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* [[Horn torus]] is a related construct where <math>c = a</math>, 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 <math>c < a</math>, 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:

<math>x = (c + a \cos v)\cos u, y = (c + a \cos v)\sin u, z = a \sin v</math>

{| class="sortable" border="1"

! Quantity !! Meaning in general !! Value

|-

| <math>E,F,G</math> for [[first fundamental form]] using this parametrization || The Riemannian metric is given by <math>ds^2 = E \, du^2 + 2F \, du \, dv + G \, dv^2</math> || {{fillin}}

|-

| <math>e,f,g</math> 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}}

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