Geometry of the parabolic subset of generically immersed 3-manifolds in \(\mathbb {R}^4\)

The parabolic subset of a 3-manifold generically immersed in \(\mathbb {R}^4\) ${\mathbb{R}}^4$ is a surface. We analyze in this study the generic geometrical behavior of such surface, considered as a submanifold of \(\mathbb {R}^4\) ${\mathbb{R}}^4$ . Typical Singularity Theory techniques based on the analysis of the family of height functions are applied in order to describe the geometrical characterizations of different singularity types.