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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 |
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.
References:
[1] |
V. I. Arnol'd,
Simple singularities of curves, Proc. Steklov Inst. Math., 226 (1999), 20-28.
|
[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.
|
[3] |
A. Castro, Chains and Monsters: From Cauchy-Riemann Geometry to Semple Towers and Singular Space Curves, PhD thesis, 2010. |
[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. |
[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. |
[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. |
[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.
|
[9] |
F. Jean,
The car with N trailers: Characterisation of the singular configurations, ESAIM: Control, Optimisation, and Calculus of Variations, 1 (1996), 241-266.
|
[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. |
[11] |
A. Kushner, V. Lychagin and V. Ruvtsov, Contact Geometry and Nonlinear Differential Equations, Cambridge University Press, Cambridge, UK, 2007.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[23] |
J. Semple,
Singularities of Space Algebraic Curves, Proceedings of the London Mathematical Society, 44 (1938), 149-174.
|
[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. |
show all references
References:
[1] |
V. I. Arnol'd,
Simple singularities of curves, Proc. Steklov Inst. Math., 226 (1999), 20-28.
|
[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.
|
[3] |
A. Castro, Chains and Monsters: From Cauchy-Riemann Geometry to Semple Towers and Singular Space Curves, PhD thesis, 2010. |
[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. |
[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. |
[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. |
[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.
|
[9] |
F. Jean,
The car with N trailers: Characterisation of the singular configurations, ESAIM: Control, Optimisation, and Calculus of Variations, 1 (1996), 241-266.
|
[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. |
[11] |
A. Kushner, V. Lychagin and V. Ruvtsov, Contact Geometry and Nonlinear Differential Equations, Cambridge University Press, Cambridge, UK, 2007.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[23] |
J. Semple,
Singularities of Space Algebraic Curves, Proceedings of the London Mathematical Society, 44 (1938), 149-174.
|
[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. |




Letter | Can be followed by | Cannot be followed by |
Letter | Can be followed by | Cannot be followed by |
Last letter in RVT code of |
Critical planes appearing in |
Last letter in RVT code of |
Critical planes appearing in |
RVT code of |
||
None | ||
None | ||
None | ||
|
||
RVT code of |
||
None | ||
None | ||
None | ||
|
||
Level |
Coordinates on |
Critical planes in |
RVT code of | |
n/a | none | n/a | ||
Level |
Coordinates on |
Critical planes in |
RVT code of | |
n/a | none | n/a | ||
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