Talk Slides (2007-04-30.ualbany) tight-CES: Tightness and Euler Characteristic

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.