Curvature, singularities and projections of smooth maps

This work is an initial attempt to extend to many-parameter families of smooth functions on a smooth manifold, and projections of smooth maps into subspaces of higher dimension, the well-known inter relations, between the space of Morse function on a smooth manifold and the space of immersions of the manifold in a cartesian space, which are given by the Gauss-maps of the immersions, and the orthogonal projections of the immersions onto lines in the Cartesian spaces. Results, both local and global, are obtained.