AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 1/15 Representation theory and the X-ray transform Projective differential geometry Michael Eastwood Australian National University
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 2/15 Summary of Lecture 1 Curvature onrp n R abcd = g ac g bd g bc g ad Curvature oncp n R abcd = g ac g bd g bc g ad + J ac J bd J bc J ad + 2J ab J cd Model embeddings µ :RP n CP n totally geodesic 2-form lemma µ ψ ab = 0 µ ψ ab = 0 Curvature lemma µ ψ abcd = 0 µ ψ abcd = 0 This Lecture! [a b] µ c = g c[a µ b]
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 3/15 Topics Tractor bundle and connection onrp n Projective differential geometry Projective tractor bundles and connections Bernstein-Gelfand-Gelfand resolutions onrp n Lie algebra cohomology Range of the Killing operator Higher Killing operators
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 4/15 Tractors on real projective space Consider the connection ont Λ 0 Λ 1 given by [ ] [ ] σ a a σ µ a a µ b + g ab σ µ b Calculate a b [a b] σ µ c σ = b σ µ b a b µ c + g bc σ a ( b σ µ b ) b µ a g ba σ = a ( b µ c + g bc σ)+g ac ( b σ µ b ) = 0 [a b] µ c g c[a µ = 0 b] 0 Flat! µ c
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 5/15 Projective differential geometry Suppose a and ˆ a are torsion-free affine connections. Def n a and ˆ a are projectively equivalent iff they have the same geodesics (as unparameterised curves). EG The round sphere is projectively flat by central projection Affine coördinate patch R n RP n is a projective equivalence
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 6/15 Projective tractors Can always choose a s.t. R ab is symmetric Then define a connection ont Λ 0 Λ 1 by [ ] [ ] σ a µ b a σ µ a a µ b + P ab σ It is projectively invariant (T.Y. Thomas) P ab 1 n 1 R ab Equivalent to a Cartan connection (É. Cartan) It is flat iff a is projectively flat Hence the flat tractor connection onrp n NB:RP n is a homogeneous space SL(n+1,R)/P
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 7/15 Back to real projective space RecallT Λ 0 Λ 1 with its flat connection. Thus,V Λ 2 T=Λ 1 Λ 2 also has a flat connection [ ] [ ] σb a a σ b µ ab. µ bc a µ bc + g ab σ c g ac σ b V Λ 1 V Λ 2 V Λ 3 V Λ 1 Λ 1 Λ 1 Λ 2 Λ 1 Λ 3 Λ 1 NB Λ 2 Λ 1 Λ 2 Λ 2 Λ 2 Λ 3 Λ 2 Λ p Λ 2 µ a bcd µ [a bc]d Λ p+1 Λ 1
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 8/15 Decompose into irreducibles Λ 1 V Λ 2 V Λ 3 V V Lie algebra cohomology! (Kostant 1961) EG: ker :Λ 2 Λ 2 Λ 3 Λ 1 ={ψ abcd =ψ [ab][cd] s.t.ψ [abc]d = 0} ={Riemann curvature tensors}! Suspend disbelief
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 9/15 BGG resolutions Diagram chasing locally exact complexes Riemannian deformation (2) Λ 0 d Λ 1 d Λ 2 d Λ 3 d Bernstein-Gelfand-Gelfand resolutions. de Rham RP n = SL(n+1,R)/ = G/P, 0. 0 where G is semisimple and P is parabolic.
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 10/15 Lie algebra cohomology u=lie algebra V=u-module 0 V Hom(u,V) Hom(Λ 2 u,v) v(x)= Xv φ(x Y)=φ([X, Y]) Xφ(Y)+Yφ(X) H p (u,v) sl(n+1,r). g=g 1 g 0 g 1 Letu=g 1 ( u =g 1 ) LetV=Λ 2 R n+1 g 1 0 V g 1 V Λ 2 g 1 V
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 11/15 Geometric import Kostant s Bott-Borel-Weil Theorem = H p (g 1,V)=,,, as SL(n,R)-modules 0 V g 1 V Λ 2 g 1 V Λ 3 g 1 V 0 V Λ 1 V Λ 2 V Λ 3 V Λ 1 Λ 1 Λ 1 Λ 2 Λ 1 Λ 3 Λ 1 ր ր ր Λ 2 Λ 1 Λ 2 Λ 2 Λ 2 Λ 3 Λ 2 Previously suspected tensor identities are justified!!
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 12/15 Back to BGG de Rham Λ 0 d Λ 1 d Λ 2 φ a φ ω a [a ω b] OnRP n for n 2, ω a = a φ [a ω b] = 0 Riemannian deformation φ a (a φ b) = Killing (2) φ a (a φ b) ω ab π( (a c) ω bd + g ac ω bd ) ω ab = (a φ b) π( (a c) ω bd + g ac ω bd )=0, where π(φ acbd =Φ [ab][cd] )
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 13/15 Higher Killing operators φ bc =φ (bc) (a φ bc) prolong using Lie algebra cohomology Λ 1 V Λ 2 V V ր ր ր ր BGG (3)
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 14/15 Summary Range of Killing operators onrp n for n 2 ω a = a φ π( a ω b )= [a ω b] = 0 ω ab = (a φ b) π( (a c) ω bd + g ac ω bd )=0 ω abc = (a φ bc) π( (a c e) ω bd f + 4g (ac e) ω bd f )=0 Combinatorics 1 1 1 1 4 1 10 9 1 20 64 1 35 259 225 1 56 784 2304......... 1 84 1974 12916 11025........
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 2 p. 15/15 THANK YOU END Of PART TWO