Higher order contact of real curves in a real hyperquadric. II

Yuli Villarroel

Archivum Mathematicum (1998)

  • Volume: 034, Issue: 3, page 361-377
  • ISSN: 0044-8753

Abstract

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Let Φ be an Hermitian quadratic form, of maximal rank and index ( n , 1 ) , defined over a complex ( n + 1 ) vector space V . Consider the real hyperquadric defined in the complex projective space P n V by Q = { [ ς ] ∈ P n V , Φ ( ς ) = 0 } . Let G be the subgroup of the special linear group which leaves Q invariant and D the ( 2 n ) - distribution defined by the Cauchy Riemann structure induced over Q . We study the real regular curves of constant type in Q , tangent to D , finding a complete system of analytic invariants for two curves to be locally equivalent under transformations of G .

How to cite

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Villarroel, Yuli. "Higher order contact of real curves in a real hyperquadric. II." Archivum Mathematicum 034.3 (1998): 361-377. <http://eudml.org/doc/248181>.

@article{Villarroel1998,
abstract = {Let $\Phi $ be an Hermitian quadratic form, of maximal rank and index $(n,1)$, defined over a complex $(n+1)$ vector space $V$. Consider the real hyperquadric defined in the complex projective space $P^nV$ by \[ Q=\lbrace [\varsigma ]\in P^nV,\;\Phi (\varsigma )=0\rbrace . \] Let $G$ be the subgroup of the special linear group which leaves $ Q $ invariant and $D$ the $(2n)-$ distribution defined by the Cauchy Riemann structure induced over $Q$. We study the real regular curves of constant type in $Q$, tangent to $D$, finding a complete system of analytic invariants for two curves to be locally equivalent under transformations of $G$.},
author = {Villarroel, Yuli},
journal = {Archivum Mathematicum},
keywords = {geometric structures on manifolds; local submanifolds; contacttheory; actions of groups; contact theory; group action; real hyperquadric; moving frame},
language = {eng},
number = {3},
pages = {361-377},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Higher order contact of real curves in a real hyperquadric. II},
url = {http://eudml.org/doc/248181},
volume = {034},
year = {1998},
}

TY - JOUR
AU - Villarroel, Yuli
TI - Higher order contact of real curves in a real hyperquadric. II
JO - Archivum Mathematicum
PY - 1998
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 034
IS - 3
SP - 361
EP - 377
AB - Let $\Phi $ be an Hermitian quadratic form, of maximal rank and index $(n,1)$, defined over a complex $(n+1)$ vector space $V$. Consider the real hyperquadric defined in the complex projective space $P^nV$ by \[ Q=\lbrace [\varsigma ]\in P^nV,\;\Phi (\varsigma )=0\rbrace . \] Let $G$ be the subgroup of the special linear group which leaves $ Q $ invariant and $D$ the $(2n)-$ distribution defined by the Cauchy Riemann structure induced over $Q$. We study the real regular curves of constant type in $Q$, tangent to $D$, finding a complete system of analytic invariants for two curves to be locally equivalent under transformations of $G$.
LA - eng
KW - geometric structures on manifolds; local submanifolds; contacttheory; actions of groups; contact theory; group action; real hyperquadric; moving frame
UR - http://eudml.org/doc/248181
ER -

References

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  1. Bredon G. E., Introduction to Compact Transformations groups, Academic. Press, New York (1972). (1972) MR0413144
  2. Cartan E., Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, I. II, Ouvres II, 2, 1231-1304; ibid III,2, 1217-1238. 
  3. Cartan E., Théorie des groupes finis el la géométrie différentielle traitées par la Methode du repère mobile, Gauthier-Villars, Paris, (1937). (1937) 
  4. Chern S. S., Moser J. K., Real hypersurfaces in complex manifolds, Acta mathematica 133(1975), 219-271. (1975) Zbl0302.32015MR0425155
  5. Chern S. S., Cowen J. M., Frenet frames along holomorphic curves, Topics in Differential Geometry, 1972-1973, 191-203. Dekker, New York, 1974. (1972) MR0361170
  6. Ehresmann C., Les prolongements d’un space fibré diferéntiable, C.R. Acad. Sci. Paris, 240(1955), 1755-1757. (1955) MR0071083
  7. Green M. L., The moving frame, Differential invariants and rigity theorems for curves in homogeneous spaces, Duke Math. Journal, Vol 45, No.4 (1978), 735-779. (1978) MR0518104
  8. Griffiths P., On Cartan’s method of Lie groups and moving frames as applied to existence and uniqueness questions in differential geometry, Duke Math. J. 41(1974), 775-814. (1974) MR0410607
  9. Hermann R., Equivalence invariants for submanifolds of Homogeneous Spaces, Math. Annalen 158(1965), 284-289. (1965) Zbl0125.39502MR0203653
  10. Hermann R., Existence in the large of parallelism homomorphisms, Trans. Am. Math. Soc. 108, 170-183 (1963). (1963) MR0151924
  11. Jensen G. R., Higher Order Contact of Submanifolds of Homogeneous Spaces, Lectures notes in Math. Vol. 610, Springer-Verlag, New York (1977). (1977) Zbl0356.53005MR0500648
  12. Jensen G. R., Deformation of submanifolds of homogeneous spaces, J. of Diff. Geometry, 16(1981), 213-246. (1981) Zbl0473.53044MR0638789
  13. Kolář I., Canonical forms on the prolongations of principle fibre bundles, Rev. Roum. Math. Pures et Appl., Bucarest, Tome XVI, No.7 (1971), 1091-1106. (1971) MR0301668
  14. Rodrigues A. M., Contact and equivalence of submanifolds of homogeneous spaces, aspects of Math. and its Applications, Elsevier Science Publishers B.V. (1986). (1986) MR2342861
  15. Villarroel Y., Differential Geometry and Applications, Proc. Conf. Brno (1996), 207-214. (1996) MR1406339
  16. Villarroel Y., Higher order contact of real curves in a real hyperquadric, Archivum mathematicum, Tomus 32(1996), 57-73. (1996) Zbl0870.53025MR1399840

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