Spring Lecture Five at the University of Arkansas p. 1/22 Conformal differential geometry and its interaction with representation theory The X-ray transform on projective space Michael Eastwood Australian National University
Spring Lecture Five at the University of Arkansas p. 2/22 Some useful facts 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 embeddingsµ :RP n CP n totally geodesic BGG OnRP n for n 2, ω a = a φ [a ω b] = 0 ω ab = (a φ b) π( (a c) ω bd + g ac ω bd )=0 &c 2-form lemmaµ ψ ab = 0 µ ψ ab = 0 Curvature lemmaµ ψ abcd = 0 µ ψ abcd = 0
Spring Lecture Five at the University of Arkansas p. 3/22 Curvature lemma ψ abcd =ψ [ab][cd] ψ [abc]d = 0 } Riemann tensor symmetries ψ abcd Γ( 0 2 0 0 0 0 0 ) (case n=4). Branch to SL(8, R) Sp(8, R) 0 2 0 0 0 0 0 = 0 2 0 0 0 1 0 0 0 0 0 0 ψ abcd = ψ abcd + φ ab J cd + θj ab J cd cf. Weyl cf. Ricci cf. Scalar Lemma:µ ψ abcd = 0 modelsµ ψ abcd = 0
Spring Lecture Five at the University of Arkansas p. 4/22 BGG detail 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..... cf. OEIS...
Spring Lecture Five at the University of Arkansas p. 5/22 Topics Funk or Radon transforms on S 2 orr 2 X-ray transform onrp 3 X-ray transform oncp n X-ray transform on functions X-ray transform on 1-forms Symplectic geometry X-ray transform on symmetric tensors Complex methods
Spring Lecture Five at the University of Arkansas p. 6/22 Funk-Radon Funk (1913) S 2 f Γ(S 2,E) γ φ(γ)= γ f Radon (1917) R 2 f Γ (R 2,E) γ φ(γ)= γ f
Spring Lecture Five at the University of Arkansas p. 7/22 Radon=Funk! F :Γ even (S 2,E) Γ even (S 2,E) Better: Γ(RP 2,E) Γ(RP 2,E) Better still: F :Γ(RP 2,E( 2)) Γ(RP 2,Ẽ( 1)) Usual affine coördinates projective equivalence!! Γ (R 2,E) Γ(RP 2,E( 2)) R F R 2 RP 2 agree!
Spring Lecture Five at the University of Arkansas p. 8/22 John (1938) The X-ray transform according to John Γ (R 3,E) f φ(γ)= Better: Γ(RP 3,E( 2)) f φ(γ)= NB invariance under SL(4,R) because Γ(RP 1,E( 2)) Γ(RP 1,Λ 1 ) R is invariant under SL(2,R). γ f γ f
Spring Lecture Five at the University of Arkansas p. 9/22 X-ray transform onrp 3 X :Γ(RP 3,E( 2)) Γ(Gr 2 (R 4 ),Ẽ[ 1]) Range? Theorem ( John) φ=x f φ=0 where :Ẽ[ 1] Ẽ[ 3] = ultrahyperbolic wave operator (SL(4,R)-invariant). Kernel?X is injective on Γ(RP n,e( 2)) As a Riemannian Γ(RP n,λ 0 ) manifold under SO(n+1) for n 2 Funk
Spring Lecture Five at the University of Arkansas p. 10/22 X-ray transform oncp n CP n γ SU(n+1)/S(U(1) U(n)) Fubini-Study metric f= smooth function oncp n γ= geodesic X f φ(γ)= f γ Questions Kernel ofx? What about ω X φ(γ)= γ ω forωa1-form? What about ω ab c a symmetric tensor?
Spring Lecture Five at the University of Arkansas p. 11/22 X-ray transform on functions Know f= 0 geodesics γ RP γ n f= 0. Suppose Funk (1913) f= 0 geodesics γ CP n. γ Then f= 0 geodesicsγ RP γ n model embedding. Hence, µ CP n for any µ f= 0 model embeddingsµ :RP n CP n. Hence f= 0, i.e.x is injective on functions oncp n cf. Helgason, The Radon Transform, 2 Corollary 2.3
Spring Lecture Five at the University of Arkansas p. 12/22 Approach (with Hubert Goldschmidt) CP n γ RP n RP n CP n induced byr n+1 C n+1 is totally geodesic. Translates by SU(n+1) too! Model Embeddings µ The X-ray transform onrp n is well-understood. Pullback of tensors underµis well-understood. Suitable global techniques oncp n are available, compatible with similar techniques (BGG) onrp n.
