The push-out space of immersed spheres

Let $f\colon M^m\to\mathbb R^{m+1}$ be an immersion of an orientable $m$-dimensional connected smooth manifold $M$ without boundary and assume that $\xi$ is a unit normal field for $f$. For a real number $t$ the map $f_{t\xi}\colon M^m\to\mathbb R^{m+1}$ is defined as $f_{t\xi}(p)=f(p)+t\xi(p)$. It is known that if $f_{t\xi}$ is an immersion, then for each $p\in M$ the number of the focal points on the line segment joining $f(p)$ to $f_{t\xi}(p)$ is a constant integer. This constant integer is called the index of the parallel immersion $f_{t\xi}$ and clearly the index lies between $0$ and $m$. In case $f\colon\mathbb S^m\to\mathbb R^{m+1}$ is an immersion, we study the presence of a component of index $\mu$ in the push-out space $\Omega(f)$. If there exists a component with index $\mu=m$ in $\Omega(f)$ then $f$ is known to be a strictly convex embedding of $\mathbb S^m$. We reveal the structure of $\Omega(f)$ when $f(\mathbb S^m)$ is convex and nonconvex. We also show that the presence of a component of index $\mu$ in $\Omega(f)$ enables us to construct a continuous field of tangent planes of dimension $\mu$ on $\mathbb S^m$ and so we see that for certain values of $\mu$ there does not exist a component of index $\mu$ in $\Omega(f)$.