Abstract
The space of all closed curves in Euclidean 3-space constitutes a Banach manifold. The existence of parallel mates to a given closed space curve depends on its total normal twist α. As a submanifold we consider the space of curves which has total normal twist an integer multiple of 2π. Also construction of special curves, like Bertrand, and helical curves with invariant total normal twist from a spherical curve will be described, and moreover, it constitutes a Banach submanifold.
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