Spring Lecture Five at the University of Arkansas p. 13/22 X-ray transform on 1-forms Know ω=0 geodesicsγ RP γ n ω=dφ. Suppose Michel (1978) ω=0 geodesics γ CP n. γ Then ω=0 geodesicsγ RP γ n model embedding. Hence, µ CP n for any µ dω=0 model embeddingsµ:rp n CP n. Hence, by 2-form lemma, dω=θj. Claimω=dφ
Spring Lecture Five at the University of Arkansas p. 14/22 1-forms cont d Theorem Forω Γ(CP n,λ 1 ), n 2, TFAE (a)ω=dφ (b) dω=0 (c) (dω) = 0 (d) dω=θj Proof (a) (b) because H 1 (CP n,r)=0. (c) (d) by definition. (b) (c) is trivial. (d) (b) If dω = θj, then 0=d 2 ω=d(θj)=dθ J dθ=0 θ=constant. But ifθ 0, then d(ω/θ)= J, a contradiction.
Spring Lecture Five at the University of Arkansas p. 15/22 Symplectic geometry Rumin-Seshadri complex Λ 0 d Λ 1 d Λ 2 d Λ 3 d d Λ n d (2) Λ 0 d Λ 1 d Λ 2 d Λ 3 d d Λ n local cohomology=r Γ(CP n,λ 0 ) d Γ(CP n,λ 1 ) d Γ(CP n,λ 2 ) is exact.
Spring Lecture Five at the University of Arkansas p. 16/22 Symplectic geometry cont d Suppose is a connection onvsuch that ( a b b a )Σ= J ab ΦΣ Φ EndV. Then we can couple the Rumin-Seshadri complex V Λ 1 V Λ 2 V It s still a complex and (ifvis a bundle oncp n ) Γ(CP n,v) Γ(CP n,λ 1 V) Γ(CP n,λ 2 V) is exact. under further mild conditions
Spring Lecture Five at the University of Arkansas p. 17/22 Symmetric 2-tensors Know (Michel (1973)) γ ω ab= 0 geodesicsγ RP n ω ab = (a φ b). Therefore (BGG), γ ω ab= 0 γ RP n π( (a c) ω bd + g ac ω bd )=0. Would like to show (Tsukamoto (1981)) γ ω ab= 0 geodesicsγ CP n ω ab = (a φ b). Know (BGG & curvature lemma) γ ω ab= 0 γ CP n (π( (a c) ω bd + g ac ω bd )) = 0. Therefore, suffices to show, oncp n ω ab = (a φ b) (π( (a c) ω bd + g ac ω bd )) = 0
Spring Lecture Five at the University of Arkansas p. 18/22 Proof of ω ab= (a φ b) ( π ( (a b) ω cd + g ab ω cd )) = 0 Consider the connection onv Λ 1 Λ 2 Λ 1 a σ b µ ab a µ bc + g ab σ c g ac σ b + J ab ρ c J ac ρ b J bc ρ a + J bc J a d σ d a ρ b + J a d µ bd It satisfies ( a b b a )Σ= J ab ΦΣ. Now unravel the exactness of Γ(CP n,v) Γ(CP n,λ 1 V) Γ(CP n,λ 2 V) by Heisenberg Lie algebra cohomology! BGG!! and further mild conditions
Spring Lecture Five at the University of Arkansas p. 19/22 Further results and summary Theorem Supposeω ab c is symmetric onrp n orcp n for n 2. γ ω ab c= 0 geodesicsγ ω ab c = (a φ b c), for some symmetric tensorφ b c. By combining pullback lemmata: curvature &c BGG complexes symplectic geometry we can bootstrap fromrp n tocp n. Therefore, It suffices to prove the Theorem forrp n.
Spring Lecture Five at the University of Arkansas p. 20/22 Complex methods dim R = 2n F 1,2 (C n+1 ) µ ν CP n Gr 2 (C n+1 ) Double fibration transform ν 1 (Gr 2 (R n+1 )) τ CP Gr 2 (R n+1 n ) F 1,2 (R n+1 ) RP n η F Gr 2 (R n+1 ) η : F CP n real blow-up alongrp n F acquires an involutive structure (cf. E & Graham)
Spring Lecture Five at the University of Arkansas p. 21/22 Further Reading T.N. Bailey and M.G. Eastwood, Zero-energy fields on real projective space, Geom. Dedicata 67 (1997) 245 258. T.N. Bailey and M.G. Eastwood, Twistor results for integral transforms, Contemp. Math. 278 (2001) 77 86. A. Čap and J. Slovák, Parabolic Geometries 1, AMS 2009. A. Čap, J. Slovák, and V. Souček, Bernstein-Gelfand-Gelfand sequences, Ann. Math. 154 (2001) 97 113. M.G. Eastwood, Complex methods in real integral geometry, Rend. Circ. Mat. Palermo, Suppl. 46 (1997) 55 71. M.G. Eastwood, Variations on the de Rham complex, Notices AMS 46 (1999) 1368 1376. M.G. Eastwood and C.R. Graham, The involutive structure on the blow-up ofr n inc n, Commun. Anal. Geom. 7 (1999) 613 626. M.G. Eastwood and H. Goldschmidt, Zero-energy fields on complex projective space, in preparation.
Spring Lecture Five at the University of Arkansas p. 22/22 THANK YOU THE END