arXiv research

This paper classifies fibrations of flat orbifolds, advancing flat 4-manifold classification.

This paper has several goals. The first idea is to study the geometric PDEs of connection-flatness, curvature-flatness, Ricci-flatness, scalar curvature-flatness in a modern and rigorous way. Although the idea is not new, our main Theorems about flatness introduce a different point of view in Differential Geometry. The…

We propose a Lie geometric point of view on flat fronts in hyperbolic space as special omega-surfaces and discuss the Lie geometric deformation of flat fronts.

Automatically identifies geometric flat outputs for robotic systems.

Two geometric tests for forward-flatness are shown to be dual.

We examine geometric representability results for various classes of equiaffine curvature operators. We show every Ricci flat algebraic curvature operator is geometrically realizable by a Ricci flat torsion free connection on the tangent bundle of some smooth manifold.

Study geodesic paths on flat surfaces, comparing length and singularity counts.

The paper classifies fibrations of 3-dimensional flat orbifolds.

New method constructs geometric flat outputs for robotic systems using symmetry.

Study of symplectically flat connections and their functionals on smooth manifolds.

The paper establishes pressure gaps for manifolds with flat subtori singularities.

We study the 8 natural GL equivariant geometric realization questions for the space of generalized algebraic curvature tensors. All but one of them is solvable; a non-zero projectively flat Ricci antisymmetric generalized algebraic curvature is not geometrically realizable by a projectively flat Ricci antisymmetric tor…

Researchers geometrically define asymptotic coordinates in General Relativity.

Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.

The study proves conditions for CAT(0) spaces with higher rank rigidity.

We compute the differential geometric invariants of cuspidal edges on flat surfaces in hyperbolic $3$-space and in de Sitter space. Several dualities of invariants are pointed out.

Study shows how to deform foliated manifolds with flat leaves, leading to geometric characterization.

Study flat manifolds' collapsed limits as flat orbifolds.

We construct transformations which take asymptotically AdS hyperbolic initial data into asymptotically flat initial data, and which preserve relevant physical quantities. This is used to derive geometric inequalities in the asymptotically AdS hyperbolic setting from counterparts in the asymptotically flat realm, whenev…

In the genus one case, we make explicit some constructions of Veech on flat surfaces and generalize some geometric results of Thurston about moduli spaces of flat spheres as well as some equivalent ones but of an analytico-cohomological nature of Deligne-Mostow, which concern the monodromy of Appell-Lauricella hypergeo…

Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.

In this paper we describe the classification of all the geometric fibrations of a closed flat Riemannian 4-manifold over a 1-orbifold.

In this paper, geometric characterizations of conformally flat and radially flat hypersurfaces in $\mathbb{S}^n \times \mathbb{R}$ and $\mathbb{H}^n \times \mathbb{R}$ are given by means of their extrinsic geometry. Under suitable conditions on the shape operator, we classify conformally flat hypersurfaces in terms of …

The paper explores properties of CR hypersurfaces and their flatness.

We construct geometric examples of N-differential graded algebras such as the algebra of differential forms of depth $N$ on an affine manifold, and $N$-flat covariant derivatives.

New geometric variant of factorization homology for conformally flat manifolds.

Extends flat submanifold properties from hyperbolic plane to symmetric spaces.

Study on deformation of weighted scalar curvature, proving geometric results and stability.

Study on how incomplete smooth metrics degenerate from asymptotically conical Ricci-flat Kähler metrics.

The study improves Wintgen inequalities for submanifolds in specific geometric spaces.

Review and generalize Haefliger's differentiable cohomology for diffeomorphisms and flat Cartan groupoids.

Study on scalar-flat Kahler 4-manifolds with a continuous symmetry.

We prove that for a given flat surface with conical singularities, any pair of geometric triangulations can be connected by a chain of flips.

In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Later, their decay assumptions were weakened by Metzger, Huang, Eichmair-M…

A seminal result in geometric group theory is that a 1-ended hyperbolic group has a locally connected visual boundary. As a consequence, a 1-ended hyperbolic group also has a path connected visual boundary. In this paper, we study when this phenomenon occurs for CAT(0) groups. We show if a 1-ended CAT(0) group with iso…

Criterion for surfaces in Heisenberg group to be graphs using flat cones.

Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichmüller theory. Geometric structures on a manifold are closely related with representations of the fundamental group and w…

In this paper, Hamiltonian monodromy is studied from the point of view of geometric quantization abd theta functions, and various differential geometric aspects thereof are dealt with, all related to holonomies of suitable flat connections.

The paper simplifies FLRW photon propagators using geometric embeddings.

Introduces new info-geometric structure for dynamics on graphs and hypergraphs.

Let $(X^n, \check{X}^n)$ be a mirror pair of an $n$-dimensional complex torus $X^n$ and its mirror partner $\check{X}^n$. Then, a simple projectively flat bundle $E(L,\mathcal{L})\rightarrow X^n$ is constructed from each affine Lagrangian submanifold $L$ in $\check{X}^n$ with a unitary local system $\mathcal{L} \righta…

We construct flat 3-webs via semi-simple geometric Frobenius manifolds of dimension three and give geometric interpretation of the Chern connection of the web. These webs turned out to be biholomorphic to the characteristic webs on the solutions of the corresponding associativity equation. We show that such webs are he…

Classifies weakly Einstein curvature tensors in 4D Euclidean space.

In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we gi…

Simplicial, piecewise-flat discretizations of manifolds provide a clear path towards curvature analysis on discrete geometries and for solutions of PDE's on manifolds of complex topologies. In this manuscript we review and expand on discrete exterior calculus methods using hybrid domains. We then analyze the geometric …

Geometrically connects theta functions and WZNW blocks.

The results of the paper concern the topological structure of complete riemannian manifolds with cyclic holonomy groups and low-dimensional orientable complete flat manifolds. We also discuss related results such as the affine classification of orientable complete flat 4-manifolds, an algebraic criterion of an affine e…

Paper presents a new triangular form for flat systems.