arXiv research

Wave fronts on certain surfaces become dense.

We characterize singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.

The paper shows how null hypersurfaces behave in Lorentz-Minkowski space.

In this note we present a description of wave front evolving from an algebraic hypersurface by means of a pull-back of the discriminantal loci of a tame polynomial via a polynomial mapping. As an application we give examples of wave fronts which define free/almost free divisors near the focal point.

We give a definition of `coherent tangent bundles', which is an intrinsic formulation of wave fronts. In our application of coherent tangent bundles for wave fronts, the first fundamental forms and the third fundamental forms are considered as induced metrics of certain homomorphisms between vector bundles. They satisf…

We give criteria for which a principal curvature becomes a bounded $C^\infty$-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularit…

Shorter proof for wave front length in Euclidean disk

The calculus correspondence has been known to exist between generic pedal evolutions and generic wave front evolutions. In this paper, we first extend the known results on the calculus correspondence to evolutions with multi-parameters, and then give applications of calculus correspondence. Moreover, we discuss the pos…

The paper defines wave-front singularities using explicit analytic functions.

Study focal surfaces of wave fronts with unbounded curvatures.

This paper describes caustics of wave fronts reflected by a surface.

We give useful and simple criteria for determining D_4 singularities of wave fronts. As an application, we investigate behaviors of singular curvatures of cuspidal edges near D_4^+ singularities.

A first order Vassiliev invariant of an oriented knot in an $S^1$-fibration and a Seifert fibration over a surface is constructed. It takes values in a quotient of the group ring of the first homology group of the total space of the fibration. It gives rise to an invariant of wave fronts on surfaces and orbifolds relat…

High frequency limit for most of wave phenomena is known as quasiclassical limit or ray optics limit. Propagation of waves in this limit is described in terms of wave fronts and rays. Wave front is a surface of constant phase whose points are moving along rays. As it appears, their motion can be described by Hamilton e…

In this paper, we discuss the recognition problem for A_k-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors' previous work "the geometry of fronts" for surfaces. The crucial point to prove our criteria for A_k…

Study compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.

It is well known that Lagrangian dynamical systems naturally arise in describing wave front dynamics in the limit of short waves (which is called pseudoclassical limit or limit of geometrical optics). Wave fronts are the surfaces of constant phase, their points move along lines which are called rays. In non-homogeneous…

In the previous paper (arXiv:0804.0701), the authors gave criteria for A_{k+1}-type singularities on wave fronts. Using them, we show in this paper that there is a duality between singular points and inflection points on wave fronts in the projective space. As an application, we show that the algebraic sum of 2-inflect…

Let $M^2$ be an oriented 2-manifold and $f:M^2\to R^3$ a $C^\infty$-map. A point $p\in M^2$ is called a singular point if $f$ is not an immersion at $p$. The map $f$ is called a front (or wave front), if there exists a unit $C^\infty$-vector field $ν$ such that the image of each tangent vector $df(X)$ $(X\in TM^2)$ is …

Recently Arnold's $\St$ and $J^{\pm}$ invariants of generic planar curves have been generalized to the case of generic planar wave fronts. We generalize these invariants to the case of wave fronts on an arbitrary surface $F$. All invariants satisfying the axioms which naturally generalize the axioms used by Arnold are …

The paper improves Zakalyukin's lemma for frontals and applies it to surface singularities.

Develops support theorem for analytic transforms in tomography.

Study describes singularities of distance squared functions on singular surfaces.

We review the properties of transversality of distributions with respect to submersions. This allows us to construct a convolution product for a large class of distributions on Lie groupoids. We get a unital involutive algebra $\cE\_{r,s}'(G,Ω^{1/2})$ enlarging the convolution algebra $C^\infty\_c(G,Ω^{1/2})$ associate…

Cuspidal edges and swallowtails are typical non-degenerate singular points on wave fronts in the Euclidean $3$-space. Their first fundamental forms belong to a class of positive semi-definite metrics called "Kossowski metrics". A point where a Kossowski metric is not positive definite is called a singular point or a se…

In [31,32,33] the Gauss-Bonnet formulas for coherent tangent bundles over compact oriented surfaces (without boundary) were proved. We establish the Gauss-Bonnet theorem for coherent tangent bundles over compact oriented surfaces with boundary. We apply this theorem to investigate global properties of maps between surf…

