# American Institute of Mathematical Sciences

September  2017, 9(3): 317-333. doi: 10.3934/jgm.2017013

## Complete spelling rules for the Monster tower over three-space

 1 Lab49, 30 St. Mary Axe, London EC3A 8EP, UK 2 Mathematics Department, De Anza College, 21250 Stevens Creek Blvd., Cupertino, CA 95014, USA 3 Department of Mathematics and Statistics, Sacramento State University, 6000 J St., Sacramento, CA 95819, USA

Received  February 2015 Revised  August 2016 Published  June 2017

The Monster tower, also known as the Semple tower, is a sequence of manifolds with distributions of interest to both differential and algebraic geometers. Each manifold is a projective bundle over the previous. Moreover, each level is a fiber compactified jet bundle equipped with an action of finite jets of the diffeomorphism group. There is a correspondence between points in the tower and curves in the base manifold. These points admit a stratification which can be encoded by a word called the RVT code. Here, we derive the spelling rules for these words in the case of a three dimensional base. That is, we determine precisely which words are realized by points in the tower. To this end, we study the incidence relations between certain subtowers, called Baby Monsters, and present a general method for determining the level at which each Baby Monster is born. Here, we focus on the case where the base manifold is three dimensional, but all the methods presented generalize to bases of arbitrary dimension.

Citation: Alex Castro, Wyatt Howard, Corey Shanbrom. Complete spelling rules for the Monster tower over three-space. Journal of Geometric Mechanics, 2017, 9 (3) : 317-333. doi: 10.3934/jgm.2017013
##### References:
 [1] V. I. Arnol'd, Simple singularities of curves, Proc. Steklov Inst. Math., 226 (1999), 20-28.   Google Scholar [2] E. Cartan, Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes, Bull. Soc. Math. France, 42 (1914), 12-48.   Google Scholar [3] A. Castro, Chains and Monsters: From Cauchy-Riemann Geometry to Semple Towers and Singular Space Curves, PhD thesis, 2010.  Google Scholar [4] A. Castro, S. Colley, G. Kennedy and C. Shanbrom, A coarse stratification of the Monster tower, arXiv: 1606.07931. [math. AG]. Google Scholar [5] A. Castro and W. Howard, A Monster tower approach to Goursat multi-flags, Differential Geom. Appl., 30 (2012), 405-427.  doi: 10.1016/j.difgeo.2012.06.005.  Google Scholar [6] A. Castro, W. Howard and C. Shanbrom, Bridges between subRiemannian geometry and algebraic geometry, Proceedings of 10th AIMS Conference on Dynamical Systems, Differential Equations, and Applications, 30 (2015), 239-247.  doi: 10.3934/proc.2015.0239.  Google Scholar [7] A. Castro, R. Montgomery and W. Howard, Spatial curve singularities and the Monster/Semple tower, Israel J. Math., 192 (2012), 381-427.  doi: 10.1007/s11856-012-0031-2.  Google Scholar [8] A. Giaro, A. Kumpera and C. Ruiz, Sur la lecture correcte d'un résultat d'Élie Cartan, C. R. Acad. Sci. Paris, 287 (1978), 241-244.   Google Scholar [9] F. Jean, The car with N trailers: Characterisation of the singular configurations, ESAIM: Control, Optimisation, and Calculus of Variations, 1 (1996), 241-266.   Google Scholar [10] A. Kumpera and J. L. Rubin, Multi-flag systems and ordinary differential equations, Nagoya Math. J., 166 (2002), 1-27.  doi: 10.1017/S0027763000008229.  Google Scholar [11] A. Kushner, V. Lychagin and V. Ruvtsov, Contact Geometry and Nonlinear Differential Equations, Cambridge University Press, Cambridge, UK, 2007.   Google Scholar [12] S. J. Li and W. Respondek, The geometry, controllability, and flatness property of the $n$-bar system, Internat. J. Control, 84 (2011), 834-850.  