Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 1/14 Conformally Fedosov Manifolds Michael Eastwood [ joint work with Jan Slovák ] Australian National University
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 2/14 Projective differential geometry Def n ˆ same geodesics (unparameterised) EG (Thales 600 BC) The round sphere is projectively flat Affine coördinate patch R n RP n is a projective equivalence Operational Def ṋ a φ b = a φ b Υ a φ b Υ b φ a
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 3/14 Conformally symplectic geometry symplectic geometry:{ conformally non-degenerate 2-form J J is closed: dj= 0 :{ non-degenerate 2-form J J is conformally closed: dj= 2α J (where α is a closed 1-form) J Ĵ= Ω2 J α ˆα=α+Υ where Υ=dlog Ω Example J (1/ x ) 2 (dx 1 dx 2 + dx 3 dx 4 + ) on R 2n {0} is invariant under dilation x λx so descends to S 1 S 2n 1
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 4/14 Combine! Recall: ˆ a φ b = a φ b Υ a φ b Υ b φ a NB: ˆ (a J b)c = (a J b)c 3Υ (a J b)c Decree (for conformally Fedosov) [J ab ] conformally symplectic [ a ] projective structure [a J bc] = 2α [a J bc] ( [a α b] = 0) (a J b)c = β (a J b)c Normalise: α a = β b. Then a J bc = 2J a[b α c] quite strong Remaining conformal change Ĵ ab = Ω 2 J ab ˆα a = α a + Υ a Υ a = a log Ω ˆ a φ b = a φ b Υ a φ b Υ b φ a
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 5/14 Fedosov gauge Locally: choose α a = 0 ( a J bc = 0) Remaining freedom Curvature Ĵ ab = Ω 2 J ab where Ω is constant (local) ˆ a = a canonically defined global connection R abcd R ab e d J ec ( a b b a )X c R ab c d X d R abcd = R [ab](cd) R [abc]d = 0 R abcd = V abcd + J ac Φ bd J bc Φ ad + J ad Φ bc J bd Φ ac + 2J ab Φ cd J-trace-free symmetric cf. Weyl + Schouten
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 6/14 Complex projective space With Fubini-Study metric so R ab c d = δ a c g bd δ b c g ad + J ad J c b J bd J c a+ 2J ab J c d R abcd = J ac g bd J bc g ad + J ad g bc J bd g ac + 2J ab g cd V abcd = 0 Φ ab = g ab Projective viewpoint (Conformal viewpoint = exercise) R ab c d = J ad J c b J bd J c a+ 2J ab J c d 3 2n 1 (δ a c g bd δ b c g ad ) + 2 n+1 2n 1 (δ a c g bd δ b c g ad ) Projective Weyl tensor!!
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 7/14 Conformal differential geometry Densities of weight w g ab ĝ ab = Ω 2 g ab ˆφ=Ω w φ Tractors T=Λ 0 [1] Λ 1 [1] Λ 0 [ 1] transform according to ˆσ ˆµ b ˆρ = σ µ b + Υ b σ ρ Υ b µ b 1 2 Υb Υ b σ, where Υ b b log Ω Υ b = g bc Υ c Tractor connection (É. Cartan / T.Y. Thomas) a σ µ b ρ = a σ µ a a µ b + g ab ρ+p ab σ a ρ P a b µ b Structure group=s0(n+1,1) flat Weyl=0 [ Cotton-York=0 ] when dim=3
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 8/14 Conformally Fedosov tractors Densities of weight w J ab Ĵab= Ω 2 J ab ˆφ=Ω w φ Tractors T=Λ 0 [1] Λ 1 [1] Λ 0 [ 1] transform according to ˆσ ˆµ b ˆρ = σ µ b + Υ b σ ρ Υ b µ b + Υ b α b σ Tractor connection in Fedosov gauge a σ µ b ρ = a σ µ a a µ b J ab ρ+φ ab σ a ρ Φ a b µ b S a σ Structure group=sp(2n+2,r), where Υ b b log Ω Υ b J bc Υ c Φ a b = J bc Φ ac S a = 1 2n+1 b Φ ab
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 9/14 Tractor curvature In Fedosov gauge ( a b b a ) σ µ c ρ = where (cf. Cotton-York) 0 V abcd µ d + Y abc σ Y abc µ c 2n 1( c Y abc V abce Φ ce )σ ρ 2J ab S c σ Φ cd µ d S c µ c 2n 1(Φ deφ de + c S c )σ Y abc a Φ bc b Φ ac + J ac S b J bc S a + 2J ab S c Bianchi d V abcd +(2n+1)Y abc = 0 so V abcd = 0 this part vanishes
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 10/14 Consequences of V abcd 0 V abcd = 0 ( a b b a )Σ=2J ab ΘΣ Bianchi [a (J bc] Θ)=0 in Fedosov gauge J [bc a] Θ=0 a Θ=0 1 0 0 Θ Γ(End(T)) Γ( 2 T) Θ= 0 Φ bc S b 0 S c 2n 1(Φde Φ de + c S c ) a Θ= 0 0 0 0 a Φ bc J ab S c J ac S b 0 Therefore V abcd = 0 a Φ bc + J ab S c + J ac S b = 0
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 11/14 Further consequences of V abcd 0 In Fedosov gauge V abcd = 0 a Θ=0 a Φ bc + J ab S c + J ac S b = 0 a Φ bc + δ a b S c + δ b c S b = 0 trace-free-part( a Φ bc = 0) mobility equations a Φ bc + δ a b S c + δ b c S b = 0 a S b + δ a b X Φ ac Φ bc = 0 a X 2Φ ab S b = 0 extremely strong!!
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 12/14 Examples with V abcd 0 CP n recall V abcd = 0 and Φ ab = g ab Θ= 1 0 0 0 Φ bc S b 0 S c 2n 1(Φde Φ de + c S c ) = 1 0 0 0 g bc 0 0 0 1 S 1 S 2n 1 locally Any more? with J ab =(1/ x ) 2 (dx 1 dx 2 + ) a = the flat connection V abcd = 0 Φ ab = 0 Θ= 1 0 0 0 0 0 0 0 0
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 13/14 (Potential) applications Integral geometry on CP n (with H. Goldschmidt) Elliptic complex on any (conformally) symplectic manifold 0 d 0 Λ 0 d Λ 1 d Λ 2 d Λ 2 d d Λ n 1 d d Λ n 1 d Λ n d Λ n d (2) V abcd = 0 OK to couple with T Smith (1976, n=2) Rumin-Seshadri Tseng-Yau Characterisation of CP n (with L. Stolovitch) Blaschke conjecture...
Workshop on Recent Developments in Conformal Geometry, Université de Nantes p. 14/14 THE END THANK YOU