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2015, 2015(special): 239-247. doi: 10.3934/proc.2015.0239

Bridges between subriemannian geometry and algebraic geometry: Now and then

Alex L Castro 1, , Wyatt Howard 2, and  Corey Shanbrom 3,

1. 

Department of Mathematics, Imperial College, London, 180 Queen's Gate, London SW7 2AZ, United Kingdom
2. 

Mathematics and Computer Science, Santa Clara University, Santa Clara, CA, United States
3. 

Department of Mathematics and Statistics, California State University, Sacramento, 6000 J St., Sacramento, CA

Received  September 2014 Revised  February 2015 Published  November 2015

We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the field of algebraic geometry emerge here organically in an attempt to elucidate the geometric structures underlying a large class of nonholonomic distributions known as Goursat constraints. Among our new results is a regularization theorem for curves stated and proved using tools exclusively from nonholonomic geometry, and a computation of topological invariants that answer a question on the global topology of our classifying space. Last but not least we present for the first time some experimental results connecting the discrete invariants of nonholonomic plane fields such as the RVT code and the Milnor number of complex plane algebraic curves.
Keywords: topological invariants., Subriemannian geometry, normal form theory, local singularities, enumerative geometry.
Mathematics Subject Classification: 53C17, 32S05, 58K40, 14N1.
Citation: Alex L Castro, Wyatt Howard, Corey Shanbrom. Bridges between subriemannian geometry and algebraic geometry: Now and then. Conference Publications, 2015, 2015 (special) : 239-247. doi: 10.3934/proc.2015.0239
References:
[1]

V. I. Arnold, Simple singularities of curves, Proc. Steklov Inst. Math., 226 (1999), 20-28. Google Scholar

[2]

V. I. Arnold, A.N. Varchenko, and S. Guzein-Sade, Singularités des Applicants Différentiables, Editions MIR, Moscow, 1986. Google Scholar

[3]

A. Castro, Chains and Monsters, Ph.D thesis, UC, Santa Cruz, 2010. Google Scholar

[4]

A. Castro and W. Howard, A Monster tower approach to Goursat multi-flags, Differential Geom. Appl., 30 (2012), 405-427. Google Scholar

[5]

A. Castro and W. Howard, A Semple-type approach to a problem of Goursat: the multi-flag case, C. R. Math. Acad. Sci. Paris, 351 (2013), 921-925. Google Scholar

[6]

A. Castro and W. Howard, Spelling rules for the Monster/Semple tower,, arXiv:1407.1824., ().   Google Scholar

[7]

A. Castro, R. Montgomery, and appendix by W. Howard, Spatial curve singularities and the Monster/Semple tower, Israel J. Math., 192 (2012), 381-427. Google Scholar

[8]

S. Colley and G. Kennedy, Triple and quadruple contact of plane curves, Enumerative Algebraic Geometry: Proceedings of the 1989 Zeuthen Symposium, Contemp. Math., 123 (1991), 31-59. Google Scholar

[9]

A. Giaro, A. Kumpera, and C. Ruiz, Sur la Lecture correcte d'un résultat d' Élie Cartan, C.R. Acad. Sci. Paris Sér. A-B, 287 (1978), A241-A244. Google Scholar

[10]

J.C. Hausmann and E. Rodriguez, Holonomy orbits of the snake charmer algorithm, Geometry and Topology of Manifolds, 76 (2007), 207-219. Google Scholar

[11]

F. Jean, The car with N trailers: characterisation of the singular configurations, Control, Optimisation, and Calculus of Variations, 1 (1996), 241-266. Google Scholar

[12]

S. Kuroki and D. Suh, Classification of complex projective towers up to dimension 8 and cohomological rigidity,, arXiv:1203.4403., ().   Google Scholar

[13]

M. Lejeune-Jalabert, Chains of points in the Semple tower, Am. J. Math., 128 (2006), 1283-1311. Google Scholar

[14]

J. Milnor and J. Stasheff, Characteristic Classes, Princeton University Press, 1974. Google Scholar

[15]

R. Montgomery and M. Zhitomirskii, Geometric appraoch to Goursat flags, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 459-493. Google Scholar

[16]

R. Montgomery and M. Zhitomirksii, Points and curves in the Monster tower, Mem. Amer. Math. Soc., 203 (2009), x+137. Google Scholar

[17]

P. Mormul, Geometric classes of Goursat flags and their encoding by small growth vectors, Central European J. Math., 2 (2004), 859-883. Google Scholar

[18]

R. Murray and S. Sastry, Nonholonomic motion planning: steering using sinusoids, IEEE Trans. Automat. Control, 38 (1993), 700-716. Google Scholar

