Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 059, 38 pages      arXiv:1205.2992      https://doi.org/10.3842/SIGMA.2014.059

Configurations of an Articulated Arm and Singularities of Special Multi-Flags

Fernand Pelletier a and Mayada Slayman b
a) Université de Savoie, Laboratoire de Mathématiques (LAMA), Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France
b) Department of Mathematical Sciences, Faculty of Sciences II, Lebanese University, Lebanon

Received January 29, 2013, in final form May 18, 2014; Published online June 05, 2014

Abstract
P. Mormul has classified the singularities of special multi-flags in terms of “EKR class'' encoded by sequences $j_1,\dots, j_k$ of integers (see [Singularity Theory Seminar, Warsaw University of Technology, Vol. 8, 2003, 87-100] and [Banach Center Publ., Vol. 65, Polish Acad. Sci., Warsaw, 2004, 157-178]). However, A.L. Castro and R. Montgomery have proposed in [Israel J. Math. 192 (2012), 381-427] a codification of singularities of multi-flags by RC and RVT codes. The main results of this paper describe a decomposition of each ''EKR'' set of depth $1$ in terms of RVT codes as well as characterize such a set in terms of configurations of an articulated arm. Indeed, an analogue description for some ''EKR'' sets of depth $2$ is provided. All these results give rise to a complete characterization of all ''EKR'' sets for $1\leq k\leq 4$.

Key words: special multi-flags distributions; Cartan prolongation; spherical prolongation; articulated arm; rigid bar.

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