arXiv research

A novel method optimizes variable-stiffness structures for better strength and weight.

We study the topology of Hamiltonian-minimal Lagrangian submanifolds N in C^m constructed from intersections of real quadrics in a work of the first author. This construction is linked via an embedding criterion to the well-known Delzant construction of Hamiltonian toric manifolds. We establish the following topologica…

New groups algebraically fibre with high-dimensional hyperbolic groups.

Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds. A complex moment-angle manifold Z is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold Z is the total space of a holomorphic bundle over a toric variet…

Study minimal Lagrangian tori on Kähler manifolds, answering questions about their existence and stability.

The article describes canonical metrics on holomorphic fibre bundles.

The aim of this paper is to demonstrate that very many Dehn fillings on a cusped hyperbolic 3-manifold yield a 3-manifold which is irreducible, atoroidal and not Seifert fibred, and which has infinite, word hyperbolic fundamental group. We establish an extension of the Thurston-Gromov $2π$ theorem by showing that if ea…

Book teaches how Lagrangian torus fibration base geometry can be read off.

The main result of this paper gives a new construction of extremal Kähler metrics on the total space of certain holomorphic submersions, giving a vast generalisation and unification of results of Hong, Fine and others. The principal new ingredient is a novel geometric partial differential equation on such fibrations, w…

This paper investigates the projectivization of real vector bundles over small covers. We first give a necessary and sufficient condition for such a projectivization to be a small cover. Then associated with moment-angle manifolds, we further study the structure of such a projectivization as a small cover. As an applic…

We prove that among four-dimensional ideal right-angled hyperbolic polytopes the 24-cell is of minimal volume and of minimal facet number. As a corollary, a dimension bound for ideal right-angled hyperbolic polytopes is obtained.

Describes spectral data for singular fibres of a specific Hitchin system.

We classify semi-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a semi-Riemannian manifold under the assumption that the dimension of the fibres is less than or equal to three and the metrics induced on fibres are negative definite. Also, we obtain the classificat…

Ozsváth and Szabó conjectured that knot Floer homology detects fibred links. We will verify this conjecture for closed 3-braids, by classifying fibred closed 3-braids. In particular, given a nontrivial closed 3-braid, either it is fibred, or it differs from a fibred link by a half twist. The proof uses Gabai's method o…

Given a (smooth) complex analytic family of compact complex manifolds, we prove that the central fibre must be Moishezon if the other fibres are Moishezon. Using a "strongly Gauduchon metric" on the central fibre whose existence was proved in our previous work on limits of projective manifolds, we show that the irreduc…

Following on from work of Dunfield, we determine the fibred status of all the unknown hyperbolic 3-manifolds in the cusped census. We then find all the fibred hyperbolic 3-manifolds in the closed census and use this to find over 100 examples each of closed and cusped virtually fibred non-fibred census 3-manifolds, incl…

We consider vector fields on knot/link complements in $S^3$ which are transverse to the fibres of a fibration of the complement over a circle. We prove that a large class of fibred knots/links, including all non-torus fibred 2-bridge knots, has the following property: any vector field transverse to the fibres of the fi…

Study calibrates high-dimensional binary classifiers using angle between estimator and true weights.

Profinite rigidity studied for algebraic fibring of groups.

The stabilisation height of a fibre surface in the 3-sphere is the minimal number of Hopf plumbing operations needed to attain a stable fibre surface from the initial surface. We show that families of fibre surfaces related by iterated Stallings twists have unbounded stabilisation height.

Study shows volume and genus unrelated for hyperbolic fibred knots.

Any leafwise connection on a fibre bundle over a foliated manifold is proved to come from a connection on this fibre bundle.

Survey on fibring in manifolds and groups, focusing on recent developments and conjectures.

Defines and parametrizes $\mathfrak{sl}(2)$-type singular fibres in symplectic and odd orthogonal Hitchin systems.

Software finds ideal polyhedra with rational dihedral angles and volume maxima.

