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There are two important classes of surfaces in the hyperbolic space. One of class consists of extrinsic flat surfaces, which is an analogous notion to developable surfaces in the Euclidean space. Another class consists of horo-flat surfaces, which are given by one-parameter families of horocycles. We use the Legendrian dualities between hyperbolic space, de Sitter space and the lightcone in the Lorentz-Minkowski 4-space in order to study the geometry of flat surfaces defined along the singular set of a cuspidal edge in the hyperbolic space. Such flat surfaces can be considered as flat approximations of the cuspidal edge. We investigate the geometrical properties of a cuspidal edge in terms of the special properties of its flat approximations.

Lagrangian equivalence among Lagrangian submanifolds and S:P+-Legendrian equivalence among graph-like Legendrian unfoldings are equivalent. We investigate r-parameter families of Lagrangian submanifolds and r-parameter families of graph-like Legendrian unfoldings. Then we show that r-parameter families of Lagrangian equivalence and r-parameter families of S:P+-Legendrian equivalence are equivalent. As an application, we give a generic classification of bifurcations of Lagrangian submanifold germs for lower dimensions.

The pedal of a curve in the Euclidean plane is a classical subject which has a singular point at the inflection point of the original curve. The primitive of a curve is a curve given by the inverse construction for making the pedal. We consider relatives of the primitive of a plane curve which we call primitivoids. We investigate the relationship of primitivoids and pedals of plane curves., Comment: arXiv admin note: text overlap with arXiv:1912.03114

In this paper, we consider the curves on the surface in the unit 3-sphere. For a regular curve on a surface in the unit 3-sphere, we have a moving frame along the curve which is called a spherical Darboux frame. We induce two special vector fields along the curve with respect to the spherical Darboux frame and investigate the singularities of extrinsic flat great circular surfaces associated with these vector fields.

A surface in the Lorentz-Minkowski $3$-space is generally a mixed type surface, namely, it has the lightlike locus. We study local differential geometric properties of such a locus on a mixed type surface. We define a frame field along a lightlike locus, and using it, we define two lightlike ruled surfaces along a lightlike locus which can be regarded as lightlike approximations of the surface along the lightlike locus. We study a relationship of singularities of these lightlike surfaces and differential geometric properties of the lightlike locus. We also consider the intersection curve of two lightlike approximations, which gives a model curve of the lightlike locus., 16 pages, 3 figures

We investigate differential geometry of Bertrand curves in 3-dimensional space form from a viewpoint of curves on surfaces. We define a special kind of surface, named geodesic surface, generated by geodesics in 3-dimensional space form. This kind of surface is nothing else, but a generalization of ruled surface in 3-dimensional Euclidean space. As results, we show that the Bertrand curve is related to the mean curvature of principal normal geodesic surface.

A world sheet in anti-de Sitter space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds. We consider the family of lightlike hypersurfaces along spacelike submanifolds in the world sheet. The locus of the singularities of lightlike hypersurfaces along spacelike submanifolds forms the caustic of the world sheet. This notion is originally introduced by Bousso and Randall in theoretical physics. In this paper we give a mathematical framework for the caustics of world sheets as an application of the theory of graph-like Legendrian unfoldings., Comment: arXiv admin note: substantial text overlap with arXiv:1503.08888

In this paper, we define generalized Sabban frames of curves in \(S^{n}\) and investigate the singularities of the spherical duals of the curves by using invariants with respect to such frames.

We consider equivalence relations among smooth map germs with respect to geometry of G-structures on the target space germ. These equivalence relations are natural generalization of right-left equivalence (i.e., A-equivalence) in the sense of Thom-Mather depending on geometric structures on the target space germ. Unfortunately, these equivalence relations are not necessarily geometric subgroups in the sense of Damon (1984). However, we have interesting applications of these equivalence relations.

We consider two types of developable surfaces along a frontal curve on an embedded surface in the Euclidean 3-space. One is called the osculating developable surface, and the other is called the normal developable surface. The frontal curve may have singular points. We give new invariants of the frontal curve which characterize singularities of the developable surfaces. Moreover, a frontal curve is a contour generator with respect to an orthogonal projection or a central projection if and only if one of these invariants is constantly equal to zero.

