Geodesic
Higher order contact of real curves in a real hyperquadric
Archivum mathematicum, Tome 32 (1996) no. 1, pp. 57-73 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $\Phi $ be an hermitian quadratic form, of maximal rank and index $(n,1)$% , defined over a complex $(n+1)$ vectorial space $V$. Consider the real hyperquadric defined in the complex projective space $P^nV$ by \[ Q=\{[\varsigma ]\in P^nV,\;\Phi (\varsigma )=0\}, \] let $G$ be the subgroup of the special linear group which leaves $Q$ invariant and $D$ the $(2n-2)$ distribution defined by the Cauchy Riemann structure induced over $Q$. We study the real regular curves of constant type in $Q$, transversal to $D$, finding a complete system of analytic invariants for two curves to be locally equivalent under transformations of $G$.
Let $\Phi $ be an hermitian quadratic form, of maximal rank and index $(n,1)$% , defined over a complex $(n+1)$ vectorial space $V$. Consider the real hyperquadric defined in the complex projective space $P^nV$ by \[ Q=\{[\varsigma ]\in P^nV,\;\Phi (\varsigma )=0\}, \] let $G$ be the subgroup of the special linear group which leaves $Q$ invariant and $D$ the $(2n-2)$ distribution defined by the Cauchy Riemann structure induced over $Q$. We study the real regular curves of constant type in $Q$, transversal to $D$, finding a complete system of analytic invariants for two curves to be locally equivalent under transformations of $G$.
Classification : 32F40, 53B25, 53C15
Keywords: geometric structures on manifolds; local submanifolds; contact theory; actions of groups 
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Villarroel, Y. Higher order contact of real curves in a real hyperquadric. Archivum mathematicum, Tome 32 (1996) no. 1, pp. 57-73. http://geodesic.mathdoc.fr/item/ARM_1996_32_1_a4/

[1] Bredon G.E.: Introduction to Compact Transformations groups. Academic Press, New York (1972). | MR

[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).

[4] Chern S. S., Moser J. K.: Real hypersurfaces in complex manifolds. Acta mathematica 133, (1975), 219-271. | MR | Zbl

[5] Chern S. S., Cowen J. M.: Frenet frames along holomorphic curves. Topics in Differential Geometry, 1972-1973, pp. 191-203. Dekker, New York, 1974. | MR

[6] Ehresmann C.: Les prolongements d’un space fibré diferéntiable. C.R. Acad. Sci. Paris, 240 (1955), 1755-1757. | MR

[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. | MR

[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. | MR

[9] Hermann R.: Equivalence invariants for submanifolds of Homogeneous Spaces. Math. Annalen 158, 284-289 (1965). | MR | Zbl

[10] Hermann R.: Existence in the large of parallelism homomorphisms. Trans. Am. Math. Soc. 108, 170-183 (1963). | MR

[11] Jensen G. R.: Higher Order Contact of Submanifolds of Homogeneous Spaces. Lectures notes in Math. Vol. 610, Springer-Verlag, New York (1977). | MR | Zbl

[12] Jensen G.R.: Deformation of submanifolds of homogeneous spaces. J. of Diff. Geometry, 16(1981), 213-246. | MR | Zbl

[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. | MR

[14] Rodrigues A. M.: Contact and equivalence of submanifolds of homogeneous spaces. Aspects of Math. and its Applications. Elsevier Science Publishers B.V. (1986). | MR | Zbl

[15] Villarroel Y.: Equivalencia de curvas. Acta Científica Venezolana. Vol. 37, No. 6, p. 625-631, (1987). | MR

[16] Villarroel Y.: Teoria de contacto y Referencial móvil. Public. Universidad Central de Venezuela. Dpto. de Matemática. 1991.