arXiv research

A compact negatively curved manifold's fundamental group isn't in a specific class of groups.

The paper studies pseudo-torsion functions of spacelike curves in Lorentz-Minkowski space.

This paper classifies fundamental groups of perforated surfaces.

Extends curve theory to non-smooth data with finite curvature and torsion.

We study the horizontally regular curves in the Heisenberg groups $H_n$. We show the fundamental theorem of curves in $H_n$ $(n\geq 2)$ and define the concept of the orders for horizontally regular curves. We also show that the curve $γ$ is of order $k$ if and only if $γ$ lies in $H_k$ but not in $H_{k-1}$ up to a Heis…

Anabelian geometry reformulated using Hodge theory for hyperbolic curves.

Study of curves in dual space with constant curvature and torsion.

The study extends removability results for quasiregular curves in Euclidean spaces.

The paper classifies compact affine quaternionic curves and surfaces.

Lecture notes on Lorentz geometry, focusing on curves and surfaces.

Extends submanifold theorem to general spaces.

In this paper, we investigate the similarity transformations in the Minkowski-n space. We study the geometric invariants of non-null curves under the similarity transformations. Besides, we extend the fundamental theorem for a non-null curve according to a similarity motion. We determine all non-null self-similar curve…

New vanishing theorems for genera derived under almost nonnegative Ricci curvature.

In this paper, we give a fully detailed exposition of computing fundamental groups of complements of line arrangements using the Moishezon-Teicher technique for computing the braid monodromy of a curve and the Van-Kampen theorem which induces a presentation of the fundamental group of the complement from the braid mono…

In this paper, we firstly introduce the group of similarity transformations in the Minkowski-3 space. We describe differential- geometric invariants of a non-lightlike curve according to the group of similarity transformations of the Minkowski 3-space. We show extension of fundamental theorem for non-lightlike curves u…

We discuss some aspects of the differential geometry of curves in Minkowski space. We establish the Serret-Frenet equations in Minkowski space and use them to give a very simple proof of the fundamental theorem of curves in Minkowski space. We also state and prove two other theorems which represent Minkowskian versions…

Proves Morse index theorem for geodesics in conic Finsler manifolds.

This paper concerns complete noncompact manifolds with nonnegative Ricci curvature. Roughly, we say that M has the loops to infinity property if given any noncontractible closed curve, C, and given any compact set, K, there exists a closed curve contained in M\K which is homotopic to C. The main theorems in this paper …

We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is prove…

The paper establishes analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.

We established existence of periodic Reeb orbits for a large class of tight contact structures on closed 3-manifolds, notably the Stein fillable structures, based on a fundamental theorem of Cliff Taubes on symplectic 4-manifolds.

The paper uses Tannakian reconstruction to understand hyperbolic log-orbi curves.

A simplicial complex is called negatively curved if all its simplices are isometric to simplices in hyperbolic space, and it satisfies Gromov's Link Condition. We prove that, subject to certain conditions, a compact graph of spaces whose vertex spaces are negatively curved 2-complexes, and whose edge spaces are points …

A theorem connects integral of second-order derivatives to function rise.

We study the local equivalence problems of curves and surfaces in three dimensional Heisenberg group via Cartans method of moving frames and Lie groups, and find a complete set of invariants for curves and surfaces. For surfaces, in terms of these invariants and their suitable derivatives, we also give a Gaussian curva…

Study shows infinitely many nonnegatively curved metrics on quotient spaces.

In this note we discuss the fundamental groups and diameters of positively Ricci curved $n$-manifolds. We use a method combining the results about equivarient Hausdorff convergence developed by Fukaya and Yamaguchi with the Ricci version of splitting theorem by Cheeger and Colding to give new information on the topolog…

The paper classifies hypersurfaces in Heisenberg groups with rotational symmetry.

The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curva…

New discrete curves defined in space forms with geometric properties.

In this paper, we study the structure of the singular set for a $C^{1}$ smooth surface in the $3$-dimensional Heisenberg group $\boldsymbol{H}_{1}$. We discover a Codazzi-like equation for the $p$-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation …

Let $M$ be a closed Riemannian manifold with a parallel 1-form $Ω$. We prove two theorems about the curve shortening flow in $M$. One is that the {\csf} $\ct$ in $M$ exists for all $t$ in $[0, \infty)$, if it satisfies $Ω(T)\geq 0$ on the initial curve $\co$. Here $T$ is the unit tangent vector on $\co$. The other one …

Develops thermodynamic formalism for quasimorphisms on negatively curved spaces.

In this paper we investigate Uludag's method for constructing new curves whose fundamental groups are central extensions of the fundamental group of the original curve by finite cyclic groups. In the first part, we give some generalizations to his method in order to get new families of curves with controlled fundamenta…

New comparison theorems for rotationally symmetric self-shrinkers help in proving the uniqueness of the Angenent torus.

Paper extends Schur's theorem to spherical curves via monotonicity.

New theorem counts curves on orbifolds.

In this note, we present a new method for computing fundamental groups of curve complements using a variation of the Zariski-Van Kampen method on general ruled surfaces. As an application we give an alternative (computation-free) proof for the fundamental group of generic $(p,q)$-torus curves.

We develop a general term structure framework taking stochastic discontinuities explicitly into account. Stochastic discontinuities are a key feature in interest rate markets, as for example the jumps of the term structures in correspondence to monetary policy meetings of the ECB show. We provide a general analysis of …

For a fundamental solution of Laplace's equation on the $R$-radius $d$-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geo…

Holomorphic curves in moduli spaces are quasi-isometrically immersed.

This paper generalizes monodromy maps for projective structures with poles.

The paper proves a generalized inverse function theorem for curved $L_\infty$ spaces.

Study shows fundamental groups of certain curved spaces are simple.

Study on null-torsion holomorphic curves in 6-sphere, focusing on their second variation.

Let $M$ be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group $Γ$ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover $\tilde M$ is a higher rank symmetric space iff $H^2_b(M;\R)\to H^2(M;\R)$ is injective (and otherwis…

The paper proves drilled bundles over graphs are virtually special cubulable.

In this paper, we show that a nontrivial compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.