Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533 Curves and surfaces in the three dimensional sphere placed in the space of quaternions Yoshihiko Tazawa Department of Natural Sciences, Tokyo Denki University, 2-2 Kanda-nishikicho, Chiyodaku, Tokyo 101, Japan [email protected] Abstract In this article we will show how to use Mathematica in dealing with curves and surfaces in the three dimensional unit sphere S^ embedded in the four dimensional Euclidian space E\ Since S* is the Lie group of unit quater- nions and at the same time it is a space of constant curvature, the analogy of the theory of curves in E^ holds. We calculate curvature and torsion of curves in S^ by Mathematica. The Gauss map v of a surface in E^ is decomposed into the two maps v+ and z/_. If the surface is contained in 5^, we can define another Gauss map z/g. We use Mathematica to visualize the shapes of the images of these Gauss maps. Finally, the meaning of these images becomes clear through the notion of the slant surface. 1 Curves The space E* is regarded as the space of quaternions with the multiplication ii — jj = kk = — 1, ij = —ji — k, jk — —kj = i, ki = —ik = j The unit sphere S^ C E^ is the Lie group of unit quaternions. Let X\ = (0,1,0,0), Xz = (0,0,1,0), %3 = (0,0,0,1) be the orthonormal frame of the tangent space T\S^ at 1 = (1,0,0,0) and Xi be the left-invariant extension of Xi over S^. By V, we denote the covariant differentiation Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533 460 Innovation In Mathematics on S*. In the geometry of S^, left translations play the analogous role to that of parallel displacements in the ordinary Euclidian space E^ (cf. Chen&Tazawa[3]). The unit tangent vector of a curve c(s) parametrized by arc-length s is expressed as The curvature, unit normal vector, unit binomial vector, and torsion are defined respectively by 1 Example 1 sin(sin(^)), cos(sin(f)), sin(f - 4^), cos(f - 4 The curvature and torsion are calculated and plotted by Mathematica in Figure 1. Figure 1: curvature and torsion Example 2 : Geodesic Let a and b be any constants. A geodesic ( great circle ) is given by Cab(t) = ( cos t, ~ sin t cos b sin a, sin t cos 6 cos a, sin t sin b ) The curvature of a geodesic vanishes. Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533 Innovation In Mathematics 461 Example 3 : Helix in S* For any constant a and 6, the curve defined by hab(t) = (asinZ, a cost, Vl - a?smbt, Vl - o? cos bt has constant curvature and torsion. Moreover, the angle between its tangent vector and the left invariant vector field Xi is constant. We call this curve a helix in 5^ (cf. Chen&Tazawa[3]). The projections to the three dimensional coordinate spaces of a helix in g^ with a = 1/64 and 6 = 1/512 are shown in Figure 2. 0.94' -0.02^ 0.02 Figure 2: projections of a helix 2 Gauss map The exterior algebra A^E^ of E^ is decomposed into two eigen spaces of the Hodge's star operator Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533 462 Innovation In Mathematics corresponding to the eigen values -hi and — 1, respectively. Let 5+, S^_ be the spheres of radius 1/\/2 in A^_E\ A^E^ centered at the origin. Then the Grassmannian manifold G(2, 4) of all oriented 2-planes in E^ is identified with the Riemannian product O(2,4) =^ xS^ of these spheres through the identification V = e\ A 62, where {GI, 62} is an orthonormal frame of a plane V G G(2, 4). For a surface / : D C E^ —> E*, the Gauss map v : D —» h?E* takes its value to be the tangent plane at each point of the surface. It is decomposed into the two maps (cf. Chen&Tazawa[2]). Example 4 (u, v] —> ( u, u^ — 3v, uv, u — v^ ) The two Gauss images of this surface are plotted in Figure 3. Figure 3: Gauss images of z/+ and z/_ Example 5 A surface z —> ( z, z*\ identifying C^ = E^. The Gauss images are shown in Figure 4. Although the image of i/+ cannot be seen in the figure, it is a singleton on the axis of the first coordinate. Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533 Innovation In Mathematics 463 0.5 Figure 4: Gauss images Example 6 : Flat torus (u,v) —> —=( cos?z, sin 16, cos?;, sin?; ) The images of z/+ and z/_ in Figure 5 are the great circles perpendicular to the axis of the first coordinate. -0 0.5 Figure 5: Gauss images of a flat torus Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533 464 Innovation In Mathematics 3 Another Gauss map If a surface is contained in the 3-sphere, / : D C E^ —> S* C E*, we can define another Gauss map as follows (cf. Chen&Tazawa[3]). Let n : D —> Tf(u,v)S^ be the unit normal vector field in S^ along the surface. Then, the spherical Gauss map 1/3 : D —> S^ C T±S^ is defined by ^ = L~* o n, that is, VQ(U,V) = Lj^^(n(u,v)). Since the left translation in S^ is analogous to the parallel displacement in E^, the spherical Gauss map is a natural generalization of the ordinary Gauss map. Example 7 The image under the spherical Gauss map of the flat torus given in Example 6 is plotted in Figure 6. Figure 6: z/g-image of a flat torus 4 Slant surfaces A surface in the 4-dimensional space can also be regarded as a surface in the 2-dimensional complex space / : D C E^ —> C^ = El We denote by J the complex structure of C^. A surface is called an a—slant surface, if the angle between JX and TpM is constant, i. e., Z (JX, TpM) = a for any X G T^M and any p G M = /(D) (cf. Chen[l]) . Let {ei, 62, 63, 64} be the canonical basis of E*. Then, the following three are equivalent (cf. Ghen&Tazawa[3]). (1) / is an a— slant surface. (2) The angle between z/+(%, f) and 4= (ei A ^2 + 63 A 64) is a. (3) The angle between i/s(u,v) and X]_ is a. A family of slant surfaces contained in the 3-sphere is obtained by the quaternionic product of a geodesic in S^ and a helix in S^ (cf. ChenfeTazawa[2])). Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533 Innovation In Mathematics 465 0.! -0.251 Figure 7: projections of a slant surface Example 8 We choose a helix h(s) in S^ parametrized by arc-length s so that its coef- ficient of the unit tangent vector satisfies /i(s) = — sin (yr/lO). We obtain such h(s) by solving a system of ordinary differential equations making use of NDSolve. We also choose a geodesic ^(t) = Cab(t) in Example 2 with a = 7T/10 and 6 = ?r/4 . Define a mapy? by (^(a;,?/, z,w) = (a;,?/, w, z) arid put f(s,t) = (p(^(t] • h(s)), where • is the quaternionic product. The projections of this surface to the coordinate spaces are shown in Figure 7. The surface is a vr/lO-slant surface and its image under v+ is contained in a circle perpendicular to the first coordinate axis as is seen in Figure 8. Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533 466 Innovation In Mathematics Figure 8: z/+-image of a slant surface As can be seen from the images under v+ , the surface of Example 4 is not a slant surface. The surface of Example 5 is a 0-slant surface ( i. e., a holomorphic surface ) and the flat torus in Example 6 is an yr/2-slant sur- face ( i. e., a totally real surface) (cf. Chen& Tazawa [2], Chenfe Tazawa [3], Tazawa[5]). References 1. Paper in a journal: 1. Chen, B.-y. Slant immersions, Bull. Austral. Math. Soc., vol 4. pp!35-147,1990. 2. Chen, B.-y. & Tazawa, Y. Slant surfaces with codimension 2, Ann. Foe. &%. TWowae, vol XI, No. 3, pp29-43,1990. 3. Chen, B.-y. & Tazawa, Y. Slant submanifolds in complex Euclidean spaces, Tokyo J. Math., vol 14, No. 1, pp 101-120, 1991. 4. Tazawa, Y. Construction of slant immersions, Bull. Inst. Math. Acad. Amm., 22, No. 2, pp!53-166,1994. 2. Edited book: 5. Tazawa, Y. Construction and visualization of slant immersions, Ge- ometry and Topology of Submanifolds, VII, World Scientific, pp257- 260,1995. .