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
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 0 0 0 0 0 s.t. λ/= 0 Real projective sphere S 5 = R 6 {0} x λx λ>0 = SL(6,R)/ λ 0 0 0 0 0 s.t. λ>0
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!!
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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 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)
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
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)/ 0 0 0 = 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= 0 0 0 1 Dually, spin bundles Λ 1 M = S S 1 1 = 0 0 1 1 1 2 1 = Λ 2 M =( 2 S Λ 2 S ) (Λ 2 S 2 S 2 3 )= 0 0 3 2 = Λ 2 + Λ 2
de Rham revisited 0 0 0 1 2 1 2 3 0 0 3 2 1 4 1 0 4 0 Road map Can view as a lune on a sphere! 7 The countries are A3 Weyl chambers! http://www.math.rug.nl/models Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 7/20
Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 8/20 Conformal deformation complex Recall tangent bundle = S 1 0 S= 0 0 0 1 1 0 1 = 4 4 0 1 0 1 2 2 2 2 6 2 1 6 1 0 4 4 Affine action of the Weyl group of A3 (λ w(λ+ρ) ρ)
Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 9/20 Conformal deformation complex cont d 1 0 1 2 2 2 4 4 0 0 4 4 2 6 2 1 6 1 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
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
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= <
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)/ 0 0 0 0 0 0 0 0 0 0 0
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!
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
Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 15/20 Differential complexes cont d Road map de Rham BGG 0 0 0 1 2 1 2 1 0 2 3 0 0 3 2 3 0 1 1 4 1 2 2 2 4 0 0 0 4 0 3 2 1 2 3 0 Deformation 1 0 1 2 2 2 3 2 1 4 4 0 0 4 4 4 1 2 2 6 2 2 3 4 6 1 0 1 6 1 4 3 2 Curvature 3 4 1
Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 16/20 G2 road map < < 0 0 < < 2 1 3 0 1 < < 2 2 6 5 2 2 < 8 4 2 < 6 2 3 < 9 4 6 < 6 1 5 0 < < 9 2 8 1 < < Business end of a vector field Deformation Cartan curvature
From to < Recall Curvature 4 4 0 0 4 4 4 1 2 Obstruction to integrability of D Vanishes on T() [D,D]=l D triggers collapse 4 4 0 a b c 0 0 < a+b c c < 0 0 0 1 2 1 2 1 0 2 3 0 0 3 2 3 0 1 1 4 1 2 2 2 4 0 0 0 4 0 3 2 1 2 3 0 0 0 < 2 1 1 0 < 5 2 4 1 < 6 2 < < < 1 < 0 4 < 1 4 0 < 4 0 < 6 1 < 5 0 < Bryant-E-Gover-Neusser Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 17/20
Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 18/20 Theorem 1 0 1 Flat model 2 2 2 3 2 1 4 4 0 0 4 4 4 1 2 2 6 2 2 3 4 6 1 0 1 6 1 4 3 2 3 4 1 0 0 0 Compare 1 2 1 2 1 0 2 3 0 0 3 2 3 0 1 1 4 1 2 2 2 4 0 0 0 4 0 3 2 1 2 3 0 Translate There are (2,3,5) geometries that do not arise from a neutral signature conformal structure via the An-Nurowski twistor construction.
Easy algorithms! Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 19/20
Analysis and Geometry in Non-Riemannian Spaces, UNSW, Sydney p. 20/20 THE END THANK YOU