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The Role of Completeness in the Type Problem for Infinitely Sheeted Planes

Any finitely sheeted plane W, or more precisely the triple as a covering surface of the finite complex plane with its projection π, is always parabolic along with its base surface . However, infinitely sheeted planes W or may or may not be parabolic even in an equivalence class by similarity of infinitely sheeted planes, where in general two multisheeted planes are similar to each other if there is a sense preserving topological mapping T of W 1 onto W 2 such that a point p ∈ W 2 is a branch point of multiplicity m ≥ 2 when and only when so is T(p) and . The purpose of this article is to give an equivalence class by similarity of infinitely sheeted planes, whose sets of projections of branch points are an arbitrarily given sequence in advance of symmetrically distributed, isolated points on the real line with respect to the origin converging only on the point at infinity such that there are both parabolic and hyperbolic (i.e. not parabolic) surfaces in and a member is parabolic if and only if is complete. This supports the hunch that complete multisheeted planes are apt to be parabolic.