doi: 10.1080/00207179.2011.569955.  Google Scholar [13] F. Luca and J. J. Risler, The maximum of the degree of nonholonomy for the car with N trailers, Proceedings of the 4th IFAC Symposium on Robot Control, Capri, (1994), 165-170.   Google Scholar [14] R. Montgomery and M. Zhitomirskii, Geometric approach to goursat flags, Ann. Inst. H. Poincaré -AN, 18 (2001), 459-493.  doi: 10.1016/S0294-1449(01)00076-2.  Google Scholar [15] R. Montgomery and M. Zhitomirskii, Points and curves in the monster tower, Memoirs of the AMS, 203 (2010), x+137 pp.  doi: 10.1090/S0065-9266-09-00598-5.  Google Scholar [16] P. Mormul, Geometric classes of Goursat flags and their encoding by small growth vectors, Central European J. Math., 2 (2004), 859-883.  doi: 10.2478/BF02475982.  Google Scholar [17] P. Mormul, Multi-dimensional Cartan prolongation and special k-flags, Geometric Singularity Theory, Banach Center Publications, 65 (2004), 157-178. doi: 10.4064/bc65-0-12.  Google Scholar [18] P. Mormul, Small growth vectors of the compactifications of the contact systems on $J^r(1,1)$, Contemporary Mathematics, 569 (2012), 123-141.  doi: 10.1090/conm/569/11247.  Google Scholar [19] P. Mormul and F. Pelletier, Special 2-flags in lengths not exceeding four: A study in strong nilpotency of distributions, arXiv: 1011.1763. [math. DG]. Google Scholar [20] F. Pelletier and M. Slayman, Articulated arm and special multi-flags, J. Math. Sci. Adv. Appl., 8 (2011), 9-41.   Google Scholar [21] F. Pelletier and M. Slayman, Configurations of an articulated arm and singularities of special multi-flags, SIGMA, 10 (2014), Paper 059, 38 pp.  doi: 10.3842/SIGMA.2014.059.  Google Scholar [22] W. Respondek, Symmetries and minimal flat outputs of nonlinear control systems, New Trends in Nonlinear Dynamics and Control and their Applications, Lecture Notes in Control and Information Science, 295 (2004), 65-86. doi: 10.1007/978-3-540-45056-6_5.  Google Scholar [23] J. Semple, Singularities of Space Algebraic Curves, Proceedings of the London Mathematical Society, 44 (1938), 149-174.   Google Scholar [24] C. Shanbrom, The Puiseux characteristic of a Goursat germ, J. Dynamical and Control Systems, 20 (2014), 33-46.  doi: 10.1007/s10883-013-9207-2.  Google Scholar

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##### References:
 [1] V. I. Arnol'd, Simple singularities of curves, Proc. Steklov Inst. Math., 226 (1999), 20-28.   Google Scholar [2] E. Cartan, Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes, Bull. Soc. Math. France, 42 (1914), 12-48.   Google Scholar [3] A. Castro, Chains and Monsters: From Cauchy-Riemann Geometry to Semple Towers and Singular Space Curves, PhD thesis, 2010.  Google Scholar [4] A. Castro, S. Colley, G. Kennedy and C. Shanbrom, A coarse stratification of the Monster tower, arXiv: 1606.07931. [math. AG]. Google Scholar [5] A. Castro and W. Howard, A Monster tower approach to Goursat multi-flags, Differential Geom. Appl., 30 (2012), 405-427.  doi: 10.1016/j.difgeo.2012.06.005.  Google Scholar [6] A. Castro, W. Howard and C. Shanbrom, Bridges between subRiemannian geometry and algebraic geometry, Proceedings of 10th AIMS Conference on Dynamical Systems, Differential Equations, and Applications, 30 (2015), 239-247.  doi: 10.3934/proc.2015.0239.  Google Scholar [7] A. Castro, R. Montgomery and W. Howard, Spatial curve singularities and the Monster/Semple tower, Israel J. Math., 192 (2012), 381-427.  doi: 10.1007/s11856-012-0031-2.  Google Scholar [8] A. Giaro, A. Kumpera and C. Ruiz, Sur la lecture correcte d'un résultat d'Élie Cartan, C. R. Acad. Sci. Paris, 287 (1978), 241-244.   Google Scholar [9] F. Jean, The car with N trailers: Characterisation of the singular configurations, ESAIM: Control, Optimisation, and Calculus of Variations, 1 (1996), 241-266.   