Travaux sur les symÃ©tries de Lie des Ã©quations aux ... - DMA - Ens

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382[Lo2001][JoPf2000] DE JONG, T.; PFISTER, G.: Local analytic geometry. Basic theory and applications,Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig,2000. xii+382 pp.[Kn2004] KNAPP, A.W.: Lie groups beyond an introduction, Progress in Mathematics,140, Birkhäuser, Basel, third edition, 2004, xviii+812 pp.[KN1963] KOBAYASHI, S.; NOMIZU, K.m: Foundations of differential geometry, I, Intersciencepublishers, John Wiley & Sons, New York, 1963. xi+329 pp.[Kr1985] KRUZHILIN, N.G.: Local automorphisms and mappings of smooth strictlypseudoconvex hypersurfaces (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 49(1985), no. 3, 566–591, 672.[Kr1987] KRUZHILIN, N.G.: Description of the local automorphism groups of real hypersurfaces,Proceedings of the International Congress of Mathematicians,Vol. 1, 2 (Berkeley, Calif., 1986), 749–758, Amer. Math. Soc., Providence,RI, 1987.[KV1987] KRUZHILIN, N.G.; VITUSHKIN, A.G.: Description of the automorphismgroups of real hypersurfaces in complex space (Russian). Investigations in thetheory of the approximation of functions, 26–69, Akad. Nauk SSSR BashkirFilial, Otdel. Fiz. Mat., Ufa, 1987.[Lie1880] LIE, S.: Theorie der Transformationsgruppen, Math. Ann. 16 (1880), 441–528.[Lie1883] LIE, S.: Klassifikation und Integration von gewöhnlichen Differentialgleichungenzwischen x, y, die eine Gruppe von Transformationen gestaten I-IV. In:Gesammelte Abhandlungen, Vol. 5, B.G. Teubner, Leipzig, 1924, pp. 240–310; 362–427, 432–448.[LS1893] LIE, S.; SCHEFFERS, G.: Vorlesungen ¨ber continuierliche Gruppen mit Geometrischenund anderen Anwendungen. (German) Nachdruck der Auflage desJahres 1893. Chelsea Publishing Co., Bronx, New York, 1971. xii+810 pp.[Lo1981] LOBODA, A.V.: Local automorphisms of real-analytic hypersurfaces (Russian),Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 3, 620–645.LOBODA, A.V.: Homogeneous strictly pseudoconvex hypersurfaces in C 3 withtwo-dimensional isotropy groups (Russian) Mat. Sb. 192 (2001), no. 12, 3–24;translation in Sb. Math. 192 (2001), no. 11-12, 1741–1761.[Lo2002] LOBODA, A.V.: Homogeneous nondegenerate surfaces in C 3 with twodimensionalisotropy groups (Russian) Mat. Sb. 192 (2001), no. 12, 3–24;translation in Sb. Math. 192 (2001), no. 11–12, 1741–1761.[Lo2003][Ma2003][Me2001][Me2003][Me2004]LOBODA, A.V.: On the determination of a homogeneous strictly pseudoconvexhypersurface from the coefficients of its normal form (Russian) Mat. Zametki73 (2003), no. 3, 453–456; translation in Math. Notes 73 (2003), no. 3-4, 419–423.MARDARE, S.: On isometric immersions of a Riemannian space under a weakregularity assumption, C. R. Acad. Sci. Paris, Sér. I 337 (2003), 785–790.MERKER, J.: On the partial algebraicity of holomorphic mappings betweentwo real algebraic sets in the complex euclidean spaces of different dimensions,Bull. Soc. Math. France 129 (2001), no. 4, 547–591.MERKER, J.: hand manuscript I, 212 pp., May – July 2003; hand manuscriptII, 114 pp., August 2003.MERKER, J.: Explicit differential characterization of the Newtonian free particlesystem in m 2 dependent variables, Acta Mathematicæ Applicandæ,to appear, 73 pp; e-print: arxiv.org/abs/math.DG/0411165.
