arXiv research

Study shows how certain foliations in unit tangent bundles behave.

We equip the whole tangent space $TM$ to a hyperbolic manifold $M$ (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of $M$ extend to isometries of $TM$ by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In t…

Researchers create metrics on hyperbolic space's tangent bundle.

We show, using two different approaches, that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian…

Projective manifolds with specific bundles are isomorphic to simpler spaces.

Combining M-algebra and hyperbolic involutory algebra extends exceptional tangent spaces to 11 dimensions.

The paper finds lower bounds for volumes of complex geometric structures.

This paper is a continuation of the previous paper of the author[M]. We show that an affine deformation space of a hyperbolic surface of type (g,b) can be parametrized by Margulis invariants and affine twist parameters with a certain decomposition of the surface, which are associated with the Fenchel-Nielsen coordinate…

For an $n$-dimensional real hyperbolic manifold $M$, we calculate the Zariski tangent space of a character variety $χ(π_1(M),SL(n+1,\mathbb R)), n>2$ at Fuchisan loci to show that the tangent space consists of cubic forms. Furthermore we prove the Weil's local rigidity theorem for uniforml hyperbolic lattices using rea…

Partial coverings of hyperbolic surfaces equidistribute with geodesics.

The paper extends Descartes' circle theorem to n-flower configurations using hyperbolic geometry.

New surfaces show horocyclic flow isn't always minimal.

For a compact riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism. However, the riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, whi…

Study of harmonic maps with extreme Kerr-like singularities.

We consider the normalized Ricci flow evolving from an initial metric which is conformally compactifiable and asymptotically hyperbolic. We show that there is a unique evolving metric which remains in this class, and that the flow exists up to the time where the norm of the Riemann tensor diverges. Restricting to initi…

Defines horocyclic evolutes, parallels, and involutes of spacelike frontals in hyperbolic 2-space.

The study limits the cohomological dimension of certain affine manifolds with partially hyperbolic holonomy groups.

The choice of approximate posterior distributions plays a central role in stochastic variational inference (SVI). One effective solution is the use of normalizing flows \cut{defined on Euclidean spaces} to construct flexible posterior distributions. However, one key limitation of existing normalizing flows is that they…

New examples of real hypersurfaces found in complex hyperbolic quadrics.

A new framework for hyperbolic neural networks using the Klein model is introduced.

Explains exceptional surgeries connecting maps and knot orbifolds.

For a non-vanishing gradient-like vector field on a compact manifold $X^{n+1}$ with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity $n$, which is the maximum possible. (Among them are trajectories that are tangent to $\partial X$ exactly $n$ times.) We prove a lower bou…

We prove that the geodesic flow on the unit tangent bundle to a hyperbolic 2-orbifold is left-handed if and only if the orbifold is a sphere with three conic points. As a consequence, on the unit tangent bundle to a 3-conic sphere, the lift of every finite collection of closed geodesics that is zero in integral homolog…

In this paper we introduce the concept of standardized call function and we obtain a new approximating formula for the Black and Scholes call function through the hyperbolic tangent. This formula is useful for pricing and risk management as well as for extracting the implied volatility from quoted options. The latter i…

Study of singular curves in a specific type of hyperbolic distribution.

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve…

Study shows shortest periodic geodesic on hyperbolic orbisphere complements figure-eight knot.

Geodesics of the same type on curved surfaces are randomly distributed.

Using global considerations, Mess proved that the moduli space of globally hyperbolic flat Lorentzian structures on $S\times\mathbb{R}$ is the tangent bundle of the Teichmüller space of $S$, if $S$ is a closed surface. One of the goals of this paper is to deepen this surprising occurrence and to make explicit the relat…

The paper proves properties of a specific Seiberg-Witten equation over 3-manifolds.

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path i…

The Hessian of the renormalized volume of geometrically finite hyperbolic $3$-manifolds without rank-$1$ cusps, computed at the hyperbolic metric $g$ with totally geodesic boundary of the convex core, is shown to be a strictly positive bilinear form on the tangent space to Teichmüller space. The metric $g$ is known fro…

We consider a projection from the center of the unit sphere to a tangent space of it, the central projection, and study two area minimizing problems of the image of a closed subset in the sphere. One of the problems is the uniqueness of the tangent plane that minimizes the area for an arbitrary fixed subset. The other …

In the course of our work on low-volume hyperbolic 3-manifolds, we came upon a linking problem for horoball necklaces in $\mathbb{H}^3$. A horoball necklace is a collection of sequentially tangent beards (i.e. spheres) with disjoint interiors lying on a flat table (i.e. a plane) such that each bead is of diameter at mo…

Extending the earlier results for analytic curve segments, in this article we describe the asymptotic behaviour of evolution of a finite segment of a C^n-smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n-manifold. In particular, we show that if the curve satisfies certain n…

We prove that the geodesic flow on the unit tangent bundle to every hyperbolic 2-orbifold that is a sphere with 3 or 4 singular points admits explicit genus one Birkhoff sections, and we determine the associated first return maps.

The paper certifies projective rigidity for once-punctured torus bundles using twisted Alexander polynomials.

The study explores dilating set properties across Euclidean and hyperbolic geometries.

Let $S$ be a surface of negative Euler characteristic and consider a finite filling collection $Γ$ of closed curves on $S$ in minimal position. An observation of Foulon and Hasselblatt shows that $PT(S) \setminus \hatΓ$ is a finite-volume hyperbolic 3-manifold, where $PT(S)$ is the projectivized tangent bundle and $\ha…

New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harm…

Asymptotics for equidistribution of circles on hyperbolic surfaces.

Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.

New hyperbolic manifolds with diverse features created.

We classify smooth locally free actions of the real affine group on closed orientable three-dimensional manifolds up to smooth conjugacy. As a corollary, there exists a non-homogeneous action when the manifold is the unit tangent bundle of a closed surface with a hyperbolic metric.

Study on focal surfaces and evolutes of framed curves in hyperbolic 3-space using Legendrian duality.

We consider the control problem where, given an orthonormal tangent frame in the hyperbolic plane or three dimensional hyperbolic space, one is allowed to transport the frame a fixed distance $r > 0$ along the geodesic in direction of the first vector, or rotate it in place a right angle. We characterize the values of …

Two flows are topologically almost commensurable if, up to removing finitely many periodic orbits and taking finite coverings, they are topologically equivalent. We prove that all suspensions of automorphisms of the 2-dimensional torus and all geodesic flows on unit tangent bundles to hyperbolic 2-orbifolds are pairwis…

A Steiner chain of length k consists of k circles, tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be …