Parabolic geometry in five dimensions

Transcription

1 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 1/20 Parabolic geometry in five dimensions Michael Eastwood [ with Katja Sagerschnig and Dennis The ] University of Adelaide

2 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 2/20 Five-dim l projective geometry Real projective space RP 5 = R 5 RP 4 (affine cell with points at infinity ) R = 6 {0} x λx λ/= 0 = SL(6,R)/ λ s.t. λ/= 0 Real projective sphere S 5 = R 6 {0} x λx λ>0 = SL(6,R)/ λ s.t. λ>0

3 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 3/20 Five-dim l projective differential geometry Def n ˆ a a same geodesics (unparameterised) EG (Thales 600 BC) the round sphere is projectively flat Affine cell R 5 RP 5 is a projective equivalence Projective motions? induced by SL(6,R)!! NB!!

4 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 4/20 Five-dim l flat models Flag manifolds SL(l+1,C)/ G/P = Dynkin diagram with l nodes All five-dimensional cases (with G simple) projective > conformal (quadric) < contact projective contact Legendrian (cf. CR geometry) < < (2,3,5) geometry (quadric)

5 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 5/20 Conformal geometry in four variables The de Rham complex in four (conformal) variables 0 Λ 0 d Λ 2 + Λ 1 Λ 3 Λ 4 0 Λ 2 Λ 2 +={ω s.t. ω=+ω} Λ 2 } in Riemannian or neutral signature ={ω s.t. ω= ω} SO (2,2) 1 2 SL(2,R) SL(2,R) spinors S(GL(2, R) GL(2, R)) conformal spinors

6 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 6/20 Conformal geometry: the flat model M= Gr 2 (R 4 ) = {Π R 4 s.t. dimπ=2} = SL(4,R)/ = G/P= S(GL(2,R) GL(2,R)) T Π M= Hom(Π,R 4 /Π)=Π R 4 /Π S 1 0 S= Dually, spin bundles Λ 1 M = S S 1 1 = = Λ 2 M =( 2 S Λ 2 S ) (Λ 2 S 2 S 2 3 )= = Λ 2 + Λ 2

7 de Rham revisited Road map Can view as a lune on a sphere! 7 The countries are A3 Weyl chambers! Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 7/20

8 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 8/20 Conformal deformation complex Recall tangent bundle = S 1 0 S= = Affine action of the Weyl group of A3 (λ w(λ+ρ) ρ)

9 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 9/20 Conformal deformation complex cont d Curved version (deformation sequence) Vector Fields X a Conformal Perturbations g ab + ǫh ab Conformal Killing operator X a h ab (a X b) trace Self-dual Weyl Curvature Anti-self-dual Weyl Curvature Second order! Bianchi/Bach

10 Twistor construction Recall Spin(2,2) SL(2,R) SL(2,R) Spin bundles TM= S S null vectors=simple vectors Segre RP 1 RP 1 RP 3 nonsingular quadric as image P(S ) M T horizontal subspace The blue lines and yellow planes are conformally invariant quadric Ravi Vakil 2007 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 10/20

11 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 11/20 Five variables geometry There are two! Let T be a 5-dimensional manifold. l D TT s.t.[d,d] l D and[l D,l D]=TT D TT s.t.[d,[d,d]]=tt (2,3,5) geometry Flat models are generalised flag manifolds F 1,2 (R 4 )={L Π R 4 s.t. diml=1, dimπ=2} L 2 Π G/P= G 2 /P= <

12 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 12/20 Twistor correspondence SL(4,R)/ F 1,2 (R 4 ) RP 3 Gr 2 (R 4 ) 0 SL(4,R)/ SL(4,R)/

13 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 13/20 An-Nurowski construction Riemannian (Σ 1,g 1 ) Surfaces (Σ 2,g 2 ) } M (Σ 1 Σ 2,g 1 g 2 ) T(M) Configuration space of Σ 1 rolling on Σ 2 Twistor structure l D (five variables) Suppose[D,D]=l D (generic) Forget l but retain D (cinq variables) EG (Bryant) spheres of radii 1 and 3: G 2 -flat (An-Nurowski) new G 2 -flat examples rolling on a plane!

14 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 14/20 Differential complexes in five variables Parabolic subgroups of SL(4, R) 1 1 lunes compatible with the A3 tiling of the sphere F 1,2 (R 4 )= a hemisphere

15 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 15/20 Differential complexes cont d Road map de Rham BGG Deformation Curvature 3 4 1

16 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 16/20 G2 road map < < 0 0 < < < < < < < < < < < < Business end of a vector field Deformation Cartan curvature

17 From to < Recall Curvature Obstruction to integrability of D Vanishes on T() [D,D]=l D triggers collapse a b c 0 0 < a+b c c < < < < 6 2 < < < 1 < 0 4 < < 4 0 < 6 1 < 5 0 < Bryant-E-Gover-Neusser Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 17/20

18 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 18/20 Theorem Flat model Compare Translate There are (2,3,5) geometries that do not arise from a neutral signature conformal structure via the An-Nurowski twistor construction.

19 Easy algorithms! Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 19/20

20 Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 20/20 THE END THANK YOU