Google Scholar [10] A. Kumpera and J. L. Rubin, Multi-flag systems and ordinary differential equations, Nagoya Math. J., 166 (2002), 1-27.  doi: 10.1017/S0027763000008229.  Google Scholar [11] A. Kushner, V. Lychagin and V. Ruvtsov, Contact Geometry and Nonlinear Differential Equations, Cambridge University Press, Cambridge, UK, 2007.   Google Scholar [12] S. J. Li and W. Respondek, The geometry, controllability, and flatness property of the $n$-bar system, Internat. J. Control, 84 (2011), 834-850.  doi: 10.1080/00207179.2011.569955.  Google Scholar [13] F. Luca and J. J. Risler, The maximum of the degree of nonholonomy for the car with N trailers, Proceedings of the 4th IFAC Symposium on Robot Control, Capri, (1994), 165-170.   Google Scholar [14] R. Montgomery and M. Zhitomirskii, Geometric approach to goursat flags, Ann. Inst. H. Poincaré -AN, 18 (2001), 459-493.  doi: 10.1016/S0294-1449(01)00076-2.  Google Scholar [15] R. Montgomery and M. Zhitomirskii, Points and curves in the monster tower, Memoirs of the AMS, 203 (2010), x+137 pp.  doi: 10.1090/S0065-9266-09-00598-5.  Google Scholar [16] P. Mormul, Geometric classes of Goursat flags and their encoding by small growth vectors, Central European J. Math., 2 (2004), 859-883.  doi: 10.2478/BF02475982.  Google Scholar [17] P. Mormul, Multi-dimensional Cartan prolongation and special k-flags, Geometric Singularity Theory, Banach Center Publications, 65 (2004), 157-178. doi: 10.4064/bc65-0-12.  Google Scholar [18] P. Mormul, Small growth vectors of the compactifications of the contact systems on $J^r(1,1)$, Contemporary Mathematics, 569 (2012), 123-141.  doi: 10.1090/conm/569/11247.  Google Scholar [19] P. Mormul and F. Pelletier, Special 2-flags in lengths not exceeding four: A study in strong nilpotency of distributions, arXiv: 1011.1763. [math. DG]. Google Scholar [20] F. Pelletier and M. Slayman, Articulated arm and special multi-flags, J. Math. Sci. Adv. Appl., 8 (2011), 9-41.   Google Scholar [21] F. Pelletier and M. Slayman, Configurations of an articulated arm and singularities of special multi-flags, SIGMA, 10 (2014), Paper 059, 38 pp.  doi: 10.3842/SIGMA.2014.059.  Google Scholar [22] W. Respondek, Symmetries and minimal flat outputs of nonlinear control systems, New Trends in Nonlinear Dynamics and Control and their Applications, Lecture Notes in Control and Information Science, 295 (2004), 65-86. doi: 10.1007/978-3-540-45056-6_5.  Google Scholar [23] J. Semple, Singularities of Space Algebraic Curves, Proceedings of the London Mathematical Society, 44 (1938), 149-174.   Google Scholar [24] C. Shanbrom, The Puiseux characteristic of a Goursat germ, J. Dynamical and Control Systems, 20 (2014), 33-46.  doi: 10.1007/s10883-013-9207-2.  Google Scholar
The three critical planes $V, T_1$, and $T_2$, and their intersections, the distinguished lines $L_1, L_2$, and $L_3$.
Critical plane configurations that can appear in the distribution above an $R$ point (top left), a $V$ or $T_1$ point (top right), a $T_2$ point (bottom left), and an $L_j$ point (bottom right).
Critical plane configuration over $p_3\in RVL_1$. The left side shows the birth of $T_1(p_3)=\delta_2^1(p_3)$ as the first prolongation of the vertical plane at level 2. The right side shows the birth of $T_2(p_3)=\delta_1^2(p_3)$ as the second prolongation of the vertical plane at level 1. These two Baby Monsters meet in $\Delta^3$, and their intersection is the distinguished line $L_2(p_3)$. See Example 2.
Critical plane configuration over $p_4\in RVL_1T_2$. This shows the birth of $T_2(p_4)=\delta_1^3(p_4)$ as the third prolongation of the vertical plane at level 1. See Example 3.
