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

More documents

Recommendations

Info

378conditions (3.29) 1 :(3.32) ⎧0 =? = −2 G j1 ,j 2 ,x j 3y + 2 G j1 ,j 3 ,x j 2y−− ∑ lG j3 ,l,x j 2 L l j 1+ ∑ lG j2 ,l,x j 3 L l j 1− G j1 ,j 2 ,y H j 3j 3 ,j 3+ G j1 ,j 3 ,y H j 2j 2 ,j 2−− 2 ∑ lG l,j3 H l j 1 ,j 2+ 2 ∑ lG l,j2 H l j 1 ,j 3− ∑ lH l j 1 ,j 2 ,x j 3H l l,l + ∑ lH l j 1 ,j 3 ,x j 2H l l,l −− 2 3 Hj 2j 2 ,j 2 ,y G j 1 ,j 3+ 2 3 Hj 3j 3 ,j 3 ,y G j 1 ,j 2− 2 3 Lj 3j 3 ,x j 3 G j 1 ,j 2+ 2 3 Lj 2j 2 ,x j 2 G j 1 ,j 3−− ∑ lL l j 1 ,x j 2G j3 ,l + ∑ lL l j 1 ,x j 3G j2 ,l−⎪⎨− 2 3 G j 1 ,j 2G j3 ,j 3M j 3+ 2 3 G j 1 ,j 3G j2 ,j 2M j 2− 4 3+ 4 ∑G j1 ,j33G j2 ,l M l − 1 ∑2ll− 1 ∑G j3 ,l H j 2j22 ,j 2L l j 1+ 1 ∑2ll+ 1 ∑G j1 ,j22Hl,l l Ll j 3− 1 ∑3llG j3 ,l H j 1j 1 ,j 1L l j 2+ 1 2G j2 ,l H j 3j 3 ,j 3L l j 1− 1 2G j1 ,j 2H j 3j 3 ,l Ll j 3+ 1 3∑G j1 ,j 2G j3 ,l M l +l∑lG j2 ,l H j 1j 1 ,j 1L l j 3−∑G j1 ,j 3Hl,l l Ll j 2+l∑lG j1 ,j 3H j 2j 2 ,l Ll j 2−⎪⎩− 1 3 G j 1 ,j 3Hj l 2 ,j 2L j 2l+ 1 3 G j 1 ,j 2Hj l 3 ,j 3L j 3l−− ∑ ∑G j2 ,p Hj l 1 ,j 3L p l + ∑ ∑G j3 ,p Hj l 1 ,j 2L p l −l pl p− ∑ ∑Hj l 1 ,j 2H p l,j 3Hp,p p + ∑ ∑Hj l 1 ,j 3H p l,j 2Hp,p.pl pl pLemma 3.33. ([Me2003, Me2004]) This first family of compatibility conditionsfor the second auxiliary system obtained by developing (3.29) 1 inlength, together with the three remaining families obtained by developing(3.29) 2 , (3.29) 3 , (3.29) 4 in length, are consequences, by linear combinationsand by differentiations, of (I’), (II’), (III’), (IV’), of Theorem 1.7.The summarized proof of Theorem 1.7 is complete.
379IV: BibliographyREFERENCES[Ar1988] ARNOL’D, V.I.: Dynamical systems. I. Ordinary differential equations andsmooth dynamical systems, Translated from the Russian. Edited by D. V.Anosov and V. I. Arnol’d. Encyclopaedia of Mathematical Sciences, vol. 1.Springer-Verlag, Berlin, 1988. x+233 pp.[Ar1968] ARTIN, M.: On the solutions of analytic equations, Invent. Math. 5 (1968),277–291.[BER1999] BAOUENDI, M.S.; EBENFELT, P.; ROTHSCHILD, L.P.: Real submanifolds incomplex space and their mappings. Princeton Mathematical Series, vol. 47,Princeton University Press, Princeton, NJ, 1999, xii+404 pp.[BJT1985] BAOUENDI, M.S.; JACOBOWITZ, H.; TREVES, F.: On the analyticity of CRmappings, Ann. of Math. 122 (1985), no. 2, 365–400.[Bel1996] BELLAÏCHE, A.: SubRiemannian Geometry, Progress in Mathematics,vol. 144, Birkhäuser Verlag, Basel/Switzerland, 1996, 1–78.[Be1979] BELOSHAPKA, V.K.: On the dimension of the group of automorphisms of ananalytic hypersurface, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 243–266;English transl. in Math. USSR-Izv. 14 (1980), 223–245.[Be1988] BELOSHAPKA, V.K.: Finite-dimensionality of the group of automorphisms ofa real-analytic surface, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 437–442;English transl. in Math. USSR-Izv. 32 (1989), 443–448.[Be1997] BELOSHAPKA, V.K.: CR-varieties of the type (1,2) as varieties of ‘super-high’[Be2002]codimension, Russian J. Math. Phys. 5 (1997), 399–404.BELOSHAPKA, V.K.: Real submanifolds in complex space: polynomial models,automorphisms, and classification problems, Uspekhi Mat. Nauk 57(2002), no. 1, 3–44; English transl. in Russian Math. Surveys 57 (2002), no. 1,1–41.[BES2005] BELOSHAPKA, V.K.; EZHOV, V.; SCHMALZ, G.: Canonical Cartanconnection and holomorphic invariants of Engel CR manifolds,arxiv.org/abs/math.CV/0508084.[BK1989]BLUMAN, G.W.; KUMEI, S.: Symmetries and differential equations, SpringerVerlag, Berlin, 1989.