Given a regular curve

*T*(*t*, *v*) = (*a*(*t*), *v* *a*'(*t*))

The pictures on this page are all projections of the tangent bundle of
the "figure eight" curve (while they may not look the same, all the
pictures come from the same surface in **R**^{4}). The
variable *v* ranges from -1 to 1. The surface is colored with
respect to the *t* variable.

(cos(t), sin(2 t)/2) |
Its tangent bundle |

Two movies of the tangent bundle rotating in **R**^{4}.

The same projection as the picture on the top of the page, colored with respect to height in the projected direction.

As one might expect, the tangent bundle is a very special surface. Obviously it is a ruled surface. Every point on the tangent bundle is parabolic, and a point (

Perhaps the most surprising property of the tangent bundle is that its Gaussian curvature and its normal curvature are identical, namely

where *k* is the curvature of *a*. This is an unusual
property. The only other examples of surfaces that I know of where
*K* = *N* everywhere are surfaces where *K* and
*N* are both identically zero (planes, cylinders, flat tori,
etc), and the symmetric self intersecting sphere (*x*, *y*,
*xz*, *yz*), *x*^{2} + *y*^{2} +
*z*^{2} = 1. Seeing how interesting the example surfaces
are, it would be interesting to classify all surfaces with this property.

This is the same projection as the
picture at the top of the page, only now *v* ranges from -10 to 10.
Note that for larger values of *v*, the surface becomes more flat.
This is predicted in the formula for the two curvatures.

One obvious surface to test is the normal bundle, defined in a similar
manner as the tangent bundle. Since a normal line is perpendicular to
the tangent line, the normal bundle will be the same as the tangent
bundle, only rotated in **R**^{4}. The rotation will
affect neither the Gaussian curvature nor the normal curvature, and so
the normal bundle also satisfies the condition *N* = *K*
everywhere, but it is not really a different surface.

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