Background on c-projective geometry

Second Kioloa Workshop on C-projective Geometry p. 1/26 Background on c-projective geometry Michael Eastwood [ following the work of others ] Australian National University

Second Kioloa Workshop on C-projective Geometry p. 2/26 Classical question Given a Riemannian metric, is there another with the same geodesics as unparameterised curves? EG (Thales circa 6 BC) the round sphere (1+y 2 )dx 2 xy dxdy+(1+x 2 )dy 2 (1+x 2 + y 2 ) 2

Second Kioloa Workshop on C-projective Geometry p. 3/26 Better (classical) question Given a projective structure, is there a metric connection in the projective class and, if so, how many? Geometric Def n Two torsion-free affine connections α and ˆ α are projectively equivalent iff they have the same geodesics as unparameterised curves Operational Def n Two torsion-free affine connections α and ˆ α are projectively equivalent iff ˆ α φ β = α φ β Υ α φ β Υ β φ α for some 1-form Υ α better derivation

Second Kioloa Workshop on C-projective Geometry p. 4/26 Tensors on oriented smooth manifolds Frame bundle principal GL + (n,r)-bundle Tangent bundle GL + (n,r) acting on R n (defining rep n ) Cotangent bundle GL + (n,r) acting on(r n ) Symmetric covariant tensors GL + (n,r) acting on 2 (R n ) Embed GL + (n,r) SL(n+1,R) λ 1 B λ B λ> * * * gl(n,r) (trace-free) raising lowering Cartan * *

Second Kioloa Workshop on C-projective Geometry p. 5/26 The (co-)tangent bundle gl(n,r) lowest weight vector 1 1 1 1 1 1 1 = = 1 1 2 1 1 = 1 1 1 1 = minus lowest weight 1 1 1 dual bundle Λ 1 2 1 =

Second Kioloa Workshop on C-projective Geometry p. 6/26 The de Rham complex In three dimensions (grad, curl, div) R Λ d Λ 1 d Λ 2 d Λ 3 Use our new notation! R 2 1 3 1 4 Already redolent of projective differential geometry! Cf. de Rham complex in four conformal dimensions 2 3 R 1 2 1 3 2 1 4 1 4

Easy algorithms! Second Kioloa Workshop on C-projective Geometry p. 7/26

Second Kioloa Workshop on C-projective Geometry p. 8/26 Real projective geometry Change of connection (contorsion) ˆ α φ β = α φ β Γ αβ γ φ γ for Γ αβ γ Λ 1 Λ 1 T change of torsion τ γ αβ = 2Γ γ [αβ] Λ 2 T Λ 1 Λ 1 T Λ 2 T coker = ker = 2 Λ 1 4 2 T = 1 1 3 2 = 1 2 1 ˆ α φ β = α φ β Υ α φ β Υ β φ α Projective differential geometry from thin air!

Second Kioloa Workshop on C-projective Geometry p. 9/26 The flat model G/P or (better) RP n = SL(n+1,R)/ S n = SL(n+1,R)/ λ for λ> a b c d e Irreducible homogeneous vector bundles

Second Kioloa Workshop on C-projective Geometry p. 1/26 PILDOs Examples Metrisability 2 2 1 2 2 T( 2) σ βγ ( α σ βγ ) Projective Killing Killing tensors!! 1 1 (2) 3 2 1 X γ ( (α β) X γ + P αβ X γ + W γ δ(α β) X δ ) d 2 d+1 d Λ 1 (2d) X βγ δ (α X βγ δ) All PILDOs BGG a b c (a+1) a 2 a+b+1 c (b+1) a a b 3 b+c+1 (c+1) a a b c 4 b

Second Kioloa Workshop on C-projective Geometry p. 11/26 Holomorphic projective geometry Repeat previous discussion in the holomorphic category! ˆ a φ b = a φ b Υ a φ b Υ b φ a for some holomorphic 1-form Υ a Crucial difference!! An (n + 1)-fold covering Do not be afraid: cf. spinors GL(n,C) / SL(n+1,C) λ 1 B λ B Upshot: insist on an(n+1) st root of the canonical bundle. Ω n =O( n 1) as on CP n (the flat model)

Second Kioloa Workshop on C-projective Geometry p. 12/26 Tensors on almost complex manifolds Reduced structure group GL(n,C) GL(2n,R) by GL(n,C) {M GL(2n,R) s.t. MJ=JM}, where, for example, J is the 2n 2n matrix Complexify ( 1 1 ). GL(n,C) GL(n,C) GL(n,C) C GL(2n,C) C 2n CT= T 1, T,1 ={v s.t. Jv= iv} {v s.t. Jv= iv}.

