Tightness by Euler Characteristic
To be tight, a surface must have the following curvature restrictions:
Sphere (
\chi = 2
)
Projective Plane (
\chi=1
)
K > 0
everywhere
Torus (
\chi = 0
)
Klein Bottle (
\chi=0
)
K = 0
everywhere
Everything else (
\chi < 0
)
K < 0
everywhere