[Bo1991] BOGGESS, A.: CR manifolds and the tangential Cauchy-Riemann complex.Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1991,xviii+364 pp.[Bo1972] BOURBAKI, N.: Groupes at algèbres de Lie, chapitre 2, Hermann, Paris, 1972.[BSW1978] BURNS, D.Jr.; SHNIDER, S.; WELLS, R.O.Jr.: Deformations of strictly pseudoconvexdomains Invent. Math. 46 (1978), no. 3, 237–253.[Ca1922][Ca1924]CARTAN, É.: Sur les équations de la gravitation d’Einstein, J. Math. pures etappl. 1 (1922), 141–203.CARTAN, É.: Sur les variétés à connexion projective, Bull. Soc. Math. France52 (1924), 205–241.[Ca1932a] CARTAN, É.: Sur la géométrie pseudo-conforme des hypersurfaces del’espace de deux variables complexes, I, Ann. Math. Pura Appl. 11 (1932),17–90.
Page 1 and 2:
Joël M E R K E RÉcole Normale Sup
Page 3 and 4:
§1. INTRODUCTIONSeveral physically
Page 5 and 6:
e a collection of m analytic second
Page 7 and 8:
7where the indices j, l 1 vary in {
Page 9 and 10:
This phenomenon could be explained
Page 11 and 12:
This yields the prolongation of the
Page 13 and 14:
X(x, y) and Y (x, y) such that it m
Page 15 and 16:
2.23. Compatibility conditions for
Page 17 and 18:
This lemma is left to the reader; a
Page 19 and 20:
simplification nor any reordering:(
Page 21 and 22:
aleza, y por otra, las organizacion
Page 23 and 24:
computational level (differential-g
Page 25 and 26:
For instance, in the case m = 2, by
Page 27 and 28:
in (3.11). The second equation that
Page 29 and 30:
Similarly, the second equation take
Page 31 and 32:
order derivatives of X and of the Y
Page 33 and 34:
Lemma 3.32. The system yxx j = F j
Page 35 and 36:
Lemma 3.45. The following quadratic
Page 37 and 38:
often be denoted by the sign “·
Page 39 and 40:
1Now, taking account of the factor
Page 41 and 42:
conditions, totally equivalent to t
Page 43 and 44:
43remaining terms afterwards:(3.71)
Page 45 and 46:
Here, the sign ≡ precisely means:
Page 47 and 48:
Next, replacing plainly (3.64) in (
Page 49 and 50:
49the order of §3.73. We get:(3.89
Page 51 and 52:
51Multiplying by −2 and reorganiz
Page 53 and 54:
⎧Θ l 1y l 2 = −Ll 1l 1 ,l 1 ,y
Page 55 and 56:
+ 1 ∑4 δj l 1Hl k 2L k k,kk− 2
Page 57 and 58:
+ 1 ∑2 δj l 1Hl k 3M l2 ,kk+ 1 4
Page 59 and 60:
In the hardest techical part of thi
Page 61 and 62:
− 1 3+ 1 3− 1 4+ 1 4∑Hl k 1H
Page 63 and 64:
− 1 3∑kL l 1l1 ,k,x Hk l 1+ 1 3
Page 65 and 66:
Here, the sign ≡ means “modulo
Page 67 and 68:
Thirdly, put j := l 2 in (3.108) wi
Page 69 and 70:
69correct. We get:(4.29)0 =?== −
Page 71 and 72:
Next, apply the operator ∑ k Ll 2
Page 73 and 74:
73Writing term by term the substrac
Page 75 and 76:
of the terms of the subgoal (4.29):
Page 77 and 78:
77+ 1 2∑kH k l 1 ,y l 2 Hl 2k15
Page 79 and 80:
79= − X xx Yx j + Y jm∑+++++l 1
Page 81 and 82:
Our first task is to compute the de
Page 83 and 84:
Replacing this expression of A k in
Page 85 and 86:
85have the continuation(5.17) −y
Page 87 and 88:
87and where thirdly (we are nearly
Page 89 and 90:
89obtain:(5.23)III :=m∑m∑l 1 =1
Page 91 and 92:
[GTW1989] GRISSOM, C.; THOMPSON, G.
Page 93 and 94:
93Nonalgebraizable real analytic tu
Page 95 and 96:
and of CR dimension m = n − d ≥
Page 97 and 98:
coordinates z ′ = 2i ln(z/z p ),
Page 99 and 100:
{x ∈ K n : |x| < ρ} for some ρ
Page 101 and 102:
(2) A K-algebraic inversion mapping
Page 103 and 104:
(1) The complex Lie algebra Hol(M,
Page 105 and 106:
Proposition 3.1. Let t ↦→ G i (
Page 107 and 108:
The first relation gives nothing, s
Page 109 and 110:
with |β∗ k | ≥ 1 and integers
Page 111 and 112:
