Tightness and Curvature
Note that
{1\over 2\pi} \int_M |K|\,dx \ge \left|{1\over 2\pi} \int_M K\,dx\right|
is a strict inequality only when
K
takes on both positive and negative values.
Theorem
: a surface in a CES is tight if, and only if, its curvature does not change sign.