[Me2005a] MERKER, J.: On the local geometry of generic submanifolds of C n and theanalytic reflection principle (Part I), Journal of Mathematical Sciences (N.Y.)125 (2005), no. 6, 751–824.[Me2005b] MERKER, J.: Étude de la régularité analytique de l’application de réflexionCR formelle, Annales Fac. Sci. Toulouse, XIV (2005), no. 2, 215–330.[MP2005] MERKER, J.; PORTEN, E.: Holomorphic extension of CR functions, envelopesof holomorphy and removable singularities, 432 pp., to appear. Downloadableat: www.cmi.univ-mrs/∼merker/index.html.[N2003] NEUT, S.: Implantation et nouvelles applications de la méthode d’équivalenced’Élie Cartan, Thèse, Université Lille 1, October 2003.[OL1979] OLVER, P.J.: Symmetry groups and group invariant solutions of partial differentialequations, J. Diff. Geom. 14 (1979), 497–542.[Ol1986] OLVER, P.J.: Applications of Lie groups to differential equations. SpringerVerlag, New York, 1986. xxvi+497 pp.[Ol1995] OLVER, P.J.: Equivalence, Invariance and Symmetries. Cambridge, CambridgeUniversity Press, 1995, xvi+525 pp.[OV1994] ONISHCHIK, A.L.; VINBERG, E.B.: Lie groups and Lie algebras, III. Encyclopædiaof mathematical sciences, 41. Springer Verlag, Berlin, 248 pp.[Pi1975] PINCHUK, S.: On the analytic continuation of holomorphic mappings (Russian),Mat. Sb. (N.S.) 98(140) (1975) no.3(11), 375–392, 416–435, 495–496.[Pi1978] PINCHUK, S.: Holomorphic mappings of real-analytic hypersurfaces (Russian),Mat. Sb. (N.S.) 105(147) (1978), no. 4, 574–593, 640; English translationin Math. USSR Sbornik 34 (1978), 503–519.[Re1993] REUTENAUER, C.: Free Lie algebras, London Mathematical Society Monograph,New Series, 7. Oxford Science Publications, The Clarendon Press, OxfordUniversity Press, New York, 1993. xviii+269 pp.[Se1931] SEGRE, B.: Intorno al problema di Poincaré della rappresentazione pseudoconforme,Rend. Acc. Lincei, VI, Ser. 13 (1931), 676–683.[Se1932] SEGRE, B.: Questioni geometriche legate colla teoria delle funzioni di duevariabili complesse, Rendiconti del Seminario di Matematici di Roma, II, Ser.7 (1932), no. 2, 59–107.[Sh1997] SHARPE, R.W.: Differential geometry; Cartan’s generalization of Klein’s Erlangenprogram, Graduate texts in mathematics, vol. 166, Springer Verlag,Berlin, 1997, xix+421 pp.[Sp1970] SPIVAK, M.: A comprehensive introduction to differential geometry, vols. 1and 2. Published by M. Spivak, Brandeis Univ., Waltham, Mass. 1970.[St1996] STANTON, N.: Infinitesimal CR automorphisms of real hypersurfaces, Amer.J. Math. 118 (1996), no. 1, 209–233.[Ste1983] STERNBERG, S.: Lectures in differential geometry, Second edition, Chelseapublishing co., New York, 1983, xviii+442 pp.[Stk1982] STORMARK, O.: On the theorem of Frobenius for complex vector fields, Ann.[Stk2000]383Scuola Norm. Sup. Pisa Cl. Sci. (4) 9 (1982), no. 1, 57–90.STORMARK, O.: Lie’s structural approach to PDE systems, Encyclopædia ofmathematics and its applications, vol. 80, Cambridge University Press, Cambridge,2000, xv+572 pp.[Su2001] SUKHOV, A.: Segre varieties and Lie symmetries, Math. Z. 238 (2001), no. 3,483–492.[Su2002] SUKHOV, A.: CR maps and point Lie transformations, Michigan Math. J. 50(2002), 369–379.
