Tightness and Euler Characteristic

 The Gauss-Bonnet theorem tells us $$\chi(M) = {1\over 2\pi} \int_M K\,dx,$$ so if $K$ doesn't change sign, then $\chi(M)$ has the same sign as $K$. Theorem: a surface in a CES is tight if, and only if, its curvature always has the same sign as its Euler characteristic.