Hokkaido Mathematical Journal 47(2) 245-267 Jun 2018 [Refereed]

Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelik...

Proceedings of the Japan Academy, Ser. A, Mathematical Sciences 94(3) 25-30 Feb 2018 [Refereed]

By Hartman--Nirenberg's theorem, any complete flat hypersurface in Euclidean space must be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples. Flat fronts are flat hypersurfaces with admissi...

Duality of singularities for spacelike CMC surfaces

Atsufumi Honda

Kobe Journal of Mathematics 34(1-2) 1-11 Dec 2017 [Refereed]

We prove the duality of singularities for spacelike CMC surfaces
in the Lorentz-Minkowski 3-space.

Journal of Geometric Analysis 3(27) 2400-2417 Jul 2017 [Refereed]

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...

Differential Geometry and its Applications 52(14) 64-77 Jun 2017 [Refereed]

In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we s...

Beiträge zur Algebra und Geometrie 58(1) 81-91 Feb 2017 [Refereed]

We derive a permutability theorem for the Christoffel, Goursat and Darboux transformations of isothermic surfaces. As a consequence we obtain a simple proof of a relation between Darboux pairs of minimal surfaces in Euclidean space, curved flats i...

Contemporary Mathematics 675 103-120 Nov 2016 [Refereed]

In a paper of do Carmo and Dajczer, associate families of Delaunay surfaces in the Euclidean 3-space are constructed explicitly and classified completely. Following the method of do Carmo-Dajczer, Sasahara gave a construction of some associate fam...

Kyushu Journal of Mathematics 70(2) 217-226 Sep 2016 [Refereed]

Murata and Umehara gave a classification of complete flat fronts in the Euclidean 3-space and proved their orientability. Here, a flat front is a flat surface (i.e., a surface where one of the principal curvatures is identically zero) with admissi...

Journal of Geometry 106(1) 185-210 Apr 2015 [Refereed]

The lightlike geometry of codimension two spacelike submanifolds in Lorentz-Minkowski space has been developed by Izumiya and Romero Fuster (Selecta Math (N.S.) 13:23-55, 2007) which is a natural Lorentzian analogue of the classical Euclidean diff...

International Journal of Mathematics 26(4) 1540008 Mar 2015 [Refereed]

In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal ...

Selecta Mathematica. New Series 20(3) 769-785 Jul 2014 [Refereed]

It is classically known that generic smooth maps of \R^2 into \R^3 admit only isolated cross cap singularities. This suggests that the class of cross caps might be an important object in differential geometry. We show that the standard cross cap f...

Tohoku Mathematical Journal, Second Series 64(2) 171-193 Jun 2012 [Refereed]

In this paper, we parametrize the space of isometric immersions of the hyperbolic plane into the hyperbolic 3-space in terms of null-causal curves in the space of oriented geodesics. Moreover, we characterize "ideal cones" (i.e., cones whose verti...

A connected regular surface in Lorentz-Minkowski 3-space is called a mixed
type surface if the spacelike, timelike and lightlike point sets are all
non-empty. Lightlike points on mixed type surfaces may be regarded as singular
points of the induce...

Calabi and Cheng-Yau's Bernstein-type theorem asserts that an entire zero
mean curvature graph in Lorentz-Minkowski -space
which admits only space-like points is a hyperplane. Recently, the third and
fourth authors p...

In the second, fourth and fifth authors' previous work, a duality on generic
real analytic cuspidal edges in the Euclidean 3-space
preserving their singular set images and first fundamental forms, was given. In
this paper, we sho...

In Lorentz-Minkowski 3-space, the fundamental theorem of spacelike curves is known, if they have spacelike, timelike or lightlike curvature vector fields. However, such a theory cannot be applied to spacelike curves with type-changing curvature ve...

A mixed type surface is a connected regular surface in a Lorentzian
3-manifold with non-empty spacelike and timelike point sets. The induced metric
of a mixed type surface is a signature-changing metric, and their lightlike
points may be regarded ...

We introduce two invariants called the secondary cuspidal curvature and the
bias on -cuspidal edges, and investigate their basic properties. While the
secondary cuspidal curvature is an analog of the cuspidal curvature of
(ordinary) cuspidal ...

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 Kosso...

Transformations and orientability of extrinsically flat surfaces

Atsufumi HONDA

Progress in Surface Theory, Mathematisches Forschungsinstitut Oberwolfach May 2013