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Systems of Clairaut type

100%

Izumiya S.

Colloquium Mathematicum

1993

tom 66

nr 2

219-226

A characterization of systems of first order differential equations with (classical) complete solutions is given. Systems with (classical) complete solutions that consist of hyperplanes are also characterized.

Caustics and wave front propagations: applications to differential geometry

63%

Izumiya S. , Takahashi M.

Banach Center Publications

2008

tom 82

nr 1

125-142

This is mainly a survey on the theory of caustics and wave front propagations with applications to differential geometry of hypersurfaces in Euclidean space. We give a brief review of the general theory of caustics and wave front propagations, which are well-known now. We also consider a relationship between caustics and wave front propagations which might be new. Moreover, we apply this theory to differential geometry of hypersurfaces, getting new geometric properties.

Formation of singularities for viscosity solutions of Hamilton-Jacobi equations

63%

Izumiya S. , Kossioris G. T.

tom 33

nr 1

127-148

Curves and surfaces in hyperbolic space

51%

Izumiya S. , Pei D. , Takahashi M.

Banach Center Publications

2004

tom 65

nr 1

107-123

In the first part (Sections 2 and 3), we give a survey of the recent results on application of singularity theory for curves and surfaces in hyperbolic space. After that we define the hyperbolic canal surface of a hyperbolic space curve and apply the results of the first part to get some geometric relations between the hyperbolic canal surface and the centre curve.

The rectifying developable and the spherical Darboux image of a space curve

51%

Izumiya S. , Katsumi H. , Yamasaki T.

tom 50

nr 1

137-149

In this paper we study singularities of certain surfaces and curves associated with the family of rectifying planes along space curves. We establish the relationships between singularities of these subjects and geometric invariants of curves which are deeply related to the order of contact with helices.