Inverting a cylinder through isometric immersions and isometric embeddings

Halpern, B.; Weaver, C.

doi:10.1090/S0002-9947-1977-0474388-1

Inverting a cylinder through isometric immersions and isometric embeddings
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by B. Halpern and C. Weaver PDF

Trans. Amer. Math. Soc. 230 (1977), 41-70 Request permission

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 $(\pi + 2)/\pi$ 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 $\pi /2$ times its height.

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