RVT Code Spelling Rules
 Letter Can be followed by Cannot be followed by $R$ $R, V$ $T_i, L_j$ $V$ $R, V, T_1, L_1$ $T_2, L_2, L_3$ $T_1$ $R, V, T_1, L_1$ $T_2, L_2, L_3$ $T_2$ $R, V, T_2, L_3$ $T_1, L_1, L_2$ $L_1$ $R, V, T_1, T_2, L_1, L_2, L_3$ $\emptyset$ $L_2$ $R, V, T_1, T_2, L_1, L_2, L_3$ $\emptyset$ $L_3$ $R, V, T_1, T_2, L_1, L_2, L_3$ $\emptyset$
 Letter Can be followed by Cannot be followed by $R$ $R, V$ $T_i, L_j$ $V$ $R, V, T_1, L_1$ $T_2, L_2, L_3$ $T_1$ $R, V, T_1, L_1$ $T_2, L_2, L_3$ $T_2$ $R, V, T_2, L_3$ $T_1, L_1, L_2$ $L_1$ $R, V, T_1, T_2, L_1, L_2, L_3$ $\emptyset$ $L_2$ $R, V, T_1, T_2, L_1, L_2, L_3$ $\emptyset$ $L_3$ $R, V, T_1, T_2, L_1, L_2, L_3$ $\emptyset$
Critical Hyperplane Configurations
 Last letter in RVT code of $p\in M^k$ Critical planes appearing in $\Delta^k(p)$ $R$ $V$ $V$ or $T_1$ $V$ and $T_1$ $T_2$ $V$ and $T_2$ $L_1, L_2,$ or $L_3$ $V, T_1$, and $T_2$
 Last letter in RVT code of $p\in M^k$ Critical planes appearing in $\Delta^k(p)$ $R$ $V$ $V$ or $T_1$ $V$ and $T_1$ $T_2$ $V$ and $T_2$ $L_1, L_2,$ or $L_3$ $V, T_1$, and $T_2$
Base Cases of Inductive Proof
 RVT code of $p_k\in M^k$ $T_{1}(p_k)$ $T_{2}(p_k)$ $\lambda V T_1^{m} L_1 T_{2} \, \, \text{for} \, \, m \geq 0$ None $\delta ^{m+3}_{k-m-3}(p_{k })$ $\lambda L_1 T_1^{m} L_1 T_{2} \, \, \text{for} \, \, m \geq 1$ None $\delta ^{m+3}_{k-m-3}(p_{k })$ $\lambda L_1 L_1 T_{2}$ None $\delta ^{3}_{k-3}(p_{k})$ $\lambda VT_1^{m} L_1L_{2} \, \, \text{for} \, \, m \geq 0$ $\delta ^{2}_{k-2}(p_{k})$ $\delta ^{m+3}_{k-m-3}(p_{k})$ $\lambda L_1T_1^{m} L_1L_{2} \, \, \text{for} \, \, m \geq 1$ $\delta ^{2}_{k-2}(p_{k})$ $\delta ^{m+3}_{k-m-3}(p_{k})$ $\lambda L_1L_1L_{2}$ $\delta ^{2}_{k-2} (p_{k})$ $\delta ^{3}_{k-3}(p_{k})$ $\lambda VT_1^{m} L_1 L_{3} \, \, \text{for} \, \, m \geq 0$ $\delta ^{1}_{k-1}(p_{k})$ $\delta ^{m + 3}_{k-m-3}(p_{k})$ $\lambda L_1T_1^{m} L_1 L_{3} \, \, \text{for} \, \, m \geq 1$ $\delta ^{1}_{k-1}(p_{k})$ $\delta ^{m + 3}_{k-m-3}(p_{k})$ $\lambda L_1L_1 L_{3}$ $\delta ^{1}_{k-1}(p_{k})$ $\delta ^{3}_{k-3}(p_{k})$
