Standard results from calculus show that one can use relatively elementary methods to study a surface if it is a surface of revolution, and classical results in differential and algebraic geometry show the usefulness of recognizing the ruled surfaces that are essentially given as parametrized families of parallel submanifolds. Of course, more general notions known as fiberings have also played important roles in geometry and topology for a long time, and in this context one of the most natural questions is whether a given space can be realized by a fibering with suitable properties. Topo-logical questions of this sort have been studied intermittently for about six decades (e.g., see [BoS], [St], [Br], [CG], [Sch], [Got], [Fe 1-2]), and in some important cases one has a fairly good understanding of the types of fiberings a space can support. For example, if the space in question is a sphere, then every smooth fibering with compact connected fibers is loosely related to one of the so-called Hopf fiberings whose fibers are spheres and whose parameter spaces are projective spaces over the complex numbers, quaternions or Cayley numbers (cf. [Br, §6]), and if the space in question is the coordinate space R~n, then no fiberings of this type exist if one insists that neither the fibers nor the base consist of a single point [BoS].
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