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geometric structures on manifolds; local submanifolds; contact theory; actions of groups
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 $G$.
[1] Bredon G.E.: Introduction to Compact Transformations groups. Academic Press, New York (1972). MR 0413144
[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 0425155 | Zbl 0302.32015
[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 0361170
[6] Ehresmann C.: Les prolongements d’un space fibré diferéntiable. C.R. Acad. Sci. Paris, 240 (1955), 1755-1757. MR 0071083
[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 0518104
[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 0410607
[9] Hermann R.: Equivalence invariants for submanifolds of Homogeneous Spaces. Math. Annalen 158, 284-289 (1965). MR 0203653 | Zbl 0125.39502
[10] Hermann R.: Existence in the large of parallelism homomorphisms. Trans. Am. Math. Soc. 108, 170-183 (1963). MR 0151924
[11] Jensen G. R.: Higher Order Contact of Submanifolds of Homogeneous Spaces. Lectures notes in Math. Vol. 610, Springer-Verlag, New York (1977). MR 0500648 | Zbl 0356.53005
[12] Jensen G.R.: Deformation of submanifolds of homogeneous spaces. J. of Diff. Geometry, 16(1981), 213-246. MR 0638789 | Zbl 0473.53044
[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 0301668
[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 2342861 | Zbl 0601.53017
[15] Villarroel Y.: Equivalencia de curvas. Acta Científica Venezolana. Vol. 37, No. 6, p. 625-631, (1987). MR 0897828
[16] Villarroel Y.: Teoria de contacto y Referencial móvil. Public. Universidad Central de Venezuela. Dpto. de Matemática. 1991.
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