Page 1 and 2:
Joël M E R K E RÉcole Normale Sup
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§1. INTRODUCTIONSeveral physically
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e a collection of m analytic second
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7where the indices j, l 1 vary in {
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This phenomenon could be explained
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This yields the prolongation of the
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X(x, y) and Y (x, y) such that it m
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2.23. Compatibility conditions for
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This lemma is left to the reader; a
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simplification nor any reordering:(
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aleza, y por otra, las organizacion
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computational level (differential-g
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For instance, in the case m = 2, by
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in (3.11). The second equation that
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Similarly, the second equation take
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order derivatives of X and of the Y
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Lemma 3.32. The system yxx j = F j
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Lemma 3.45. The following quadratic
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often be denoted by the sign “·
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1Now, taking account of the factor
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conditions, totally equivalent to t
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43remaining terms afterwards:(3.71)
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Here, the sign ≡ precisely means:
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Next, replacing plainly (3.64) in (
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49the order of §3.73. We get:(3.89
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51Multiplying by −2 and reorganiz
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⎧Θ l 1y l 2 = −Ll 1l 1 ,l 1 ,y
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+ 1 ∑4 δj l 1Hl k 2L k k,kk− 2
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+ 1 ∑2 δj l 1Hl k 3M l2 ,kk+ 1 4
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In the hardest techical part of thi
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− 1 3+ 1 3− 1 4+ 1 4∑Hl k 1H
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− 1 3∑kL l 1l1 ,k,x Hk l 1+ 1 3
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Here, the sign ≡ means “modulo
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Thirdly, put j := l 2 in (3.108) wi
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69correct. We get:(4.29)0 =?== −
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Next, apply the operator ∑ k Ll 2
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73Writing term by term the substrac
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of the terms of the subgoal (4.29):
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77+ 1 2∑kH k l 1 ,y l 2 Hl 2k15
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79= − X xx Yx j + Y jm∑+++++l 1
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Our first task is to compute the de
Page 83 and 84:
Replacing this expression of A k in
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85have the continuation(5.17) −y
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87and where thirdly (we are nearly
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89obtain:(5.23)III :=m∑m∑l 1 =1
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[GTW1989] GRISSOM, C.; THOMPSON, G.
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93Nonalgebraizable real analytic tu
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and of CR dimension m = n − d ≥
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coordinates z ′ = 2i ln(z/z p ),
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{x ∈ K n : |x| < ρ} for some ρ
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(2) A K-algebraic inversion mapping
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(1) The complex Lie algebra Hol(M,
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Proposition 3.1. Let t ↦→ G i (
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The first relation gives nothing, s
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with |β∗ k | ≥ 1 and integers
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Clearly, the left hand side is an a
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field X 1 ′ := h ∗ (X 1 ). Taki
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which yields after differentiating
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117Im w∆ n(ρ 4 )∆ n(ρ 1 )∆
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[∂ i,ei H ei (t ′ )] t ′ =He
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p = (w p , z p , ζ p , ξ p ) ∈
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Finite nondegeneracy is interesting
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such that |ω j i ∗ ,β∗ i(t,
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for k even, we have Γ k (z (k) ) =
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that for every local holomorphic se
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h(t) = ˜H(t, J l 0µ 0¯h(0)) with
Page 133 and 134:
infinite families of pairwise non b
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functions w(z)). The coefficients R
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Then the Lie criterion states that
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y k ′ = λk k y k and w ′ = µ
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The last statements of Corollaries
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[Sha2000] SHAFIKOV, R.: Analytic co
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Here x = (x 1 , . . ., x n ) ∈ K
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Consequently, for the case κ = 2,
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are devoted to provide a general on
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for l = 1, . . .,m such that the lo
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Lemma 8.1. The following conditions
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155namely X ←→ X + X ←→ X .
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In terms of Sussmann’s approach [
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x κ−1 χ κ−1 + O(|x| κ ) + O
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Observe that these expressions are
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where the first term I involves onl
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I 5 = ∑ i 1I 6 = ∑ i 1 ,i 2I 7
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U 2 U κ−1 , we obtain the five f
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and U 2 U κ−1 , we obtain the fi
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following equations for j = 1, . .
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linear equations:(7.28) ⎧0 = Rx j
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175R j containing at least one part
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177[3] :∑σ∈S κ−1κδ l,....
