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 torus has $\chi(M)=0$, and our previous examples have zero curvature at every point, hence in this case $\tau(M)=0$, and so the inequality above is sharp. A surface immersed in a CES is tight provided $\tau(M) = |\chi(M)|$.