Conformal geodesics and geometry of the 3rd order ODEs systems Alexandr Medvedev Masaryk University Trest, 2012 Supported by the project CZ.1.07/2.3.00/20.0003 of the Operational Programme Education for Competitiveness of the Ministry of Education, Youth and Sports of the Czech Republic. Alexandr Medvedev Geometry of the 3rd order ODEs systems The problem Characterize the geometry of the 3rd order ODEs system 000 00 0 yi (x) = fi (yj (x); yk (x); yl (x); x); 1 ≤ i; j; k; l ≤ m: (1) under point transformations. The number of equations will be at most 2. Historical remark I Single ODE of 3rd order was studied by S.Chern I System of 2nd order ODE was studied by M.Fels, D.Grossman a aa Order Numberaa 2 3 aa 1 E.Cartan S.Chern ≥ 2 M.Fels, D.Grossman ??? Alexandr Medvedev Geometry of the 3rd order ODEs systems ODE as a Surface in Jet Space Consider J3(M) be a 3 order jet space of the curves on M. Let's use the following coordinates on the jet space J3(M): 0 0 00 00 000 000 x; y1; :::; ym; p1 = y1; :::; pm = ym; q1 = y1 ; qm = ym; y1 ; :::; ym : We can represent equation as surface E in the J3(M). In coordinates we have the following equations that define E : 000 E = fyi = fi (qj ; pk ; yl ; x)g: (2) The lifts of the solutions of E are integral curves of 1-dimensional distribution E = T E\ C(J3(M)) @ @ @ i @ E = h + pi + qi + f i; @x @yi @pi @qi Alexandr Medvedev Geometry of the 3rd order ODEs systems Filtered Manifolds I We call a manifold M a filtered manifold if it is supplied with a filtration C −i of the tangent bundle: TM = C −k ⊃ C −k+1 ⊃ · · · ⊃ C −1 ⊃ C 0 = 0 (3) where [C −i ; C −1] ⊂ C −i−1: I With every filtered manifold we associate a filtration gr TM of the following form: m M gr TM = gr−i TM; (4) i=1 −i −i+1 where gr−i TM = C =C : I Space gr TM has a nilpotent Lie algebra structure. I Let m be a nilpotent Lie algebra. Filtered manifold has type m, if for every point p Lie algebra gr TpM is eqaual to m. Alexandr Medvedev Geometry of the 3rd order ODEs systems Filtered Manifolds Distribution C −1 for the manifold E has two components: I 1-dimensial distribution @ @ @ i @ E = + pi + qi + f ; @x @yi @pi @qi I m-dimensional vertical distribution @ V = : @qi A filtered manifold associated with system (1) is C −1 ⊂ C −2 ⊂ C −3 = T E: @ C −2 = C −1 ⊕ @pi Alexandr Medvedev Geometry of the 3rd order ODEs systems Universal Prolongation For every nilpotent Lie algebra m there exists unique universal prolongation g(m) = ⊕i2Zgi ; that: 1. gi = mi ; for all i < 0 2. From [X ; g−1] = 0; X 2 gi ; i ≥ 0 it is follows that X = 0: Universal prolongation g(m) for the equation (1) is called a symbol of the differential equation (1). Fact (Tanaka,Marimoto) For every filtered manifold we can build a normal Cartan connection of type (g; h), where h = ⊕i≥0gi . Fact (Tanaka,Marimoto) All invariants of the filtered manifold arise from the curvature tensor of the normal Cartan connection. Fundamental system of invariants described by positive part of cohomology space 2 H (g−; g). Alexandr Medvedev Geometry of the 3rd order ODEs systems Symbol Lie algebra Universal prolongation g of Lie algebra m: m g = (sl2(R) × glm(R)) i (V2 ⊗ R ): We fix the following basis in the Lie algebra sl2 0 1 1 0 0 0 x = ; h = ; y = : 0 0 0 −1 1 0 We fix basis e0; e1; e2 of module V2 with relations xei = ei−1: g1 = hyi; g0 = hglmi; m m m g−1 = hxi + he2 ⊗ R i; g−2 = he1 ⊗ R i; g−3 = he0 ⊗ R i: "Parabolic" subalgebra: h = g1 + g0 Alexandr Medvedev Geometry of the 3rd order ODEs systems Cartan Connection i i i We call co-frame !x ;!−1;!−2;!−3 associated with equation (1) if −2 i t C =< !−3 > −1 i i t C =< !−3;!−2 > i i i t E =< !−3;!−2;!−1 > i i i t V =< !−3;!−2;!x > Let π : P !E be principle H-bundle. We say that Cartan connection ! i i i x h i j h ! = !−3v0 ⊗ei +!−2v1 ⊗ei +!−1v2 ⊗ei +! x +! h +!j ei +! h: on a principal H-bundle is associated to equation (1), if for any ∗ ∗ i ∗ i ∗ i local section s of π the set fs !x ; s !