We explore algebraic characterizations of 2-knots whose associated knot manifolds fibre over lower-dimensional orbifolds, and consider also some issues related to the groups of higher-dimensional fibred knots.

We consider Schrödinger operators $H=-Δ_{g_\varepsilon} + V$ on a fibre bundle $M\stackrelπ{\to}B$ with compact fibres and a metric $g_\varepsilon$ that blows up directions perpendicular to the fibres by a factor ${\varepsilon^{-1}\gg 1}$. We show that for an eigenvalue $λ$ of the fibre-wise part of $H$, satisfying a l…

In this paper we give a generalisation of the Radius Rigidity theorem of F.Wilhelm. This is done by showing that if a Riemannian submersion of $S^{15}$ with 7-dimensional fibres has at least one fibre which is a great sphere then all the fibres are so. Some weaker than known conditions which force the existence of such…

Study low energy resolvent behavior on fibred boundary metrics.

Knot Floer homology matches fixed point Floer for fibred knots.

We define the pull-back of a smooth principal fibre bundle, and show that it has a natural principal fibre bundle structure. Next, we analyse the relationship between pull-backs by homotopy equivalent maps. The main result of this article is to show that for a principal fibre bundle over a paracompact manifold, there i…

Formula calculates homology groups of Milnor fibres for real hypersurface singularities.

We develop some techniques to study the adiabatic limiting behaviour of Calabi-Yau metrics on the total space of a fibration, and obtain strong control near the singular fibres by imposing restrictions on the singularity types. We prove a uniform lower bound on the metric up to the singular fibre, under fairly general …

Study finds lower bounds for energy on fibred manifolds using fiberwise symmetrization.

Real Milnor fibres become contractible after attaching handles, matching classical results.

Given a closed 3--manifold $Y$, we show that the Heegaard Floer homology determines whether $Y$ fibres over the circle with a fibre of negative Euler characteristic. This is an analogue of an earlier result about knots proved by Ghiggini and the author.

We introduce the fibred toric varieties as equivariant $\mathbb{C}P^r$ bundles over lower dimensional toric varieties. An equivalent characterization is that the natural morphisms on them degenerate to bundle projections in the context of variation of toric varieties as GIT quotients. Our main observation is that these…

In a fibre bundle, natural derivatives of a section are defined as tangent vector fields on the image of a section of the fibre bundle. A local extension to vector fields in the tangent bundle leads to a direct proof of the formula expressing the curvature of a connection in terms of covariant derivatives. The result i…

A review of the parallel transport (translation) in fibre bundles is presented. The connections between transports along paths and parallel transports in fibre bundles are examined. It is proved that the latter ones are special cases of the former.

Let $X$ be a compact manifold with boundary. Suppose that the boundary is fibred, $φ:\pa X\longrightarrow Y,$ and let $x\in\CI(X)$ be a boundary defining function. This data fixes the space of `fibred cusp' vector fields, consisting of those vector fields $V$ on $X$ satisfying $Vx=O(x^2)$ and which are tangent to the f…

We get obstructions to existence of semi-Riemannian submersions with totally umbilic fibres.

We get obstructions to existence of semi-Riemannian submersions with totally umbilic fibres.

Paper studies curvature in Finsler geometry, proving curvature constancy under isotropy.

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all…

Classifies 3D spaces using specific invariants.

This work has the purpose of applying the concept of Geometric Calculus (Clifford Algebras) to the Fibre Bundle description of Quantum Mechanics. Thus, it is intended to generalize that formulation to curved spacetimes [the base space of the fibre bundle in question] in a more natural way.

It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle is multiplicative if the fundamental group of the base acts trivially on the cohomology ring of the fibre with real coefficients, in which case the signature of the total space equals the product of the signatures of base and fibre. Hambl…

We put together some of the efforts by several people of making aspects of fibre bundle theory into algebra. The initiator of these efforts was Charles Ehresmann, who put the notion of groupoid and groupoid action in the focus of fibre bundle theory in general, and in connection theory in particular.