We consider a developable surface normal to a surface along a curve on the surface. We call it a normal developable surface along the curve on the surface. We investigate the uniqueness and the singularities of such developable surfaces. We discover two new invariants of curves on a surface that characterize these singularities.

As an application of the theory of graph-like Legendrian unfoldings, relations of the hidden structures of caustics, and wave front propagations are revealed.

In this paper, we define generalized Sabban frames of non-lightlike curves on $$S_{1}^{n}$$ and investigate the singularities of de Sitter dual hypersurfaces of timelike Sabban curves.

In this study, we construct one-parameter families of new extrinsic differential geometries on spacelike submanifolds of codimension two in Lorentz–Minkowski space. Moreover, we give our previous results as special cases of these spacelike submanifolds of codimension two. Furthermore, we investigate spacelike curves in Lorentz–Minkowski 3-space from different viewpoints as another special case.

We consider a developable surface tangent to a surface along a curve on the surface. We call it an osculating developable surface along the curve on the surface. We investigate the uniqueness and the singularities of such developable surfaces. We discover two new invariants of curves on a surface which characterize these singularities. As a by-product, we show that a curve is a contour generator with respect to an orthogonal projection or a central projection if and only if one of these invariants constantly equal to zero.

We can use a moving frame, as in the case of regular plane curves in the Euclidean plane, in order to define the arc-length parameter and the Frenet formula for non-lightlike regular curves in the Lorentz-Minkowski plane. This leads naturally to a well defined evolute associated to non-lightlike regular curves without inflection points in the Lorentz-Minkowski plane. However, at a lightlike point the curve shifts between a spacelike and a timelike region and the evolute cannot be defined by using this moving frame. In this paper, we introduce an alternative frame, the lightcone frame, that will allow us to associate an evolute to regular curves without inflection points in the Lorentz-Minkowski plane. Moreover, under appropriate conditions, we shall also be able to obtain globally defined evolutes of regular curves with inflection points. We investigate here the geometric properties of the evolute at lightlike points and inflection points.

This is a half survey article on the recent development of the theory of graph-like Legendrian unfoldings which is the sequel to the previous surveys. 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 Legendrian submanifolds. In particular, natural equivalence relations among graph-like Legendrian unfoldings are introduced and geometric properties of these equivalence relations are investigated. Although this is a survey article, some new original results and proofs for some implicitly known results are given.

For a regular curve on a surface, we have a moving frame along the curve which is called the Darboux frame. We induce three special vector fields along the curve associated to the Darboux frame and investigate their singularities as an application of the theory of spherical dualities. Moreover, characterizations of isophotic curves on a surface are given by using one of the three special vector fields.

We investigate a special timelike surfaces in Anti de Sitter 3-space. We call such a timelike surface an Anti de Sitter horospherical flat surface which belongs to a class of surfaces given by one parameter families of Anti de Sitter horocycle. We give a generic classification of singularities and study the geometric properties of such surfaces from the viewpoint of Legendrian singularity theory.

We introduce the totally absolute lightcone curvature for a spacelike submanifold with general codimension and investigate global properties of this curvature. One of the consequences is that the Chern–Lashof type inequality holds. Then the notion of lightlike tightness is naturally induced.

We study timelike hypersurfaces in anti-de Sitter space from the viewpoint of the Lagrangian/ Legendrian singularity theory.

In this paper, we consider hypersurfaces in the unit lightlike sphere. The unit sphere can be canonically embedded in the lightcone and de Sitter space in Minkowski space. We investigate these hypersurfaces in the framework of the theory of Legendrian dualities between pseudo-spheres in Minkowski space. (C) 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

We consider the singularities of lightlike hypersurfaces along spacelike submanifolds in Lorentz–Minkowski space of general codimension. As an application of the theory of Legendrian singularities, we investigate the geometric meanings of the singularities of lightlike hypersurfaces in terms of the contact of spacelike submanifolds with lightcones.

