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