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)|. |