Second Kioloa Workshop on C-projective Geometry p. 13/26 Covering subgroup of SL(n+1,C) λ B Homomorphism GL(n,C) given by λ B λ 1 B. Recall: it is an(n+1)-fold covering.

Second Kioloa Workshop on C-projective Geometry p. 14/26 Representations GL(n,C) C GL(n,C) GL(n,C) Lie algebra (case n=5), and decorate p a b c d q e f g h and optionally insist that p+2a+3b+4c+5d q+ 2e+3f+ 4g+ 5h mod 6. External tensor product p a b c d q e f g h but keeps track of real structure.

Second Kioloa Workshop on C-projective Geometry p. 15/26 Conjugate bundles Conjugate of p a b c d q e f g h is q e f g h p a b c d Prototype 1 1 CT= T 1, T,1 = 1 1 Dual Λ 1 = Λ,1 Λ 1, = 2 1 2 1

Second Kioloa Workshop on C-projective Geometry p. 16/26 2-tensors and 3-forms Λ 2 = Λ 1,1 (Λ 2, Λ,2 ) 2 1 Λ 2 = 2 1 3 1 ( Cc) 2 1 2 Λ 1 = 2 1 4 2 ( Cc) Λ 3 =(Λ 2,1 Λ 1,2 ) (Λ 2, Λ,2 ) 3 1 Λ 3 =( 2 1 4 1 Cc) ( Cc)

Second Kioloa Workshop on C-projective Geometry p. 17/26 The de Rham complex Remark in advance: c-projective geometry is 1 -graded (integrable case) Λ, Λ 1, Λ 2, Λ,1 Λ 1,1 Λ,2 2 1 3 1 2 1 2 1 2 1 3 1

Second Kioloa Workshop on C-projective Geometry p. 18/26 Torsion Λ 2 T decomposes into five R-irreducibles 2 1 2 ( 1 1 1 ( 2 1 Cc) Cc) 2 1 1 ( Cc) 3 1 ( 1 1 Cc)

Second Kioloa Workshop on C-projective Geometry p. 19/26 Complex connections Λ 1 gl(n,c) decomposes into six R-irreducibles 2 1 3 ( 1 1 1 ( 2 1 Cc) Cc) 2 1 1 ( Cc) 3 2 1 ( Cc)

Second Kioloa Workshop on C-projective Geometry p. 2/26 Compare! 2 1 ( 3 2 1 Cc) ( Λ 1 gl(n,c) Λ 2 TM Cc) 3 1 ( 1 1 Cc), where Γ αβ γ 2Γ [αβ] γ. 3 1 ( 1 1 Cc) the Nijenhuis tensor

Second Kioloa Workshop on C-projective Geometry p. 21/26 C-projective geometry (from thin air) 2 1 ( 3 2 1 Cc) ( Λ 1 gl(n,c) Λ 2 TM Cc) 3 1 ( 1 1 Cc), where Γ αβ γ 2Γ [αβ] γ. 2 1 ( Cc) C-projective freedom

Second Kioloa Workshop on C-projective Geometry p. 22/26 C-projective freedom In real money ˆ α φ β = α φ β 1 2 (Υ αφ β +Υ β φ α J α γ Υ γ J β δ φ δ J β γ Υ γ J α δ φ δ ) In complex money (barred and unbarred indices) ˆ a φ b = a φ b Υ a φ b Υ b φ a ˆ āφ b = āφb ( Λ 1, Λ,1 Λ 1, ) ˆ āφ b= āφ b Υāφ b Υ bφā ˆ a φ b= a φ b

Second Kioloa Workshop on C-projective Geometry p. 23/26 Deformation complex 1 1 ( 3 2 1 Cc) ( 4 1 1 1 Cc) ( Cc) 1 1 ( 2 1 Cc) 3 2 1 ( 2 1 Cc) 1 1 ( 3 1 Cc) vector fields infinitesimal deformations harmonic curvature

Second Kioloa Workshop on C-projective Geometry p. 24/26 C-metrisability complex 1 1 2 1 1 1 3 1 1 1 1 2 1 1 2 1 1 2 1 1 or 1 3 1 1 1 1 2 1 1 ( 1 3 1 1 Cc) ( 1 Cc) 2 1 1 2 1 1

Second Kioloa Workshop on C-projective Geometry p. 25/26 Parabolic viewpoint Curvature Tractors Prolongation Kähler metrics Kähler-Einstein metrics Hamiltonian two-forms... and more besides...

Second Kioloa Workshop on C-projective Geometry p. 26/26 THE END THANK YOU