Inverting a cylinder through isometric immersions and isometric embeddings

Abstract

It is shown that a right circular cylinder can be turned inside out through immersions which preserve its flat Riemannian metric if and only if its diameter is greater than its height.Such a cylinder can be turned inside out through embeddings which preserve its flat Riemannian metric provided its diameter is greater than (w + 2)/ir times its height.A flat Möbius strip has an immersion into Euclidean three dimensional space which preserves its Riemannian metric if and only if its length is greater than ir/2 times its height.§ §2-10.For each isometry /: Ch -» R3 we define an element G(f) which lies in Z for h > 1, and in Z2 for h < 1, and which is invariant under isometric Presented to the Society, August 25, 1970 under the title Bending and immersing strips and bands; received by the editors September 9, 1975.AMS (MOS) subject classifications (1970).