The push-out space of immersed spheres
Y. Kaya
article (Uluslararası, Hakemli) | Yazar(lar): Y. Kaya | Yıl: 2006 | Dergi/Kurum: SIBERIAN MATHEMATICAL JOURNAL | İndeks: WOS.SCI | Anahtar Kelimeler: parallel immersion, index of a parallel immersion, push-out space — Özet: Let f : M-m --> Rm+1 be an immersion of an orientable m-dimensional connected smooth manifold M
Let f : M-m --> Rm+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) : M-m --> Rm+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 is an element of 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 : S-m --> Rm+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 S-m. We reveal the structure of Omega(f) when f (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 S-m and so we see that for certain values of mu there does not exist a component of index mu in Omega(f).