arXiv research

Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.

Study of double complexes on Iwasawa manifold yields 3 isomorphism types.

This paper equates geometric structures to algebraic ones for degree 2 manifolds.

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…

Constructs higher-dimensional twistor spaces for complex manifolds.

The study finds infinitely many counterexamples to a generalized Double Soul Conjecture.

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 …

Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.

Research shows how complexity of h-cobordisms affects double points in 4-manifolds.

Study simplifies homotopy groups of 6D manifolds.

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…

Survey of global geometry for double field theory.

In a previous paper, we have shown that the geometry of double field theory has a natural interpretation on flat para-Kähler manifolds. In this paper, we show that the same geometric constructions can be made on any para-Hermitian manifold. The field is interpreted as a compatible (pseudo-)Riemannian metric. The tangen…

New invariants from quantum group theory for hyperbolic 3-manifolds.

New bounds on knot complexity defined via band number and surface diagrams.

Generalizes fully augmented links to doubled 3-manifolds with geometric bounds.

This text is meant to be a brief overview of the topics announced in the title and is based on my talk in Vienna (August/September 2007). It does not contain new results (except probably for a remark concerning Q-manifold homology, which I wish to elaborate elsewhere). "Mackenzie theory" stands for the rich circle of n…

Ancient caloric functions on manifolds with polynomial growth are studied under volume doubling barrier.

Uncountably many contractible 3-manifolds with double 3-space property and those without are identified.

Vaisman algebroid explains gauge symmetry in DFT.

Standard trisection diagrams found for Mazur type 4-manifolds.

Study evaluates manifold alignment methods for noisy double pendulum dynamics.

The study of double coset growth in specific groups confirms a conjecture about generic 3-manifolds.

Formulae for volumes of double twist knots found.

The paper extends Hodge-de Rham and Lichnérowicz Laplacians to double forms and proves vanishing theorems.

The study finds generalized torsion elements in 3-manifolds from specific knot surgeries.

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

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…

The paper explores conditions for positive scalar curvature on manifolds with boundaries and their doubles.

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

Revisits the Gauss-Bonnet formula using double forms.

We consider a homology sphere $M_n(K_1,K_2)$ presented by two knots $K_1,K_2$ with linking number 1 and framing $(0,n)$. We call the manifold {\it Matsumoto's manifold}. We show that there exists no contractible bound of $M_n(T_{2,3},K_2)$ if $n<2τ(K_2)$ holds. We also give a formula of Ozsváth-Szabó's $τ$-invariant as…

The tetrus is a sort of big brother to the tripus, W.P. Thurston's example of a compact hyperbolic 3-manifold with totally geodesic boundary. We describe a sixfold cover of the double of the tetrus, itself a double, which fibers over the circle with fiber a closed surface of genus 19. We also record arithmeticity of th…

Introduces symplectic hopfoids and their relation to double Lie groupoids.

Motivated by the celebrated Schoen-Yau-Gromov-Lawson surgery theory on metrics of positive scalar curvature, we construct a double manifold associated with a minimal isoparametric hypersurface in the unit sphere. The resulting double manifold carries a metric of positive scalar curvature and an isoparametric foliation …

The study explores cohomological invariants and decomposes them into irreducible parts, focusing on zigzags.

Study of 4-manifolds with shadow-complexity one, identifying all such manifolds.

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, consta…

Double field theory was developed by theoretical physicists as a way to encompass $T$-duality. In this paper, we express the basic notions of the theory in differential-geometric invariant terms, in the framework of para-Kaehler manifolds. We define metric algebroids, which are vector bundles with a bracket of cross se…

Study shows bounds on knot groups and nonembeddability of certain open manifolds.

Liouville theorem for minimal graphs on manifolds with specific properties.

The paper finds left orderable slopes for specific double twist knots.

Generalizes double transgression formulas on complex manifolds.

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…

New invariant detects non-homotopy equivalent 4-manifolds.

Study on involution in double jet bundles.

We introduce a new operation, double point surgery, on immersed surfaces in a 4-manifold, and use it to construct knotted configurations of surfaces in many 4-manifolds. Taking branched covers, we produce smoothly exotic actions of Z/m x Z/n on simply connected 4-manifolds with complicated fixed-point sets.

We develop some theory of double fibration transforms where the cycle space is a smooth manifold and apply it to complex projective space.