We consider projections of a smooth and regular surface M in the Minkowski 3-space along lightlike directions to a fixed transverse plane. The lightlike directions in can be parametrized by a circle on the lightcone and the resulting 1-parameter family of projections can be considered as viewing M along a special 'camera motion'. The associated 1-parameter families of contour generators and apparent contours reveal some aspects of the extrinsic and intrinsic geometry of M. We characterize geometrically the generic -codimension ≤1 singularities of a given projection and consider their bifurcations in the family of projections. We show that the families of contour generators and apparent contours are solutions of certain first order ordinary differential equations and obtain their generic local configurations.

In this paper, we consider a special class of the surfaces in 3-sphere dened by oneparameter families of great circles. We give a generic classication of singularities of such surfaces and investigate the geometric meanings from the view point of spherical geometry. In this paper we investigate a special class of surfaces in 3-sphere which are called great circular surfaces. We say that a surface in 3-sphere is a great circular surface if it is given by a oneparameter family of great circles (cf., §4). On the other hand, there appeared two kinds of curvatures in the previous theory of surfaces in 3-sphere, One is called the extrinsic Gauss curvature Ke and another is the intrinsic Gauss curvature KI. The intrinsic Gauss curvature is nothing but the Gauss curvature defined by the induced Riemannian metric on the surface. The relation between these curvatures is known that Ke = KI − 1. We can show that an extrinsic flat surface is (at least locally) parametrized as a great circular surface (cf., Theorem 3.3). Such a surface is an extrinsic flat great circular surface (briefly, we call an E-flat great circular surface). This is one of the motivation to investigate great circular surfaces. In Euclidean space, surfaces with the vanishing Gauss curvature are developable surfaces which belong to a special class of ruled surfaces [5, 6]. Therefore, the notion of great circular surfaces is one of the analogous notions with ruled surfaces in 3-sphere. In this paper, we study geometric properties and singularities of great circular surfaces. However, there is the canonical double covering π : S 3 −→ RP 3 onto the projective space. A great circle corresponds to a projective line in RP 3 , so that the singularities of great circular surfaces are the same as those of ruled surfaces. There are a lot of researches on developable surfaces in R 3 ⊂ RP 3 from the view point of Projective differential geometry [2, 4, 12, 16]. We investigate the singularities of great circular surfaces from the view point of spherical geometry (i.e, SO(4)invariant geometry). For any smooth curve A : I −→ SO(4) in the rotation group SO(4), we can define a parametrization FA of a great circular surface M = Image FA in 3-sphere. We can easily show

We study horo-tight immersions of manifolds into hyperbolic spaces. The main result gives several characterizations of horo-tightness of spheres, answering a question proposed by Cecil and Ryan. For instance, we prove that a sphere is horo-tight if and only if it is tight in the hyperbolic sense. For codimension bigger than one, it follows that horo-tight spheres in hyperbolic space are metric spheres. We also prove that horo-tight hyperspheres are characterized by the property that both of its total absolute horospherical curvatures attend their minimum value. We also introduce the notion of weak horo-tightness: an immersion is weak horo-tight if only one of its total absolute curvature attends its minimum. We prove a characterization theorem for weak horo-tight hyperspheres.

We study the horospherical geometry of submanifolds in hyperbolic space. The main result is a formula for the total absolute horospherical curvature of M, which implies, for the horospherical geometry, the analogues of classical inequalities of the Euclidean Geometry. We prove the horospherical Chern–Lashof inequality for surfaces in 3-space and the horospherical Fenchel and Fary–Milnor's theorems.

We define and study the asymptotic, characteristic, and principal-direction fields associated to a self-adjoint operator on a smooth surface M endowed with a metric g which is singular along a smooth curve on M.

We study orthogonal projections of embedded surfaces M in H3+ (−1) along horocycles to planes. The singularities of the projections capture the extrinsic geometry of M related to the lightcone Gauss map. We give geometric characterizations of these singularities and prove a Koenderink-type theorem that relates the hyperbolic curvature of the surface to the curvature of the profile and of the normal section of the surface. We also prove duality results concerning the bifurcation set of the family of projections.

We study timelike surfaces in Anti de Sitter 3-space as an application of singularity theory. We define two mappings associated to a timelike surface which are called a Anti de Sitter nullcone Gauss image and a Anti de Sitter torus Gauss map. We also define a family of functions named the Anti de Sitter null height function on the timelike surface. We use this family of functions as a basic tool to investigate the geometric meanings of singularities of the Anti de Sitter nullcone Gauss image and the Anti de Sitter torus Gauss map.

