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.

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