The push-out space of immersed spheres

Abstract

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