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        3. Tangential symmetries for space curves

For closed smooth curves in Euclidean 3-space the most general type of tangential symmetry described above will be too general to lead to conclusions on the shape of the curve. Hence we start immediately with the notions of parallelism and self-parallelism [FR]:

The exterior parallelism of two smooth closed curves c1, c2: S1 ® E3 is defined by the following condition: For every parameter t Î S1 the affine spaces normal to c1 at c1(t) and c2 at c2(t) coincide. This condition has been shown to be equivalent to the condition that both curves are connected by a parallel section of their normal bundles (see [W2]), i.e. there is a smooth normal vector field e1 along c1 such that

c2 (t) = c1 (t) + le1 (t)     and    prn (Ñc'_1 (t) e1 ) = 0                        (1)

for all t Î S1, where c'1 (t) denotes the tangent vector field of c1 as usual and prn denotes the orthogonal projection to the corresponding normal (vector) space of c1. Previous investigations of this notion for curves could be found in the paper [CR2] by F.J. Craveiro de Carvalho and S.A. Robertson and in [W3].

Hence the existence of a parallel mate for c : S1 ® E3 has been reduced to the search for a global parallel normal vector field along c. Generally, these vector fields only exist locally along c. Parallel transfer of the normal plane along one period of c with respect to the normal connection leads to a rotation of the normal plane, which is characterized (up to integer multiples of 2p) by an oriented angle a(c), which we call the total normal twist of c. For Frenet curves this quantity is given by their total torsion up to integer multiples of 2p. Looking at general orthonormal frame fields { T, e1, e2 } along c, where T denotes the unit tangent field of c, and setting

w12 = < Ñe1, e2 > =  - < Ñe2, e1 > = - w21,                        (2)

we get for the total normal twist of c (up to integer multiples of 2p)

a(c) =  ò S 1 w21(t) dt                                                                  (3)

where there is no need to parametrize c with arclength.

A self-parallelism of c is given by a diffeomorphism d: S 1 ®S 1 such that c and c·d are parallel in the exterior sense. This is the notion of tangential symmetry to be discussed now. Clearly only closed curves with vanishing total twist possibly will admit such a tangential symmetry. The variety of these curves has been studied in much detail in a joint paper with T.F. Mersal [MW]. But already in a previous paper [W3] curves with non-trivial tangential symmetries have been related to a center curve with total normal twist being a rational multiple of 2p in the following way: Take the non-vanishing normal vector to the center curve, connecting the center curve and the original curve at some point, and apply normal parallel transfer to this vector along several periods of the center curve, until the trace of the endpoint of this vector will lead to a closed curve. This will restore the original curve. Moreover, starting the same procedure with any curve, where the total normal twist is a rational multiple of 2p, and with a suitable normal vector such that the construction will avoid singularities, we get a curve exhibiting Zn as its group of tangential symmetries, where n is the number of periods until the trace of the end point of the vector will provide a closed curve.

This gives a fairly clear picture of space curves admitting this kind of symmetries. But there are other advantages related to the visual perception of these curves. There is an obvious one, coming from the kinematic interpretation of the parallel transfer in the normal bundle of a curve. Consider a normal frame as a rigid two-dimensional figure, moving without acceleration freely along the curve, with the constraint to remain in the normal plane forever. Then the motion will be described by the parallel transfer in the normal bundle of the curve. Hence, considering the intersection of a tangentially symmetric curve, having Zn as its symmetry group, with its normal plane as the vertex set of a regular n-gon, this motion of the frame along the central curve will preserve this figure after one period, though a permutation of the vertices may have happened.

Interpreting the original curve as a figure located on a tube around the central curve, it carries information for the visualization of the center curve as follows: The tube may be taken as a more solid image of the center curve. The twist of the center curve may be visualized by drawing families of curves on the surface of this tube. The most appropriate curves for this will be those obtained by parallel transfer in the normal bundle of the center curve, and they will be closed curves only, if they exhibit a tangential symmetry. Clearly, then the surface of the torus bounding the tube can be foliated by curves with this kind of tangential symmetry. Furthermore, if another profile than a disk should be taken to thicken the center curve to a solid body, this only will be possible if the tangential symmetry of the original curve is respected by this profile. This can be observed in many images where closed space curves are displayed.

The more restricted form of tangential symmetry which is given by the notion of transnormality has been studied by M.C. Irwin [I], and several geometric results have been obtained for them in [W4]. Within the current context the only interesting result is, that in this case the symmetry group can be Z2 only, and that overmore the central curve cannot avoid singularities. There are no further conclusions than those of the preceding paragraph resp. preceding section for this case.

There also are a lot of considerations concerning tangential symmetries of curves in higher-dimensional spaces. In particular, the case of Minkowski 4-space is of special interest for Relativity. But this is beyond the goal of this presentation.