 RVT code of $p_k\in M^k$ $T_{1}(p_k)$ $T_{2}(p_k)$ $\lambda V T_1^{m} L_1 T_{2} \, \, \text{for} \, \, m \geq 0$ None $\delta ^{m+3}_{k-m-3}(p_{k })$ $\lambda L_1 T_1^{m} L_1 T_{2} \, \, \text{for} \, \, m \geq 1$ None $\delta ^{m+3}_{k-m-3}(p_{k })$ $\lambda L_1 L_1 T_{2}$ None $\delta ^{3}_{k-3}(p_{k})$ $\lambda VT_1^{m} L_1L_{2} \, \, \text{for} \, \, m \geq 0$ $\delta ^{2}_{k-2}(p_{k})$ $\delta ^{m+3}_{k-m-3}(p_{k})$ $\lambda L_1T_1^{m} L_1L_{2} \, \, \text{for} \, \, m \geq 1$ $\delta ^{2}_{k-2}(p_{k})$ $\delta ^{m+3}_{k-m-3}(p_{k})$ $\lambda L_1L_1L_{2}$ $\delta ^{2}_{k-2} (p_{k})$ $\delta ^{3}_{k-3}(p_{k})$ $\lambda VT_1^{m} L_1 L_{3} \, \, \text{for} \, \, m \geq 0$ $\delta ^{1}_{k-1}(p_{k})$ $\delta ^{m + 3}_{k-m-3}(p_{k})$ $\lambda L_1T_1^{m} L_1 L_{3} \, \, \text{for} \, \, m \geq 1$ $\delta ^{1}_{k-1}(p_{k})$ $\delta ^{m + 3}_{k-m-3}(p_{k})$ $\lambda L_1L_1 L_{3}$ $\delta ^{1}_{k-1}(p_{k})$ $\delta ^{3}_{k-3}(p_{k})$
Summary of Example 2: $RVL_1$
 Level $i$ Coordinates on $M^i$ $\mathbb P\Delta^{i-1} = F_i$ coordinates Critical planes in $\Delta^i$ RVT code of $p_i$ $0$ $(x, y, z)$ n/a none n/a $1$ $(x, y, z, u_1, v_1)$$u_1=\frac{dy}{dx}, v_1=\frac{dz}{dx} [dx : dy : dz] V(p_1)=\delta_1^0 p_1=(p_0, l_0)\in R$$l_0 \subset \Delta^0=T_{p_0}M^0$ $2$ $(x, y, z, u_1, v_1, u_2, v_2)$$u_2=\frac{dx}{du_1}, v_2=\frac{dv_1}{du_1} [dx : du_1 : dv_1] V(p_2)=\delta_2^0,$$T_1(p_2)=\delta_1^1$ $p_2=(p_1, l_1)\in RV$$l_1 \subset V(p_1) \subset \Delta^1 3 (x, y, z, u_1, v_1, u_2, v_3, u_3, v_3)$$u_3=\frac{du_1}{dv_2}, v_3=\frac{du_2}{dv_2}$ $[du_1 : du_2 : dv_2]$ $V(p_3)=\delta_3^0,$$T_1(p_3)=\delta_2^1,$$T_2(p_3)=\delta_1^2$ $p_3=(p_2, l_2)\in RVL_1$$l_2 = L_1(p_2) \subset \Delta^2  Level i Coordinates on M^i \mathbb P\Delta^{i-1} = F_i coordinates Critical planes in \Delta^i RVT code of p_i 0 (x, y, z) n/a none n/a 1 (x, y, z, u_1, v_1)$$u_1=\frac{dy}{dx}, v_1=\frac{dz}{dx}$ $[dx : dy : dz]$ $V(p_1)=\delta_1^0$ $p_1=(p_0, l_0)\in R$$l_0 \subset \Delta^0=T_{p_0}M^0 2 (x, y, z, u_1, v_1, u_2, v_2)$$u_2=\frac{dx}{du_1}, v_2=\frac{dv_1}{du_1}$ $[dx : du_1 : dv_1]$ $V(p_2)=\delta_2^0,$$T_1(p_2)=\delta_1^1 p_2=(p_1, l_1)\in RV$$l_1 \subset V(p_1) \subset \Delta^1$ $3$ $(x, y, z, u_1, v_1, u_2, v_3, u_3, v_3)$$u_3=\frac{du_1}{dv_2}, v_3=\frac{du_2}{dv_2} [du_1 : du_2 : dv_2] V(p_3)=\delta_3^0,$$T_1(p_3)=\delta_2^1,$$T_2(p_3)=\delta_1^2 p_3=(p_2, l_2)\in RVL_1$$l_2 = L_1(p_2) \subset \Delta^2$
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