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.