Differential Geometry and Partial Differential Equations as an application of Singularity theory

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In this research project, we applied Singularity Theory to some areas in Mathematics such as Differential Geometry, Symplectic Geometry, non-linear Partial Differential Equations etc, so that we have obtained several results. Moreover, we have obtained related results on some boundary areas such as Astrophysics etc. Especially, we applied Singularity Theory to Differential Geometry of submanifolds in several kinds of space forms. Then we constructed new geometries (Horospherical Geometry, Slant Geometry) and induced some new invariants. We also clarified the geometric meanings of these invariants. As a result, we have a geometric characterization of the singularities andthe shape of event horizons