# Changes

## Ring torus

, 06:53, 12 August 2011
no edit summary
* [[Horn torus]] is a related construct where $c = a$, i.e., the case of a circle being revolved about its tangent line. This is topologically not even a manifold.
* [[Spindle torus]] is a related construct where $c < a$, i.e., the case of a circle being revolved about a line intersecting it. A spindle torus has an ''inner'' and ''outer'' surface respectively called a [[lemon]] (the surface of revolution of a [[circular lens]]) and an [[apple]]. The spindle torus is topologically not even a manifold, but, taken individually, the lemon and the apple are topologically both 2-spheres.

==Fundamental forms and curvatures==

For the table below, we consider the parametric description:

$x = (c + a \cos v)\cos u, y = (c + a \cos v)\sin u, z = a \sin v$

{| class="sortable" border="1"
! Quantity !! Meaning in general !! Value
|-
| $E,F,G$ for [[first fundamental form]] using this parametrization || The Riemannian metric is given by $ds^2 = E \, du^2 + 2F \, du \, dv + G \, dv^2$ || {{fillin}}
|-
| $e,f,g$ for [[second fundamental form]] using this parametrization || {{fillin}} || {{fillin}}
|-
| [[principal curvature]]s using this parametrization || eigenvalues of the [[shape operator]] || {{fillin}}
|-
| [[mean curvature]] || arithmetic mean of the principal curvatures = half the trace of the shape operator || {{fillin}}
|-
| [[Gaussian curvature]] || product of the principal curvatures = determinant of shape operator || {{fillin}}
|}

==Verification of theorems==

===Gauss-Bonnet theorem===

{{fillin}}