Vol. 57, No. 1, 1975

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Developable surfaces in hyperbolic space

Esther Portnoy

Vol. 57 (1975), No. 1, 281–288

A developable surface in hyperbolic space is a ruled surface (the trace of a moving hyperbolic line) whose tangent hyperbolic plane is constant along elements. This paper discusses the following hyperbolic analogs of theorems on developable surfaces in euclidean space:

1. A ruled surface is developable if and only if the tangent to the directrix, the unit vector giving the direction of the element, and the covariant derivative of the latter along the directrix, are linearly dependent.

2. A developable surface consists of portions of cones, tangential surfaces, and geodesic cylinders (to be defined).

3. A developable surface is applicable on a hyperbolic plane.

4. A flat surface in hyperbolic space (a surface whose intrinsic curvature is the same as that of a hyperbolic plane) is necessarily developable.

Mathematical Subject Classification 2000
Primary: 53A35
Received: 1 April 1974
Published: 1 March 1975
Esther Portnoy