Historical Note

Classically, tightness is defined in terms of the total absolute curvature:

\tau(f)={1\over 2\pi}\int_M |K|\,dA.

It is possible to show that \tau(f)\ge 4-\chi(M), and when equality holds, f is tight.

This definition is only valid for smooth immersions, but the two-piece property makes sense even in the polyhedral case.

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