−1; s !−2; s !−3g is an adapted co-frame on E. Alexandr Medvedev Geometry of the 3rd order ODEs systems Normal Form Let the curvature have a form i i i x h i j y Ω = Ω−3v0 ⊗ei +Ω−2v1 ⊗ei +Ω−2v2 ⊗ei +Ω x +Ω h+Ωj ei +Ω y j Let's Ωk [!1;!2] be the coefficients of the structure function of the curvature Ω. Theorem There exist the unique normal Cartan connection associated with equation (1) with the following conditions on curvature: I all components of degree 0 and 1 is equal to 0; i i i I in degree 2 we have Ωh[!x ^ !−1] = 0, Ωj [!x ^ !−1] = 0, i i i Ωx [!x ^ !−2] = 0, Ω−1[!x ^ !−2] = 0; i i I in degree 3 we have Ωy [!x ^ !−1] = 0, Ωh[!x ^ !−2] = 0, i i Ωj [!x ^ !−2] = 0 i I in degree 4 it is Ωy [!x ^ !−2] = 0 Alexandr Medvedev Geometry of the 3rd order ODEs systems Serre spectral sequence We use Serre spectral sequence to compute cohomology space. Theorem 2 The space of cohomology classes H (g−; g) is direct sum of two 1;1 0;2 1;1 0;2 spaces E2 and E2 , where E2 and E2 has the following form 1;1 1 1 E2 = H (Rx; H (V ; g)); (5) 0;2 0 2 E2 = H (Rx; H (V ; g)): (6) 1;1 0;2 Description of E2 and E2 ` Let Vm be irreducible sl2-module and v0 and vm such, that x:v0 = 0 and y:vm = 0: Then 0 H (Rx; Vm) = Rv0 1 ∗ H (Rx; Vm) = Rx ⊗ vm: Alexandr Medvedev Geometry of the 3rd order ODEs systems 1;1 Structure of the space E2 Theorem 1;1 The space E2 has the following form ∗ 2 x ⊗ (Ry ⊗ gl(W ) + Ry ⊗ sl(W )): (7) 1;1 Elements ' 2 E2 which have form ': Rx ! Ry ⊗ sl(W ) have 2 degree 2 and elements which have form ': Rx ! Ry ⊗ gl(W ) have degree 3 1;1 I The space E2 describes so called Wilczynski invariants. 1;1 I The space E2 corresponds to torsion. Alexandr Medvedev Geometry of the 3rd order ODEs systems 0;2 Structure of the space E2 Theorem 0;2 The space E2 has the following parts in direct sum decomposition: Space Degree 2 ∗ V6 ⊗ ^ (W ) ⊗ W -1 2 ∗ V4 ⊗ S0 (W ) ⊗ W 0 2 ∗ V4 ⊗ ^ (W ) ⊗ W 0 2 ∗ V2 ⊗ ^0W ⊗ W 1 2 ∗ V0 ⊗ S0 (W ) ⊗ W 2 2 ∗ V0 ⊗ S (W ) 4 V2, m = 2 3 2 ∗ where Vm is dimension m + 1 sl2-module and S0 (W ) ⊗ W and 2 ∗ ^0(W ) ⊗ W is traceless part of corresponding spaces Alexandr Medvedev Geometry of the 3rd order ODEs systems Explicit formula We have the following correspondence between cohomological 2 2 classes H (^ g−; g) and fundamental invariants: Degree Space Invariant 0 2 ∗ ∗ 1 v2 ⊗ ^ W ⊗ W =V2 ⊗ W ≡ 0 ∗ 2 x ⊗ Ry ⊗ sl(W ) W2 0 2 ∗ 2 v0 ⊗ S (W ) ⊗ W I2 ∗ 2 3 x ⊗ Ry ⊗ gl(W ) W3 0 2 ∗ 4 v0 ⊗ S (W ) I4 0 3 v2 if m = 2 ≡ 0 Alexandr Medvedev Geometry of the 3rd order ODEs systems Explicit formula Theorem There are 4 fundamental invariants: @f i d @f i 1 @f i @f k W = tr − + 2 0 @pj dx @qj 3 @qk @qj @2f i I = tr 2 0 @qj @qk i 2 i i i k i k @f 1 d @f 1 @f 3 2 @f d @f 1 d @f @f W3 = − − ( ) + + @y j 3 dx2 @qj 27 @qk 9 @qk dx @qj 9 dx @qk @qj k @ d @ @f @Hk I4 = Hj + k (Hl l ) + @qk dx @q @q @pj @2f i where Hj = tr @qj @qk Alexandr Medvedev Geometry of the 3rd order ODEs systems Conformal geodesics An arbitrary conformal geometry is defined by equivalence class of Riemannian metrics. Definition Conformal geodesic on a conformal manifold M is a curve on M; which development in the flat space is a circle. Example ... Pm y_ y¨ y j=1 j j i = 3¨yi Pm 2 ; i = 1;:::; m: 1 + j=1 y_j Theorem (Yano) In order that an infinitesimal transformation of the manifold M carry every conformal geodesic into conformal geodesic, it is necessary and sufficient that the transformation be a conformal motion. Alexandr Medvedev Geometry of the 3rd order ODEs systems Correspondence Space Construction Penrose transform Consider a semisimple Lie group G with two parabolic subgroups P1 and P2. Assume, that P1 \ P2 is also parabolic. Then a natural double fibration from G=P1 \ P2 to G=P1 and G=P2 defines a correspondence between G=P1 and G=P2. Correspondence space let G be a semisimple Lie group with two parabolic subgroups Q ⊂ P ⊂ G. Consider a parabolic Cartan geometry (G! N;!) of type (G; P), where G is principal P-bundle.