The Tangent Bundle

Given a regular curve a(t) in R2, we can think of its tangent bundle as a regular surface in R4, parametrized by

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 R4). 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 R4.

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 (t, v) is a flat inflection if t is an inflection point of a(t). The binormal surface of a tangent bundle degenerates to the curve (n(t), 0), where n(t) is the unit normal vector to a(t).

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), x2 + y2 + z2 = 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 R4. 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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