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)|.