Tightness in CES

We saw that, for smooth surfaces in
$\R^3$, tightness was originally defined as having
minimal total absolute curvature, $\tau(M)$. We
can do the same for surfaces in CES.
Note that
$$\tau(M)={1\over 2\pi} \int_M |K|\,dx \ge \left|{1\over 2\pi} \int_M K\,dx\right| = |\chi(M)|.$$
A surface immersed in a CES is tight provided $\tau(M) = |\chi(M)|$. |