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.