Sculpting with Fiber Bundle and Foliation Hang Li, Zhipei Yan May, 2016 1 Theory Definition 1.1. For space E(total space), B(base), F (fiber) and surjection π : E ! B (1) if there is an open neighborhood U of π(x) (x 2 E) such that π−1(U) and U × F are homeomorphic, (E; B; F; π) is a fiber bundle. And E has the form of product space B × F . Figure 1: Base and Fiber Here are two kinds of fiber bundle on a torus. Figure 2: Trivial Fibers Figure 3: Nontrivial Fibers 1 Definition 1.2. fUig is an atlas of an n-dimensional manifold U. For each chart Ui, define n φi : Ui ! R (2) If Ui \ Uj 6= ;, define −1 'ij = φj ◦ φi (3) And if 'ij has the form of 1 2 'ij = ('ij(x);'ij(x; y)) (4) 1 n−p n−p 'ij : R ! R (5) 2 n p 'ij : R ! R (6) then for any stripe si : x = ci in Ui there must be a stripe in Uj sj : x = cj such that si and sj are the same curve in Ui \ Uj and can be connected. The maximal connection is a leaf. This structure is called a p-dimensional foliation F of an n-dimensional manifold U. Property 1.1. All the fiber bundles compose a group GF = (fFig; ◦). If we regard the fiber bundle(1-manifold) as a parameterization on a surface(2-manifold), we can prove it easily. As an example, we use the parameterization on a torus. fu; vg ! f(R + r cos v) cos u; (R + r cos v) sin u; r sin vg (7) Then we can get the trivial fibers in Fig. 2. fu; vg = ft; constg (8) To generalize the creating method of fiber bundle, it's necessary to define the transformation functions. Definition 1.3. A transformation function F : [0; 1]2 ! [0; 1]2 for fiber bundle must be: • F is C1([0; 1]2) • F is bijection • F (u; 0) = F (u; 1), F (0; v) = F (1; v) Fu(u; 0) = Fu(u; 1), Fu(0; v) = Fu(1; v) Fv(u; 0) = Fv(u; 1), Fv(0; v) = Fv(1; v) An example is F (u; v) = fu; fv + u=2gg. The {·} inside means the fractional part. Generally, transformation functions always use {·} to make sure that F have the properties listed above. In the following parts, we will omit it. With transformation functions, we can easily get any fiber in any fiber bundle: F : fu; vg = ft; constg ! fu0; v0g = F (u; v) ! f(R + r cos v0) cos u0; (R + r cos v0) sin u0; r sin v0g (9) And Property 1.2. For GF , we have following properties: • 8Fi;Fj, Fi ◦ Fj is defined as 0 0 0 0 0 0 0 Fj ◦ Fi : fu ; v g = Fj(Fi(u; v)) ! f(R + r cos v ) cos u ; (R + r cos v ) sin u ; r sin v g (10) • Identity element is I : I(u; v) = fu; vg (11) • 8Fi, −1 0 0 −1 0 0 0 0 0 Fi : fu ; v g = Fi (u; v) ! f(R + r cos v ) cos u ; (R + r cos v ) sin u ; r sin v g (12) 2 2 Result As an example, assume u F (u; v) = fu; v + g (13) 1 2 F2(u; v) = fu; v + sin 2πug (14) 1 F (u; v) = fu + sin 2πv; vg (15) 3 2 F = F3 ◦ F2 ◦ F1 (16) 1.0 0.8 0.6 1.0 0.4 0.5 0.0 2 -0.5 -1.0 0.2 0 -2 0 -2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Figure 4: A fiber of I Figure 5: A fiber of I 1.0 0.8 0.6 1.0 0.4 0.5 0.0 2 -0.5 -1.0 0.2 0 -2 0 -2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Figure 6: Five fibers of I Figure 7: Five fibers of I 3 1.0 0.8 0.6 1.0 0.4 0.5 0.0 2 -0.5 -1.0 0.2 0 -2 0 -2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Figure 8: A fiber of F1 Figure 9: A fiber of F1 1.0 0.8 0.6 1.0 0.4 0.5 0.0 2 -0.5 -1.0 0.2 0 -2 0 -2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Figure 10: Five fibers of F1 Figure 11: Five fibers of F1 4 1.0 0.8 0.6 1.0 0.4 0.5 0.0 2 -0.5 -1.0 0.2 0 -2 0 -2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Figure 12: A fiber of F2 Figure 13: A fiber of F2 1.0 0.8 0.6 1.0 0.4 0.5 0.0 2 -0.5 -1.0 0.2 0 -2 0 -2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Figure 14: Five fibers of F2 Figure 15: Five fibers of F2 5 1.0 0.8 0.6 1.0 0.4 0.5 0.0 2 -0.5 -1.0 0.2 0 -2 0 -2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Figure 16: A fiber of F3 Figure 17: A fiber of F3 1.0 0.8 0.6 1.0 0.4 0.5 0.0 2 -0.5 -1.0 0.2 0 -2 0 -2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Figure 18: Five fibers of F3 Figure 19: Five fibers of F3 6 1.0 0.8 0.6 1.0 0.4 0.5 0.0 2 -0.5 -1.0 0.2 0 -2 0 -2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Figure 20: A fiber of F2 ◦ F1 Figure 21: A fiber of F2 ◦ F1 1.0 0.8 0.6 1.0 0.4 0.5 0.0 2 -0.5 -1.0 0.2 0 -2 0 -2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Figure 22: Five fibers of F2 ◦ F1 Figure 23: Five fibers of F2 ◦ F1 7 1.0 0.8 0.6 1.0 0.4 0.5 0.0 2 -0.5 -1.0 0.2 0 -2 0 -2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Figure 24: A fiber of F3 ◦ F2 ◦ F1 Figure 25: A fiber of F3 ◦ F2 ◦ F1 1.0 0.8 0.6 1.0 0.4 0.5 0.0 2 -0.5 -1.0 0.2 0 -2 0 -2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Figure 26: Five fibers of F3 ◦ F2 ◦ F1 Figure 27: Five fibers of F3 ◦ F2 ◦ F1 8 Figure 28: Base and Fiber 9.