κ, μ)$-manifolds.

We consider contact structures on simply-connected 5-manifolds which arise as circle bundles over simply-connected symplectic 4-manifolds and show that invariants from contact homology are related to the divisibility of the canonical class of the symplectic structure. As an application we find new examples of inequival…

New Bochner technique for foliations with non-negative Ricci curvature.

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…

The paper explores existence and non-existence of constant scalar curvature and extremal Sasaki metrics.

Here we investigate some geometric properties of the contactomorphism group $\mathcal{D}_θ(M)$ of a compact contact manifold with the $L^2$ metric on the stream functions. Viewing this group as a generalization to the $\mathcal{D}(S^1)$, the diffeomorphism group of the circle, we show that its sectional curvature is al…

Study nondegenerate fibrations of Euclidean spaces and their relation to sphere fibrations.

We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odd-dimensiona…

Introduces new stability concept for Fano fibrations.

Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.

In this paper, we examine mapping class group relations of some symplectic manifolds. For each $n\geq 1$ and $k \geq 1$, we show that the $2n$-dimensional Weinstein domain $W = \{f=δ\} \cap B^{2n+2}$, determined by the degree $k$ homogeneous polynomial $f\in \mathbb{C}[z_0,\dots,z_n]$, has a Boothby-Wang type boundary …

Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.

Paper proves achiral Lefschetz fibrations from non-orientable Lefschetz fibrations.

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism type…

A fibration of $\mathbb{R}^3$ by oriented lines is given by a unit vector field $V : \mathbb{R}^3 \to S^2$, for which all of the integral curves are oriented lines. A line fibration is called skew if no two fibers are parallel. Skew fibrations have been the focus of recent study, in part due to their close relationship…

Same genus-2 fibration structures for specific types found by different researchers.

Constructs new coassociative fibrations for G2 manifolds.

A smooth fibration of $\mathbb{R}^3$ by oriented lines is given by a smooth unit vector field $V$ on $\mathbb{R}^3$, for which all of the integral curves are oriented lines. Such a fibration is called skew if no two fibers are parallel, and it is called nondegenerate if $\nabla V$ vanishes only in the direction of $V$.…

We show that generalized broken fibrations in arbitrary dimensions admit rank-2 Poisson structures compatible with the fibration structure. After extending the notion of wrinkled fibration to dimension 6 we prove that these wrinkled fibrations also admit compatible rank-2 Poisson structures. In the cases with indefinit…

Study fibrations over $S^2$ with same singularities, showing monodromies are equivalent up to direct sums.

The paper proves finiteness for stable Lagrangian fibrations with a given divisor.

Constructs Lefschetz fibrations with slopes near 2.

The paper bounds the first Betti number and discusses properties of Lefschetz fibrations.

We classify the Seifert fibrations of any given lens space L(p,q). We give an algorithmic construction of a Seifert fibration of L(p,q) over the base orbifold S^2(m,n) with the coprime parts of m and n arbitrarily prescribed. This algorithm produces all possible Seifert fibrations, and the equivalences between the resu…

We show that there exists a non-trivial simplified broken Lefschetz fibration which has infinitely many homotopy classes of sections. We also construct a non-trivial simplified broken Lefschetz fibration which has a section with non-negative square. It is known that no Lefschetz fibration satisfies either of the above …

Examines properties of holomorphic fibrations in complex geometry.

A fibration of a Riemannian manifold is fiberwise homogeneous if there are isometries of the manifold onto itself, taking any given fiber to any other one, and preserving fibers. Examples are fibrations of Euclidean n-space by parallel n-planes, and the Hopf fibrations of the round n-sphere by great n-spheres. In this …

The paper embeds 4-manifolds into CP^2 x CP^1 using Lefschetz fibrations.

It is known that an arbitrary smooth, oriented 4-manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. Th…

Study how singular fibrations affect Poisson cohomology in 4D.