# On geometry of curves of flags of constant type

Open Mathematics (2012)

- Volume: 10, Issue: 5, page 1836-1871
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topBoris Doubrov, and Igor Zelenko. "On geometry of curves of flags of constant type." Open Mathematics 10.5 (2012): 1836-1871. <http://eudml.org/doc/269813>.

@article{BorisDoubrov2012,

abstract = {We develop an algebraic version of Cartan’s method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space W with respect to the action of a subgroup G of GL(W). Under some natural assumptions on the subgroup G and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure linear algebra. The scope of applicability of the theory includes geometry of natural classes of curves of flags with respect to reductive linear groups or their parabolic subgroups. As simplest examples, this includes the projective and affine geometry of curves. The case of classical groups is considered in more detail.},

author = {Boris Doubrov, Igor Zelenko},

journal = {Open Mathematics},

keywords = {Curves and submanifolds in flag varieties; Equivalence problem; Bundles of moving frames; Graded Lie algebras; Tanaka prolongation; sl2-representations; curves and submanifolds in flag varieties; equivalence problem; bundles of moving frames; graded Lie algebras; -representations},

language = {eng},

number = {5},

pages = {1836-1871},

title = {On geometry of curves of flags of constant type},

url = {http://eudml.org/doc/269813},

volume = {10},

year = {2012},

}

TY - JOUR

AU - Boris Doubrov

AU - Igor Zelenko

TI - On geometry of curves of flags of constant type

JO - Open Mathematics

PY - 2012

VL - 10

IS - 5

SP - 1836

EP - 1871

AB - We develop an algebraic version of Cartan’s method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space W with respect to the action of a subgroup G of GL(W). Under some natural assumptions on the subgroup G and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure linear algebra. The scope of applicability of the theory includes geometry of natural classes of curves of flags with respect to reductive linear groups or their parabolic subgroups. As simplest examples, this includes the projective and affine geometry of curves. The case of classical groups is considered in more detail.

LA - eng

KW - Curves and submanifolds in flag varieties; Equivalence problem; Bundles of moving frames; Graded Lie algebras; Tanaka prolongation; sl2-representations; curves and submanifolds in flag varieties; equivalence problem; bundles of moving frames; graded Lie algebras; -representations

