Realization of smooth functions on surfaces as height functions

A criterion describing all functions with finitely many critical points on two-dimensional surfaces that can be height functions corresponding to some immersions of the surface in three-dimensional Euclidean space is established. It is proved that each smooth deformation of a Morse function on the surface can be realized as the deformation of the height function induced by a suitable deformation of the immersion of the surface in $\mathbb R^3$. A new proof of the well-known result on the path connectedness of the space of all smooth immersions of a two-dimensional sphere in $\mathbb R^3$ obtained. A new description of an eversion of a two-dimensional sphere in $\mathbb R^3$ is given. Generalizations of S. Matveev's result on the connectedness of the space of Morse functions with fixed numbers of minima and maxima on a closed surface are established.