The theory of graph-like Legendrian unfoldings and its applications

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 Legendrian submanifolds. Although this is a survey article, some new original results and the corrected proofs of some results are given.Comment: 30 pages,9 figure

Differential Geometry from the viewpoint of Lagrangian or Legendrian singularity theory

This is a half survey on the classical results of extrinsic differential geometry of hypersurfaces in Euclidean space from the view point of Lagrangian or Legendrian singularity theory. Many results in this paper have been already obtained in some articles. However, we can discover some new information of geometric properties of hypersurfaces from this point of view

Total lightcone curvatures of spacelike submanifolds in Lorentz-Minkowski space

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

Timelike hypersurfaces in de Sitter space and Legendrian singularities

We construct a basic framework for the study of extrinsic differential geometry on timelike hypersurfaces from the view point of the theory of Legendrian singularities. As an application, we study the contact of timelike hypersurfaces with flat totally umbilic timelike hypersurfaces in de Sitter space

Geometry of world sheets in Lorentz-Minkowski space (Theory of singularities of smooth mappings and around it)

"Theory of singularities of smooth mappings and around it". November 25～29, 2013. edited by Takashi Nishimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.A world sheet in Lorentz-Minkowski space is a timelike submanifold consisting of a oneparameter 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