We prove generalizations of the isoperimetric inequality for both spherical and hyperbolic wave fronts (i.e. piecewise smooth curves which may have cusps). We then discuss "bicycle curves" using the generalized isoperimetric inequalities. The euclidean model of a bicycle is a unit segment AB that can move so that it re…

We present an axiomatic/synthetic account of the Huygens Principle of wave fronts. The primitive notions are "touching", and (a weak notion of ) metric. The paper simplifies some of the exposition of the author's "Metric spaces and SDG", Theory and Appl. of Categories 32 (2017), 803-822

We analyze the inverse problem, originally formulated by Dix in geophysics, of reconstructing the wave speed inside a domain from boundary measurements associated with the single scattering of seismic waves. We consider a domain $\tilde M$ with a varying and possibly anisotropic wave speed which we model as a Riemannia…

A world sheet in Lorentz-Minkowski space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds in Lorentz-Minkowski space. In this paper we investigate differential geometry of world sheets in Lorentz-Minkowski space as an application of the theory of big wave fronts.

We prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover, we show an explicit formula of the non-holomorphic hyperbolic Gauss map via ano…

Study geometric isomorphisms between spacetime solutions using paracausal metrics.

Motivated by the study of wave fronts in anisotropic media, we propose an incidence geometry of anisotropic spheres in a Finsler-Minkowski space. An anisotropic version of the Laguerre functional is considered. In some circumstances, this functional can be used to determine that two wavefronts observed at distinct time…

Two wave fronts $W_1$ and $W_2$ that originated at some points of the manifold $M^n$ are said to be causally related if one of them passed through the origin of the other before the other appeared. We define the causality relation invariant $CR (W_1, W_2)$ to be the algebraic number of times the earlier born front pass…

In a previous work, the authors gave a definition of `front bundles'. Using this, we give a realization theorem for wave fronts in space forms, like as in the fundamental theorem of surface theory. As an application, we investigate the behavior of principal singular curvatures along A_2-singularities of hypersurfaces w…

In this paper, we give a definition of coherent tangent bundles of space form type, which is a generalized notion of space forms. Then, we classify their realizations in the sphere as a wave front, which is a generalization of a theorem of O'Neill and Stiel: any isometric immersion of the n-sphere into the (n+1)-sphere…

This is mainly a survey article on the recent development of the theory of graph-like Legendrian unfoldings and its applications. The notion of big Legendrian submanifolds was introduced by Zakalyukin for describing the wave front propagations. Graph-like Legendrian unfoldings belong to a special class of big Legendria…

We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate relation between a kind of boundedness of Gaussian curvatures near cuspidal edges an…

A spherical periscope in multi-dimensional space is a system of two ideal mirrors that reflect the rays emanating from a fixed point to the rays coming back to the same point, and a reversed periscope is a system of two mirrors that reflect the rays having a fixed direction to the rays having the opposite direction. We…

Reconstructing Finsler manifolds from sphere data.

In a joint work with Saji, the second and the third authors gave an intrinsic formulation of wave fronts and proved a realization theorem of wave fronts in space forms. As an application, we show that the following four objects are essentially same; * conformally flat n-manifolds (n>=3) with admissible singular points …

We describe the space of isometric immersions from the Lorentz plane $L^2$ into the 3-dimensional anti-de Sitter space, and solve several open problems of this context raised by M. Dajczer and K. Nomizu in 1981. We also obtain from the above result a description of the space of Lorentzian flat tori isometrically immers…

We show that for a large class of contact 3-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's $J^+$-type invariants of wave fronts on a surface $F$ is isomorphic to the group of Vassiliev invariant…

In this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3-space known as frontals. Also, we characterize in a simple way these singular surfaces and its fundament…

It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a (wave) front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We shall give easily-computable criteria for a singular point on a…

Let $M^m$ be an oriented manifold, let $N^{m-1}$ be an oriented closed manifold, and let $p$ be a point in $M^m$. For a smooth map $f:N^{m-1} \to M^m, p \not\in Im f,$ we introduce an invariant $awin_p(f)$ that can be regarded as a generalization of the classical winding number of a planar curve around a point. We show…

We define cuspidal curvature $κ_c$ (resp. normalized cuspidal curvature $μ_c$) along cuspidal edges (resp. at swallowtail singularity) in Riemannian $3$-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product $κ_Π$ called the product curva…

We show that the resolvent of the Laplacian on SL(3,$\mathbb{R}$)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of $\mathbb{C}$. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue op…