Conformal and CR Geometry from the Parabolic Viewpoint

Transcription

1 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 1/13 Conformal and CR Geometry from the Parabolic Viewpoint Michael Eastwood Australian National University

2 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 2/13 The flat model: conformal case S n stereographic projection conformal R n R n x 1 x 2 +4 trough 4x x 2 4 S n

3 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 3/13 Conformal group R n+2 SO(n+1,1) acts on S n by conformal transformations semisimple S n = SO(n+1,1)/P flat model of conformal differential geometry {generators} S n parabolic S n RP n+1 R n =(SO(n) R n )/SO(n) flat model of Riemannian differential geometry

4 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 4/13 The flat model: CR case S 2n+1 CP n+1 { z 0 2 = z z n 2 + z n+1 2 } semisimple S 2n+1 = SU(n+1,1)/P flat model of CR geometry parabolic flat conformal geometry Fefferman 1976

5 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 5/13 Parabolic geometry Parabolic invariant theory..., Fefferman Parabolic geometries, Graham 1990 ditto and canonical Cartan connections, Čap & Schichl 2000 ditto I: background and general theory, Čap & Slovák 2009 Parabolic geometry G/P { G semisimple P parabolic Klein v. Cartan EG Conformal: SO(n+1,1) preserves 2x 0 x +x x 2 n 0 SO(n+1,1)/ block upper triangular

6 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 6/13 Flat models aka flag manifolds RP 2 ={L R 3 dim R L=1} F 1,2 (R 3 )={L P R 3 dim R L=1,dim R P= 2} L P RP 2 = SL(3,R)/ 0 0 F 1,2 (R 3 )=SL(3,R)/ 0 0 0

7 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 7/13 Complex flag manifolds SL(n+1,C)/ = Dynkin diagram with n nodes Lagrangian Grassmannians chss Quadrics chss chss Exceptional examples chss

8 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 8/13 Real forms F 1,2 (C 3 )=SL(3,C)/ F 1,2 (R 3 )=SL(3,R)/ = = (abuse notation) S 3 = SU(2,1)/P= (severely abuse notation) flat model of 3-dim l CR geometry flat model of...

9 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 9/13 geometry M= µ RP 2 = F 1 (R 3 ) F 1,2 (R 3 ) ν F 2 (R 3 )=RP 2 TM H= T 0,1 T 1,0 contact Lagrangian or para-cr define J H H by J T 0,1 Id and J T 1,0 Id NB flat model J 2 = Id (cf. J 2 = Id for CR geometry) T 0,1 = span{ / x} T 1,0 = span{ / t+x / y} symmetries SL(3,R) or sl(3,r) for the flat model Lie 1888 dimension 8 in general Tresse 1896 none generically Similarly for CR geometry Cartan 1924, 1932

10 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 10/13 Invariant connections and ambience M= G/P homogeneous bundlesv P -modules V G-module T restrict to P and induce to G/P tractor bundle T with G-invariant flat connection conformal geometry Tracey Thomas vector tensor spinor tractor twistor A.P. Hodges 1991 curved analogues, e.g. in the conformal case SO(n+1,1)-module R n+2 standard tractors Cartan connection ambience Fefferman- Graham Čap- Gover

11 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 11/13 Invariant differential operators 3-dim l CR case dim l CR case BGG complex Rumin complex Hirachi operator curved adjoint BGG sequence Akahori-Garfield-Lee CR deformation complex (E+Seshadri)

12 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 12/13 Where next? beyond parabolic? prolongation of overdetermined systems on (Riemannian) manifolds (Branson-Čap-E-Gover) contact manifolds (E-Gover) BGG machinery/ Lie algebra cohomology (Kostant) filtered manifolds (Neusser) differential complexes (Bryant-E-Gover-Neusser,..., Calin-Chang,... ) EG Rumin-Seshadri complex (Tseng-Yau) for M 2n symplectic 0 Λ 0 Λ 1 Λ 2 Λ n second order! 0 Λ 0 Λ 1 Λ 2 Λ n elliptic conformally Fedosov manifolds (E-Slovák) +projective

13 Workshop on Conformal and CR Geometry at the Banff International Research Station p. 13/13 THE END THANK YOU