Clearly, the left hand side is an a
Page 113 and 114:
field X 1 ′ := h ∗ (X 1 ). Taki
Page 115 and 116:
which yields after differentiating
Page 117 and 118:
117Im w∆ n(ρ 4 )∆ n(ρ 1 )∆
Page 119 and 120:
[∂ i,ei H ei (t ′ )] t ′ =He
Page 121 and 122:
p = (w p , z p , ζ p , ξ p ) ∈
Page 123 and 124:
Finite nondegeneracy is interesting
Page 125 and 126:
such that |ω j i ∗ ,β∗ i(t,
Page 127 and 128:
for k even, we have Γ k (z (k) ) =
Page 129 and 130:
that for every local holomorphic se
Page 131 and 132:
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 and 184:
[5] F. ENGEL; LIE, S.: Theorie der
Page 185 and 186:
185Nonrigid sphericalreal analytic
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
Page 235 and 236:
Differentiating the first equation
Page 237 and 238:
Example 1.21. (Continued) With n =
Page 239 and 240:
defined by the graphed equations:((
Page 241 and 242:
for some two local analytic maps Π
Page 243 and 244:
Proof. Let l = l ( x i , y j , yβ(
Page 245 and 246:
complementary views on the same obj
Page 247 and 248:
Classification problem 3.11. Classi
Page 249 and 250:
Definition 4.10. L is an infinitesi
Page 251 and 252:
Convention 5.2. The letters R will
Page 253 and 254:
Assuming that dimSYM(E 1 ) = 8, tak
Page 255 and 256:
Restarting from §4.1, let ϕ a Lie
Page 257 and 258:
Here, R j l 1 ,kare universal polyn
Page 259 and 260:
Lemma 6.34. Let M be a submanifold
Page 261 and 262:
where the expressions D β,β1 are
Page 263 and 264:
Then the sum L + L is tangent to M
Page 265 and 266:
Lemma 7.11. ([CM1974, BER1999, Me20
Page 267 and 268:
where(7.33) ∆ := ∑1kn∂ 2∂x
Page 269 and 270:
with r, s being unspecified functio
Page 271 and 272:
and moreover, all other, higher ord
Page 273 and 274:
To conclude, we replace X xx so obt
Page 275 and 276:
Definition 8.43. A real analytic hy
Page 277 and 278:
Lemma 8.59. The function b depends
Page 279 and 280:
where the letter r denotes an unspe
Page 281 and 282:
(8.80) ⎧Y 1,1,1 = Y x 1 x 1 x 1 +
Page 283 and 284:
(X 1 , X 2 , Y , Xy 1,X 2x, Y 2 x 1
Page 285 and 286:
espect to x 1 and (8.96) 2 with res
Page 287 and 288:
For the two unknowns Xyy 1 and X 2x
Page 289 and 290:
Redeveloping the determinant, the v
Page 291 and 292:
291are obtained by equating to zero
Page 293 and 294:
Let (z 0 , c 0 ) = (x 0 , y 0 , a 0
Page 295 and 296:
we deduce that Γ 5 is submersive (
Page 297 and 298:
11.4. Regularity and jet parametriz
Page 299 and 300:
Cramer’s rule, we get(11.13) ⎧
Page 301 and 302:
in K[a, z] n+m and in K[x, c] p+m .
Page 303 and 304:
of the cancellation properties (11.
Page 305 and 306:
305II: Explicit prolongations of in
Page 307 and 308:
Define then the transformed jet ϕ
Page 309 and 310:
Let λ ∈ N be an arbitrary intege
Page 311 and 312:
311[Ol1986], [BK1989]):(Y(1.31)⎧
Page 313 and 314:
+ [Y y 2 − 2 X xy ] y 1 y 2 + [
Page 315 and 316:
for some nonnegative integers A, B,
Page 317 and 318:
Once the correct theorem is formula
Page 319 and 320:
By the classical formula for the de
Page 321 and 322:
The induction formulas are⎧(3.4)
Page 323 and 324:
We have to compute:( ) ⎛ ⎞∑
Page 325 and 326:
Next, we gather the underlined term
Page 327 and 328: + ∑k 1 ,k 2 ,k 3 ,k 4[−δ k 1,k
Page 329 and 330: 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: are a consequence of (I’), (I”)
Page 381 and 382: [G1989] GARDNER, R.B.: The method o
Page 383 and 384: [Me2005a] MERKER, J.: On the local
show all

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

Create successful ePaper yourself

Delete template?

Save as template?