Applications to differential geometry and singularity theory of differential equations

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As applications of singularity theory, we have studied a qualitative theory for implicit ordinary differential equations. We gave a generic classification of semi-local first order ordinary differential equations of Clairaut type. Moreover, we defined complete solutions for singular first order ordinary differential equations and gave its an existence condition. We also gave types of completely integrable implicit second order ordinary differential equations.As applications to differential geometry, we have studied a theory of Legendre curves. We gave the existence and the uniqueness for Legendre curves via Legendre curvatures. By using the Legendre curvature, we investigated the evolutes and the involutes. Moreover, we gave a generic classifications of tangent surfaces associated with G2 and D4 geometries.