Using the Legnedrian duarities between surfaces in pseudo-spheres in Lorentz-{Minkow}{ski} 4-space, we study various kind of flat surfaces in pseudo-spheres. We consider a surface in the pseudo-sphere and its dual surface. Flatness of a surface is defined by the degeneracy of the dual surface similar to the case for the Gauss map of a flat surface in the Euclidean space. We study singularities of these flat surfaces and dualities of singularities.

We define the notions of (S 1 × S 2 )-nullcone Legendrian Gauss maps and S + 2 -nullcone Lagrangian Gauss maps on spacelike surfaces in anti de Sitter 4-space. We investigate the relationships between singularities of these maps and geometric properties of surfaces as an application of the theory of Legendrian/Lagrangian singularities. By using S + 2 -nullcone Lagrangian Gauss maps, we define the notion of S + 2 -nullcone Gauss-Kronecker curvatures and show a Gauss-Bonnet type theorem as a global property. We also introduce the notion of horospherical Gauss maps which have geometric properties different from those of the above Gauss maps. As a consequence, we can say that anti de Sitter space has much richer geometric properties than the other space forms such as Euclidean space, hyperbolic space, Lorentz-Minkowski space and de Sitter space.

The lightlike hypersurfaces in Lorentz–Minkowski space are of special interest in Relativity Theory. In particular, the singularities of these hypersurfaces provide good models for the study of different horizon types. We introduce the notion of flatness for these hypersurfaces and study their singularities. The classification result asserts that a generic classification of flat lightlike hypersurfaces is quite different from that of generic lightlike hypersurfaces.

We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing normal curvature) in Minkowski space. We use the analysis of their contacts with hyperplanes and hyperquadrics in order to get some global information on them. As a consequence we obtain new versions of Carathéodory's and Loewner's conjectures on spacelike surfaces in 4-dimensional Minkowski space and 4-flattenings theorems for closed spacelike curves in 3-dimensional Minkowski space.

We show four Legendrian dualities between pseudo-spheres in Minkowski space as a basic theorem. We can apply such dualities for constructing extrinsic differential geometry of spacelike hypersurfaces in pseudo-spheres. In this paper we stick to spacelike hypersurfaces in the lightcone and establish an extrinsic differential geometry which we call the lightcone differential geometry.

The Lorentzian space form with the positive curvature is called de Sitter space which is an important subject in the theory of relativity. In this paper we consider spacelike curves in de Sitter 3-space. We define the notion of lightlike surfaces of spacelike curves in de Sitter 3-space. We investigate the geometric meanings of the singularities of such surfaces.

We study in this paper orthogonal projections in a hyperbolic space to hyperhorospheres and hyperplanes. We deal in more details with the case of embedded surfaces $M$ in $H^3_+(-1)$. We study the generic singularities of the projections of $M$ to horospheres and planes. We give geometric characterizations of these singularities and prove duality results concerning the bifurcation sets of the families of projections. We also prove Koenderink type theorems that give the curvature of the surface in terms of the curvatures of the profile and the normal section of the surface.

We consider extrinsic differential geometry on spacelike hypersurfaces in Minkowski pseudo-spheres (hyperbolic space, de Sitter space and the lightcone). In the previous paper [S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, Preprint] we have shown a basic Legendrian duality theorem between pseudo-spheres. We define the spacelike parallels by using the basic Legendrian duality theorem. This definition unifies the notions of parallels of spacelike hypersurfaces in pseudo-spheres. We also define the evolute as the locus of singularities of the spacelike parallels. These notions are investigated as applications of Lagrangian or Legendrian singularity theory. We consider geometric properties of non-singular spacelike hypersurfaces corresponding to singularities of spacelike parallels or evolutes.

We introduce the notion of the lightcone Gauss–Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss–Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.

A circular surface is a one-parameter family of standard circles in ℝ3. In this paper some correspondences between the properties of circular surfaces and those of classical ruled surfaces are investigated. Singularities of circular surfaces are also studied.