[19]

F. Pelletier and M. Slayman, Configuration of an articulated arm and singularities of special multi-flags, SIGMA, 10 (2014), 1-38. Google Scholar

[20]

C. Shanbrom, The Puiseux characteristic of a Goursat germ, J. Dynamical and Control Systems,20 (2014), 33-46. Google Scholar

[21]

K. Shibuya and K. Yamaguchi, Drapeau theorem for differential systems, Differential Geom. Appl., 27 (2009), 793-808. Google Scholar

[22]

R. Thom, Quelques propriétés globales des variétés, Comment. Math. Helv., 28 (1954), 17-86. Google Scholar

[23]

C. Wall, Singular Points of Plane Curves, London Mathematical Society Student Texts, 2004. Google Scholar

show all references

References:
[1]

V. I. Arnold, Simple singularities of curves, Proc. Steklov Inst. Math., 226 (1999), 20-28. Google Scholar

[2]

V. I. Arnold, A.N. Varchenko, and S. Guzein-Sade, Singularités des Applicants Différentiables, Editions MIR, Moscow, 1986. Google Scholar

[3]

A. Castro, Chains and Monsters, Ph.D thesis, UC, Santa Cruz, 2010. Google Scholar

[4]

A. Castro and W. Howard, A Monster tower approach to Goursat multi-flags, Differential Geom. Appl., 30 (2012), 405-427. Google Scholar

[5]

A. Castro and W. Howard, A Semple-type approach to a problem of Goursat: the multi-flag case, C. R. Math. Acad. Sci. Paris, 351 (2013), 921-925. Google Scholar

[6]

A. Castro and W. Howard, Spelling rules for the Monster/Semple tower,, arXiv:1407.1824., ().   Google Scholar

[7]

A. Castro, R. Montgomery, and appendix by W. Howard, Spatial curve singularities and the Monster/Semple tower, Israel J. Math., 192 (2012), 381-427. Google Scholar

[8]

S. Colley and G. Kennedy, Triple and quadruple contact of plane curves, Enumerative Algebraic Geometry: Proceedings of the 1989 Zeuthen Symposium, Contemp. Math., 123 (1991), 31-59. Google Scholar

[9]

A. Giaro, A. Kumpera, and C. Ruiz, Sur la Lecture correcte d'un résultat d' Élie Cartan, C.R. Acad. Sci. Paris Sér. A-B, 287 (1978), A241-A244. Google Scholar

[10]

J.C. Hausmann and E. Rodriguez, Holonomy orbits of the snake charmer algorithm, Geometry and Topology of Manifolds, 76 (2007), 207-219. Google Scholar

[11]

F. Jean, The car with N trailers: characterisation of the singular configurations, Control, Optimisation, and Calculus of Variations, 1 (1996), 241-266. Google Scholar

[12]

S. Kuroki and D. Suh, Classification of complex projective towers up to dimension 8 and cohomological rigidity,, arXiv:1203.4403., ().   Google Scholar

[13]

M. Lejeune-Jalabert, Chains of points in the Semple tower, Am. J. Math., 128 (2006), 1283-1311. Google Scholar

[14]

J. Milnor and J. Stasheff, Characteristic Classes, Princeton University Press, 1974. Google Scholar

[15]

R. Montgomery and M. Zhitomirskii, Geometric appraoch to Goursat flags, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 459-493. Google Scholar

[16]

R. Montgomery and M. Zhitomirksii, Points and curves in the Monster tower, Mem. Amer. Math. Soc., 203 (2009), x+137. Google Scholar

[17]

P. Mormul, Geometric classes of Goursat flags and their encoding by small growth vectors, Central European J. Math., 2 (2004), 859-883. Google Scholar

[18]

R. Murray and S. Sastry, Nonholonomic motion planning: steering using sinusoids, IEEE Trans. Automat. Control, 38 (1993), 700-716. Google Scholar

[19]

F. Pelletier and M. Slayman, Configuration of an articulated arm and singularities of special multi-flags, SIGMA, 10 (2014), 1-38. Google Scholar

[20]

C. Shanbrom, The Puiseux characteristic of a Goursat germ, J. Dynamical and Control Systems,20 (2014), 33-46. Google Scholar

[21]

K. Shibuya and K. Yamaguchi, Drapeau theorem for differential systems, Differential Geom. Appl., 27 (2009), 793-808. Google Scholar

[22]

R. Thom, Quelques propriétés globales des variétés, Comment. Math. Helv., 28 (1954), 17-86. Google Scholar

[23]

C. Wall, Singular Points of Plane Curves, London Mathematical Society Student Texts, 2004. Google Scholar

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