arXiv research

Survey of global geometry for double field theory.

For compact real manifolds, a new double conformal invariant is constructed using the Wodzicki residue and the $d$ operator in the framework of Connes. In the flat case, we compute this double conformal invariant, and in some special cases, we also compute this double conformal invariants. For complex manifolds, a new …

Characterizes a specific type of Courant algebroid with a Calabi-Yau structure.

Constructs cmc doublings of minimal surfaces via min-max theory.

The paper constructs triangulations for double twist knots using geometric methods.

We define an abstract notion of double Lie algebroid, which includes as particular cases: (1) the double Lie algebroid of a double Lie groupoid in the sense of the author, such as the iterated tangent bundle of an ordinary manifold, and various iterated tangent/cotangent constructions in symplectic and Poisson geometry…

Study extra-special quotients of surface braid groups and construct double Kodaira fibrations.

We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures on a vector bundle A and its dual A* is a double Lie algebroid if and only if (…

We construct closed embedded minimal surfaces in the round three-sphere, resembling two parallel copies of the equatorial two-sphere, joined by small catenoidal bridges symmetrically arranged either along two parallel circles of the equator, or along the equatorial circle and the poles. To carry out these constructions…

Paper proves existence of minimal doublings on surfaces with specific properties.

New minimal surface doublings of Clifford Torus improve bounds on minimal surfaces in S^3.

Researchers found all embeddings of Kuratowski graphs on a double torus.

Extends double symplectic groupoids to transitive Courant algebroids.

The paper introduces colorings and invariants for twisted links and shows how double coverings can be equivalent.

A new double quasi-Poisson bracket on surface groups.

Standard trisection diagrams found for Mazur type 4-manifolds.

In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if $Σ$ is a closed minimal surface embedded in a Riemannian three-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given a suitable family of LD sol…

A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation and basic examples come from Analytical Mechanics, where double affine bundles hav…

We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional $\mathbb Z$-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles over supermanifolds, double vector bundles, iterated constructions like $TTM$, e…

We establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space $\R^6_\st$ with exactly on…

We construct a C-space associated with every closed 3-form on a spacetime $M$ and show that it depends on the class of the form in $H^3(M, Z)$. We also demonstrate that C-spaces have a relation to generalized geometry and to gerbes. C-spaces are constructed after introducing additional coordinates at the open sets and …

We show how the double vector bundle structure of the manifold of double velocities, with its submanifolds of holonomic and semiholonomic double velocities, is mirrored by a structure of holonomic and semiholonomic subgroups in the principal prolongation of the first jet group. We use the actions of these groups to con…

Complete minimal surfaces with any genus found in a specific 3-manifold.

Constructs minimal hypersurfaces in S^4(1) by doubling equatorial S^3.

In the first part of the paper we discuss the current status of the application of the gluing methodology to doubling and desingularization constructions for minimal surfaces in Riemannian three-manifolds. In particular a doubling construction for equatorial spheres in $S^3(1)$ is announced. Aspects of the current unde…

In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theor…

We introduce the notion of a symplectic hopfoid, which is a "groupoid-like" object in the category of symplectic manifolds where morphisms are given by canonical relations. Such groupoid-like objects arise when applying a version of the cotangent functor to the structure maps of a Lie groupoid. We show that such object…

We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We show that the Lie algebroid structure of an LA--groupoid may be prolonged to the Lie algebroid of its Lie groupoid structure; in the case of a double groupoid this prolonged structure for either …

We construct and study a family of double-periodic almost entire solutions of the maximal surface equation. The solutions are parameterized by a submanifold of $3\times 3$-matrices (the so-called generating matrices). We show that the constructed solutions are either space-like or of mixed type with the light-cone type…

We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated l…

In this paper we give a procedure to construct hypersymplectic structures on $R^{4n}$ beginning with affine-symplectic data on $R^{2n}$. These structures are shown to be invariant by a 3-step nilpotent double Lie group and the resulting metrics are complete and not necessarily flat. Explicit examples of this constructi…

We develop new algebraic methods refining the Witt group of linking forms and Ranicki's torsion algebraic L-groups into double Witt groups and double L-groups. At each prime ideal of the underlying ring, our double Witt groups capture infinitely many more integral signatures of the linking form than the single Witt gro…

The study of quotient structures in multi-graded bundles, including double vector bundles.

The canonical involution of a double (=iterated) tangent bundle may be dualized in different ways to yield relations between the Tulczyjew diffeomorphism, the Poisson anchor associated with the standard symplectic structure on the cotangent space,and the reversal diffeomorphism. We show that the constructions which yie…

Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the o…

Research decouples Lie algebroids using bicocycle double cross product theory.

New invariant detects non-homotopy equivalent 4-manifolds.

Defines and extends flat pseudo-Riemannian F-Lie algebras.

On a Poisson manifold endowed with a Riemannian metric we will construct a vector field that generalizes the double bracket vector field defined on semi-simple Lie algebras. On a regular symplectic leaf we will construct a generalization of the normal metric such that the above vector field restricted to the symplectic…

In earlier work of NK new closed embedded smooth minimal surfaces in the round three-sphere $\mathbb{S}^3(1)$ were constructed, each resembling two parallel copies of the equatorial two-sphere $\mathbb{S}^2_{eq}$ joined by small catenoidal bridges, with the catenoidal bridges concentrating along two parallel circles, o…

New groups from surface braids help create complex geometric shapes.

The paper defines new knot genera and finds bounds for stabilization distances.

Synthetic construction of Hopf fibration in 4D space.

The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minim…

Constructs special Kähler structures on Lie groups.

Let $S$ be a compact oriented surface with boundary together with finitely many marked points on the boundary, and let $S^\circ$ be the same surface equipped with the opposite orientation. We consider the double $S_\mathcal{D}$ obtained by gluing the surfaces $S$ and $S^\circ$ along corresponding boundary components. W…

We use symplectic reduction to give a new construction of the core $C$ of a symplectic double groupoid $D$ as the common leaf space of characteristic foliations associated to various coisotropic submanifolds of $D$. In the case of the cotangent double groupoid of a Lie groupoid $G$, the canonical relations arising from…

New algebra for twice-punctured torus curves.