UR - http://eudml.org/doc/269813

ER -

## References

top- [1] Agrachev A.A., Feedback-invariant optimal control theory and differential geometry II. Jacobi curves for singular extremals, J. Dynam. Control Systems, 1998, 4(4), 583–604 http://dx.doi.org/10.1023/A:1021871218615 Zbl0972.49014
- [2] Agrachev A.A., Gamkrelidze R.V., Feedback-invariant optimal control theory and differential geometry I. Regular extremals, J. Dynam. Control Systems, 1997, 3(3), 343–389 http://dx.doi.org/10.1007/BF02463256 Zbl0952.49019
- [3] Agrachev A., Zelenko I., Principle invariants of Jacobi curves, In: Nonlinear Control in the Year 2000, 1, Lecture Notes in Control and Inform. Sci., 258, Springer, 2000, 9–21 Zbl1239.53059
- [4] Agrachev A., Zelenko I., Geometry of Jacobi curves. I, J. Dynam. Control Systems, 2002, 8(1), 93–140 http://dx.doi.org/10.1023/A:1013904801414
- [5] Agrachev A., Zelenko I., Geometry of Jacobi curves. II, J. Dynam. Control Systems, 2002, 8(2), 167–215 http://dx.doi.org/10.1023/A:1015317426164
- [6] Cartan E., La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repere mobile, Cahiers Scientifiques, 18, Gauthier-Villars, Paris, 1937 Zbl63.1227.02
- [7] Derksen H., Weyman J., Quiver representations, Notices Amer. Math. Soc., 2005, 52(2), 200–206 Zbl1143.16300
- [8] Doubrov B., Projective reparametrization of homogeneous curves, Arch. Math. (Brno), 2005, 41(1), 129–133 Zbl1122.53029
- [9] Doubrov B., Generalized Wilczynski invariants for non-linear ordinary differential equations, In: Symmetries and Overdetermined Systems of Partial Differetial Equations, IMA Vol. Math. Appl., 144, Springer, New York, 2008, 25–40 http://dx.doi.org/10.1007/978-0-387-73831-4_2
- [10] Doubrov B.M., Komrakov B.P., Classification of homogeneous submanifolds in homogeneous spaces, Lobachevskii J. Math., 1999, 3, 19–38 Zbl0964.53035
- [11] Doubrov B., Machida Y., Morimoto T., Linear equations on filtered manifolds and submanifolds of flag varieties (manuscript)
- [12] Doubrov B., Zelenko I., A canonical frame for nonholonomic rank two distributions of maximal class, C. R. Acad. Sci. Paris, 2006, 342(8), 589–594 http://dx.doi.org/10.1016/j.crma.2006.02.010 Zbl1097.58002
- [13] Doubrov B., Zelenko I., On local geometry of non-holonomic rank 2 distributions, J. Lond. Math. Soc., 2009, 80(3), 545–566 http://dx.doi.org/10.1112/jlms/jdp044 Zbl1202.58002
- [14] Doubrov B., Zelenko I., On local geometry of rank 3 distributions with 6-dimensional square, preprint available at http://arxiv.org/abs/0807.3267 Zbl1202.58002
- [15] Doubrov B., Zelenko I., Geometry of curves in parabolic homogeneous spaces, preprint available at http://arxiv.org/abs/1110.0226 Zbl1278.53047
- [16] Eastwood M., Slovák J., Preferred parameterisations on homogeneous curves, Comment. Math. Univ. Carolin., 2004, 45(4), 597–606
- [17] Fels M., Olver P.J., Moving coframes: I. A practical algorithm, Acta Appl. Math., 1998, 51(2), 161–213 http://dx.doi.org/10.1023/A:1005878210297 Zbl0937.53012
- [18] Fels M., Olver P.J., Moving coframes: II. Regularization and theoretical foundations, Acta Appl. Math., 1999, 55(2), 127–208 http://dx.doi.org/10.1023/A:1006195823000 Zbl0937.53013
- [19] Fulton W., Harris J., Representation Theory, Grad. Texts in Math., 129, Springer, New York, 1991 Zbl0744.22001
- [20] Gabriel P., Unzerlegbare Darstellungen I, Manuscripta Math., 1972, 6, 71–103 http://dx.doi.org/10.1007/BF01298413
- [21] Gel’fand I.M., Lectrures on Linear Algebra, Interscience Tracts in Pure and Applied Mathematics, 9, Interscience, New York-London, 1961
- [22] Green M.L., The moving frame, differential invariants and rigidity theorem for curves in homogeneous spaces, Duke Math. J., 1978, 45(4), 735–779 http://dx.doi.org/10.1215/S0012-7094-78-04535-0 Zbl0414.53039
- [23] Griffiths P., On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J., 1974 41, 775–814 http://dx.doi.org/10.1215/S0012-7094-74-04180-5 Zbl0294.53034
- [24] Humphreys J.E., Introduction to Lie Algebras and Representation Theory, 3rd printing, Grad. Texts in Math., 9, Springer, New York-Berlin, 1980 Zbl0447.17002
- [25] Jacobson N., Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, 10, Interscience, New York-London, 1962 Zbl0121.27504
- [26] Lie S., Theory der Transformationgruppen, 3, Teubner, Leipzig, 1893
- [27] Marí Beffa G., Poisson brackets associated to the conformal geometry of curves, Trans. Amer. Math. Soc., 2005, 357(7), 2799–2827 http://dx.doi.org/10.1090/S0002-9947-04-03589-5 Zbl1081.37042
- [28] Marí Beffa G., On completely integrable geometric evolutions of curves of Lagrangian planes, Proc. Roy. Soc. Edinburgh Sect. A, Math., 2007, 137(1), 111–131 Zbl1130.37032
- [29] Marí Beffa G., Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds, Ann. Inst. Fourier (Grenoble), 2008, 58(4), 1295–1335 http://dx.doi.org/10.5802/aif.2385 Zbl1192.37099
- [30] Marí Beffa G., Moving frames, geometric Poisson brackets and the KdV-Schwarzian evolution of pure spinors, Ann. Inst. Fourier (Grenoble) (in press) Zbl1245.53066
- [31] Ovsienko V., Lagrange Schwarzian derivative and symplectic Sturm theory, Ann. Fac. Sci. Toulouse Math., 1993, 2(1), 73–96 http://dx.doi.org/10.5802/afst.758 Zbl0780.34004
- [32] Se-ashi Y., On differential invariants of integrable finite type linear differential equations, Hokkaido Math. J., 1988, 17(2), 151–195
- [33] Se-ashi Y., A geometric construction of Laguerre-Forsyth’s canonical forms of linear ordinary differential equations, In: Progress in Differential Geometry, Adv. Stud. Pure Math., 22, Kinokuniya, Tokyo, 1993, 265–297 Zbl0842.34015
- [34] Tanaka N., On differential systems, graded Lie algebras and pseudogroups, J. Math. Kyoto. Univ., 1970, 10, 1–82
- [35] Tanaka N., On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J., 1979, 6(1), 23–84 Zbl0409.17013
- [36] Vinberg È.B., The Weyl group of a graded Lie algebra, Izv. Akad. Nauk SSSR Ser. Mat., 1976, 40(3), 488–526 (in Russian) Zbl0363.20035
- [37] Vinberg È.B., Classification of homogeneous nilpotent elements of a semisimple graded Lie algebra, Trudy Sem. Vektor. Tenzor. Anal., 1979, 19, 155–177 (in Russian) Zbl0431.17006
- [38] Wilczynski E.J., Projective Differential Geometry of Curves and Ruled Surfaces, Teubner, Leipzig, 1906 Zbl37.0620.02
- [39] Zelenko I., Complete systems of invariants for rank 1 curves in Lagrange Grassmannians, In: Differential Geometry and its Applications, Prague, August 30–September 3, 2004, Matfyzpress, Prague, 2005, 367–382 Zbl1110.53062
- [40] Zelenko I., Li C., Parametrized curves in Lagrange Grassmannians, C. R. Math. Acad. Sci. Paris, 2007, 345(11), 647–652 http://dx.doi.org/10.1016/j.crma.2007.10.034 Zbl1130.53042
- [41] Zelenko I., Li C., Differential geometry of curves in Lagrange Grassmannians with given Young diagram, Differential Geom. Appl., 2009, 27(6), 723–742 http://dx.doi.org/10.1016/j.difgeo.2009.07.002 Zbl1177.53020
- [42] Sophus Lie’s 1880 Transformation Group Paper, Lie Groups: Hist., Frontiers and Appl., 1, Math Sci Press, Brookline, 1975

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.