CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd - PowerPoint PPT Presentation

Outline Introduction and Problem Solution ODE/CR-manifolds with additional symmetries CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England August 2, 2006 CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem Solution ODE/CR-manifolds with additional symmetries Introduction and Problem Three geometric structures Linearisation Problem from CR geometry Solution A priori information on automorphism Shear invariant ODE ODE/CR-manifolds with additional symmetries CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem Three geometric structures Solution Linearisation Problem from CR geometry ODE/CR-manifolds with additional symmetries Second order ODE, CR-manifolds of 3-dim manifolds ( C 3 ) complex 2 hol. dir. fields Z 1 , Z 2 dim 6, CR-dim=2 ( ⇔ 2 foliations by hol. holomorphic D = span( Z 1 , Z 2 ) curves) J | Z 1 = i, J | Z 2 = − i non-involutivity y ′′ = B ( x , y , y ′ ) special Levi form [ Z 1 , Z 2 ] �∈ span( Z 1 , Z 2 ) curvature condition ( x , y , p = dy dx ) ∂ Z 1 = ∂ p ∂ x + p ∂ ∂ ∂ y + B ∂ Z 2 = ∂ p 2 nd foliation encodes all structure Embedding into C 4 : x = 1, ˙ ˙ y = p , y = φ ( x , c , d ) p = B ( x , y , p ) ˙ w 2 = φ (¯ ¯ z 2 , z 1 , w 1 ) CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem Three geometric structures Solution Linearisation Problem from CR geometry ODE/CR-manifolds with additional symmetries Mappings (prolonged) point mappings that pre- CR-mappings transformations serve Z 1 and Z 2 up to scale CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem Three geometric structures Solution Linearisation Problem from CR geometry ODE/CR-manifolds with additional symmetries Most symmetric objects y = cx + d F (2 , 1) w 1 − ¯ w 2 = z 1 ¯ z 2 y ′′ = 0 2 i ւ ց ( CP (2)) ∗ (polarisation of CP (2) Im w = | z 2 | ) PSL(3 , C ) acts as polarisa- acts by projective induced action tion of SU(2 , 1) transformations CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem Three geometric structures Solution Linearisation Problem from CR geometry ODE/CR-manifolds with additional symmetries Problem from CR geometry Sphere Im w = | z | 2 can be characterised by the property that there exist non-trivial automorphisms Φ with Φ(0) = 0 and d Φ(0) = id, namely z + aw z �→ az − ( r + i | a | 2 ) w 1 − 2 i ¯ w w �→ az − ( r + i | a | 2 ) w 1 − 2 i ¯ Is the analogous statement true for (elliptic) CR manifolds of codimension 2? What symmetries can appear? Describe manifolds with symmetries. CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem Three geometric structures Solution Linearisation Problem from CR geometry ODE/CR-manifolds with additional symmetries Easy: ◮ for given B ⇒ (infinitesimal) automorphisms ◮ for given (infinitesimal) automorphism ⇒ B infinitesimal automorphisms: ξ ∂ ∂ x + η ∂ ∂ y + φ ∂ ∂ p � � with φ = ∂η ∂η ∂ y − ∂ξ − p 2 ∂ξ ∂ x + p ∂ y . ∂ x Solve B − ∂ 2 η ξ ∂ B ∂ x + η∂ B ∂ y + φ∂ B � � 2 ∂ξ ∂ x + 3 p ∂ξ ∂ y − ∂η ∂ p + ( ∂ x ) 2 ∂ y � ∂ 2 ξ 2 ∂ 2 ξ + p 3 ∂ 2 ξ � � � ( ∂ x ) 2 − 2 ∂η ∂η + p 2 + p ∂ x ∂ y − ( ∂ y ) 2 = 0 ( ∂ y ) 2 ∂ x ∂ y Difficulty: Don’t know neither B nor ξ, η . CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem A priori information on automorphism Solution Shear invariant ODE ODE/CR-manifolds with additional symmetries Power series methods, normal form (adapted to defining equation): ◮ such CR-manifold is torsion-free ◮ φ is (algebraic) deformation of z 1 w 1 z 1 �→ w 1 �→ , 1 − 2 i ¯ az 1 1 − 2 i ¯ az 1 z 2 �→ z 2 + aw 2 , w 2 �→ w 2 which corresponds to p x �→ x + ty , y �→ y , p �→ 1 + tp . Cartan geometry, normal form (adapted to symmetry): ◮ in normal coordinates φ is exactly as above ◮ hence, has the same topology (curve of fixed points y = p = 0) CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem A priori information on automorphism Solution Shear invariant ODE ODE/CR-manifolds with additional symmetries Consequence y ∂ B ∂ x − p 2 ∂ B ∂ p + 3 pB = 0 Solution B ( x , y , p ) = F ( y , x − x p ) p 3 Due to regularity, 3 � f j ( y )( y − px ) 3 − j p j . B ( x , y , p ) = j =0 Question: Can they be equivalent to B = 0?? CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem A priori information on automorphism Solution Shear invariant ODE ODE/CR-manifolds with additional symmetries Theorem (Ezhov, S., 2005) There are local coordinates x , y , p such that B takes the reduced form B = f 0 ( y )( y − px ) 3 + f 1 ( y ) p ( y − px ) 2 . Two reduced forms are equivalent if and only if they are equivalent under c 1 x c 2 y x �→ y �→ 1 − cy , 1 − cy , i.e. the remaining freedom in coordinate choice consists of three complex parameters c 1 , c 2 , c. CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem A priori information on automorphism Solution Shear invariant ODE ODE/CR-manifolds with additional symmetries Idea of proof. Case 1: y ∂ ∂ x is the only shear-symmetry. Then preserving reduced form requires preserving y ∂ ∂ x . Such mappings satisfy a pair of second order ODE ( ⇒ 4 parameters). But one parameter corresponds to the one-parametric shear symmetry group and c 1 x c 2 y x �→ 1 − cy , y �→ 1 − cy is known to preserve the reduced form. Case 2: There is a second shear symmetry. ⇒ Study ODE with more symmetries. CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem Solution ODE/CR-manifolds with additional symmetries Which of those ODE/CR-manifolds have additional automorphisms? S. Lie classified ODE by (infinitesimal) symmetries 8 symmetries ⇒ y ′′ = 0 3 symmetries ⇒ short list of ODE 2 symmetries ⇒ y ′′ = f ( y ′ ) (for ∂ x , ∂ ∂ ∂ y ) or y ′′ = f ( y ′ ) ∂ y , x ∂ ∂ ∂ x + y ∂ (for ∂ y ) x 1 symmetry ⇒ y ′′ = f ( x , y ′ ). CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem Solution ODE/CR-manifolds with additional symmetries Theorem (Ezhov, S.) y ′′ = 0 and y ′′ = ( y − xy ′ ) 3 are (up to equivalence) the only ODE with more than one shear symmetry. y ′′ = ( y − xy ′ ) 3 is SL(2 , C ) invariant. The ODE with exactly two symmetries with fixed point 0 are equivalent to y ′′ = y k ( y − xy ′ ) 3 y ′′ = y ℓ y ′ ( y − xy ′ ) 2 + Cy 2 ℓ +2 ( y − xy ′ ) 3 . or The additional automorphisms are ( k + 2) x ∂ ∂ x − 2 y ∂ ( ℓ + 2) x ∂ ∂ x − y ∂ resp. ∂ y . ∂ y CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem Solution ODE/CR-manifolds with additional symmetries The corresponding CR manifolds z 2 ) 2 = 0 w 1 + w 2 z 2 1 ¯ 2 − (¯ w 2 − z 1 ¯ √ � dy w 2 = z 1 ¯ ¯ z 2 + k + 2 w 1 ¯ z 2 � z k +2 y 2 1 + w 2 1 ¯ 2 dy � is a hypergeometric function, which satisfies q z k +2 y 2 1+ w 2 1 ¯ 2 non-linear ODE ⇒ (apparently new) relation CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England

Outline Introduction and Problem Solution ODE/CR-manifolds with additional symmetries Shear invariant ODE with a second transitive symmetry: The examples from above shifted in x -direction (by power series methods, comparison of parameters) But corresponding coordinate transformations is highly transcendental. Example: f 0 ( y ) ≡ 0 and f 1 ( y ) satisfies � ′′ � f 1 = − 2 f 1 . 3 f 1 + yf ′ 1 CR-manifolds, Pseudo product structures and 2 nd order ODE Gerd Schmalz University of New England