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On complete solutions and complete singular solutions of second order ordinary differential equations

100%
Takahashi M.
Colloquium Mathematicum
|
2007
|
tom 109
|
nr 2
271-285
EN
A complete solution of an implicit second order ordinary differential equation is defined by an immersive two-parameter family of geometric solutions on the equation hypersurface. We show that a completely integrable equation is either of Clairaut type or of first order type. Moreover, we define a complete singular solution, an immersive one-parameter family of singular solutions on the contact singular set. We give conditions for existence of a complete solution and a complete singular solution of implicit second order ordinary differential equations.
2
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Caustics and wave front propagations: applications to differential geometry

63%
Izumiya S., Takahashi M.
Banach Center Publications
|
2008
|
tom 82
|
nr 1
125-142
EN
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.
3
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Curves and surfaces in hyperbolic space

51%
Izumiya S., Pei D., Takahashi M.
Banach Center Publications
|
2004
|
tom 65
|
nr 1
107-123
EN
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.
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