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In this paper, we define a perspective projection of a given immersed n-dimensional hypersurface as a C^∞ map via a C^∞ immersion from the given n-manifold to S^$n+1$, and characterize when and only when such a perspective projection is nonsingular. In order to obtain such characterizations, we consider an immersion from an n-dimensional manifold to S^$n+1$. We first obtain equivalent conditions for a given point P of S^$n+1$ to be outside the union of tangent great hyperspheres of a given immersed n-dimensional manifold r(N) in S^$n+1$ (Theorem 2.4). It turns out that if such a point P exists then the given manifold N must be diffeomorphic to S^n and in the case that n\geq2 the given immersion r:N→S^$n+1$ must be an embedding. Then, we obtain characterizations of a perspective projection of a given immersed n-dimensional manifold to be non-singular. Next, we obtain one more equivalent condition in terms of hedgehogs when the given N is S^n and the given immersion is an embedding (Theorem 3.3). We also explain why we consider these equivalent conditions for an embedding r:S^n→S^$n+1$ instead of an embedding r^~S^n→R^$n+1$ in terms of hedgehogs.
- Hokkaido Mathematical Journal
Hokkaido Mathematical Journal 40 (3), 361-373, 2011-10
Department of Mathematics, Hokkaido University