The push-out space of immersed spheres |
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Authors: | Y Kaya |
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Institution: | (1) Zonguldak Karaelmas University, Zonguldak, Turkey |
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Abstract: | Let f: M m → ? 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 → ? 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). |
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Keywords: | parallel immersion index of a parallel immersion push-out space |
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