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

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184[28] TRESSE, A.: Détermination des invariants ponctuels de l’équation différentielle dusecond ordre y ′′ = ω(x, y, y ′ ), Hirzel, Leipzig, 1896.
185Nonrigid sphericalreal analytic hypersurfaces in C 2Joël MerkerAbstract. A Levi nondegenerate real analytic hypersurface M of C 2 representedin local coordinates (z,w) ∈ C 2 by a complex defining equation of the formw = Θ(z,z,w) which satisfies an appropriate reality condition, is spherical if andonly if its complex graphing function Θ satisfies an explicitly written sixth-orderpolynomial complex partial differential equation. In the rigid case (known before),this system simplifies considerably, but in the general nonrigid case, its combinatorialcomplexity shows well why the two fundamental curvature tensors constructedby Élie Cartan in 1932 in his classification of hypersurfaces have, since then, neverbeen reached in parametric representation.arxiv.org/abs/0910.1694/Table of contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2. Segre varieties and partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 220.3. Geometry of associated submanifolds of solutions. . . . . . . . . . . . . . . . . . . . . . . . .224.4. Effective differential characterization of sphericality in C 2 . . . . . . . . . . . . . . . . 226.5. Some complete expansions: examples of expression swellings. . . . . . . . . . . . . .210.§1. INTRODUCTIONA real analytic hypersurface M in C 2 is called spherical at one of itspoints p if there exists a nonempty open neighborhood U p of p in C 2such that M ∩ U p is biholomorphic to a piece of the unit sphere S 3 ={(z, w): |z| 2 + |w| 2 = 1 } . When M is connected, sphericality at onepoint is known to propagate all over M, for it is equivalent to the vanishingof two certain real analytic curvature tensors that were constructed byÉlie Cartan in [3]. However, the intrinsic computational complexity, in theCauchy-Riemann (CR for short) context, of Élie Cartan’s algorithm to derivean absolute parallelism on some suitable eight-dimensional principalbundle P → M prevents from controlling explicitly all the appearing differentialforms. As a matter of fact, the effective computation, in terms of adefining equation for M, of the two fundamental differential invariants thevanishing of which characterizes sphericality, appears nowhere in the literature(see e.g. [23, 5, 11] and the references therein as well), except notablywhen one makes the assumption that, in some suitable local holomorphic
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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
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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
Page 135 and 136: functions w(z)). The coefficients R
Page 137 and 138: Then the Lie criterion states that
Page 139 and 140: y k ′ = λk k y k and w ′ = µ
Page 141 and 142: The last statements of Corollaries
Page 143 and 144: [Sha2000] SHAFIKOV, R.: Analytic co
Page 145 and 146: Here x = (x 1 , . . ., x n ) ∈ K
Page 147 and 148: Consequently, for the case κ = 2,
Page 149 and 150: are devoted to provide a general on
Page 151 and 152: for l = 1, . . .,m such that the lo
Page 153 and 154: Lemma 8.1. The following conditions
Page 155 and 156: 155namely X ←→ X + X ←→ X .
Page 157 and 158: In terms of Sussmann’s approach [
Page 159 and 160: x κ−1 χ κ−1 + O(|x| κ ) + O
Page 161 and 162: Observe that these expressions are
Page 163 and 164: where the first term I involves onl
Page 165 and 166: I 5 = ∑ i 1I 6 = ∑ i 1 ,i 2I 7
Page 167 and 168: U 2 U κ−1 , we obtain the five f
Page 169 and 170: and U 2 U κ−1 , we obtain the fi
Page 171 and 172: following equations for j = 1, . .
Page 173 and 174: linear equations:(7.28) ⎧0 = Rx j
Page 175 and 176: 175R j containing at least one part
Page 177 and 178: 177[3] :∑σ∈S κ−1κδ l,....
Page 179 and 180: the values of the partial derivativ
Page 181 and 182: also appears some derivatives Q l
Page 183: [5] F. ENGEL; LIE, S.: Theorie der
Page 187 and 188: origin:{ ( ∣ ∣ )AJ 4 1∣∣∣
Page 189 and 190: I 2 characterizes equivalence to w
Page 191 and 192: y means of a fundamental, elementar
Page 193 and 194: 2) When M is Levi nondegenerate at
Page 195 and 196: 195holding in C { z ′ , z ′ , w
Page 197 and 198: Fortunately for our present purpose
Page 199 and 200: We notice passim that S ≡ Q x (no
Page 201 and 202: for a certain local K-analytic new
Page 203 and 204: Conversely, if y xx (x) = F ( x, y(
Page 205 and 206: 205But we may also express the dual
Page 207 and 208: Then thanks to a straightforward ap
Page 209 and 210: and we then expand carefully the re
Page 211 and 212: explicit computation, so let us rew
Page 213 and 214: 213+T ah3Q 2 a Q b∆(aa|b)∆(b|bb
Page 215 and 216: 215Vanishing Hachtroudi curvaturean
Page 217 and 218: to fix the ideas, it will be assume
Page 219 and 220: then by replacing the(so obtained)v
Page 221 and 222: and also in addition the fact that
Page 223 and 224: and this defines without ambiguity
Page 225 and 226: the leaves of this local foliation
Page 227 and 228: Then by differentiating with respec
Page 229 and 230: ∂Here, the coefficients of the2 T
Page 231 and 232: 231for (7.28):( )∂□ ·∂a □
Page 233 and 234: 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
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must be equal to the number of indi
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Y i1 ,i 2 ,i 3 ,i 4written in one o
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335conclusion, we have shown that (
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337Theorem 3.73. For every κ 1 an
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Theorem 3.79. For every integer κ
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+++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3
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343form:(4.12)κ+1Yκ j = Y jx κ +
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eing related to the number 2 in the
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Appying the chain rule, we may comp
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Secondly:(5.3)Y j i 1 ,i 2= Y jx i
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As explained before the statement o
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g j i 1 ,...,i λand similarly for
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with respect to the variables x i 1
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Assuming F = F(x, y x ) to be indep
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359(III’) ⎧0 = ∑ (δ k 2)j 3H
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Open problem 1.17. For n = 2 establ
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⎧∣ ⎨X 1 xX 1+ y x 1 y x 1 ·1
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eplacing the third column by the se
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⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣
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e written under the specific form:(
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Indeed, the collection of equations
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3.12. Principal unknowns. As there
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375Sixthly:(3.22) ⎧δ k 1j 1Θ n+
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are a consequence of (I’), (I”)
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379IV: BibliographyREFERENCES[Ar198
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[G1989] GARDNER, R.B.: The method o
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[Me2005a] MERKER, J.: On the local
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