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the values of the partial derivativ
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also appears some derivatives Q l
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[5] F. ENGEL; LIE, S.: Theorie der
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185Nonrigid sphericalreal analytic
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origin:{ ( ∣ ∣ )AJ 4 1∣∣∣
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I 2 characterizes equivalence to w
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y means of a fundamental, elementar
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2) When M is Levi nondegenerate at
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195holding in C { z ′ , z ′ , w
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Fortunately for our present purpose
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We notice passim that S ≡ Q x (no
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for a certain local K-analytic new
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Conversely, if y xx (x) = F ( x, y(
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205But we may also express the dual
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Then thanks to a straightforward ap
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and we then expand carefully the re
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explicit computation, so let us rew
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213+T ah3Q 2 a Q b∆(aa|b)∆(b|bb
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215Vanishing Hachtroudi curvaturean
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to fix the ideas, it will be assume
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then by replacing the(so obtained)v
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and also in addition the fact that
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and this defines without ambiguity
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the leaves of this local foliation
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Then by differentiating with respec
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∂Here, the coefficients of the2 T
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231for (7.28):( )∂□ ·∂a □
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233Lie symmetriesof partial differe
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Differentiating the first equation
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Example 1.21. (Continued) With n =
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defined by the graphed equations:((
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for some two local analytic maps Π
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Proof. Let l = l ( x i , y j , yβ(
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complementary views on the same obj
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Classification problem 3.11. Classi
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Definition 4.10. L is an infinitesi
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Convention 5.2. The letters R will
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Assuming that dimSYM(E 1 ) = 8, tak
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Restarting from §4.1, let ϕ a Lie
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Here, R j l 1 ,kare universal polyn
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Lemma 6.34. Let M be a submanifold
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where the expressions D β,β1 are
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Then the sum L + L is tangent to M
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Lemma 7.11. ([CM1974, BER1999, Me20
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where(7.33) ∆ := ∑1kn∂ 2∂x
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with r, s being unspecified functio
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and moreover, all other, higher ord
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To conclude, we replace X xx so obt
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Definition 8.43. A real analytic hy
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Lemma 8.59. The function b depends
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where the letter r denotes an unspe
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(8.80) ⎧Y 1,1,1 = Y x 1 x 1 x 1 +
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(X 1 , X 2 , Y , Xy 1,X 2x, Y 2 x 1
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espect to x 1 and (8.96) 2 with res
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For the two unknowns Xyy 1 and X 2x
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Redeveloping the determinant, the v
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291are obtained by equating to zero
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Let (z 0 , c 0 ) = (x 0 , y 0 , a 0
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we deduce that Γ 5 is submersive (
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11.4. Regularity and jet parametriz
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Cramer’s rule, we get(11.13) ⎧
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in K[a, z] n+m and in K[x, c] p+m .
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of the cancellation properties (11.
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305II: Explicit prolongations of in
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Define then the transformed jet ϕ
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Let λ ∈ N be an arbitrary intege
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311[Ol1986], [BK1989]):(Y(1.31)⎧
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+ [Y y 2 − 2 X xy ] y 1 y 2 + [
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for some nonnegative integers A, B,
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Once the correct theorem is formula
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By the classical formula for the de
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The induction formulas are⎧(3.4)
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We have to compute:( ) ⎛ ⎞∑
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Next, we gather the underlined term
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+ ∑k 1 ,k 2 ,k 3 ,k 4[−δ k 1,k
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Correspondingly, we identify the se
Page 331 and 332: must be equal to the number of indi
Page 333 and 334: Y i1 ,i 2 ,i 3 ,i 4written in one o
Page 335 and 336: 335conclusion, we have shown that (
Page 337 and 338: 337Theorem 3.73. For every κ 1 an
Page 339 and 340: Theorem 3.79. For every integer κ
Page 341 and 342: +++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3
Page 343 and 344: 343form:(4.12)κ+1Yκ j = Y jx κ +
Page 345 and 346: eing related to the number 2 in the
Page 347 and 348: Appying the chain rule, we may comp
Page 349 and 350: Secondly:(5.3)Y j i 1 ,i 2= Y jx i
Page 351 and 352: As explained before the statement o
Page 353 and 354: g j i 1 ,...,i λand similarly for
Page 355 and 356: with respect to the variables x i 1
Page 357 and 358: Assuming F = F(x, y x ) to be indep
Page 359 and 360: 359(III’) ⎧0 = ∑ (δ k 2)j 3H
Page 361 and 362: Open problem 1.17. For n = 2 establ
Page 363 and 364: ⎧∣ ⎨X 1 xX 1+ y x 1 y x 1 ·1
Page 365 and 366: eplacing the third column by the se
Page 367 and 368: ⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣
Page 369 and 370: e written under the specific form:(
Page 371 and 372: Indeed, the collection of equations
Page 373 and 374: 3.12. Principal unknowns. As there
Page 375 and 376: 375Sixthly:(3.22) ⎧δ k 1j 1Θ n+
Page 377 and 378: are a consequence of (I’), (I”)
Page 379 and 380: 379IV: BibliographyREFERENCES[Ar198
Page 381: [G1989] GARDNER, R.B.: The method o
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