The X-ray transform: part II

Transcription

1 The 34th Czech Winter School in Geometry and Physics, Srní p. 1/13 The X-ray transform: part II Michael Eastwood [ Toby Bailey Robin Graham Hubert Goldschmidt ] Lionel Mason Rod Gover Laurent Stolovitch Eduard Čech Institute The simplest X-ray transform is a version of the Radon transform in three dimensions. One starts with suitably decaying function of three variables and integrates it over the lines in Euclidean three-space obtaining a function on the four-dimensional space of lines. This transform is often named after John who identified its range in There are many variations on this theme! There is a compactified version, due to Funk in There is a complex version, due to Bateman in Nowadays, there are all sorts of X-ray transforms and the purpose of these lectures will be to describe the links between them and to use representation theory and differential geometry to establish their range and kernel in various cases.

2 The 34th Czech Winter School in Geometry and Physics, Srní p. 2/13 References T.N. Bailey and M.G. Eastwood, Zero-energy fields on real projective space, Geom. Dedicata 67 (1997) Srní 1996 M.G. Eastwood, Complex methods in real integral geometry, Rend. Circ. Mat. Palermo, Suppl. 46 (1997) T.N. Bailey, M.G. Eastwood, A.R. Gover, L.J. Mason, The Funk transform as a Penrose transform, Math. Proc. Camb. Phil. Soc. 125 (1999) M.G. Eastwood and C.R. Graham, The involutive structure on the blow-up of R n in C n, Commun. Anal. Geom. 7 (1999) M.G. Eastwood and A.R. Gover, The BGG complex on projective space, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011) 18 pp. Srní 2014 M.G. Eastwood and H. Goldschmidt, Zero-energy fields on complex projective space, Jour. Diff. Geom. 94 (2013)

3 The 34th Czech Winter School in Geometry and Physics, Srní p. 3/13 Recall Lecture I X-ray transform RP 3 α F 1,2 (R 4 ) correspondence β Gr 2 (R 4 ) x α(β 1 (x)) Gr 2 (R 4 geodesic on RP ) 3 Th m J There is an exact sequence 0 Γ(RP 3, E( 2)) X Γ(Gr 2 (R 4 ), Ẽ[ 1]) Γ(Gr 2 (R 4 ), Ẽ[ 3]). Penrose transform CP 3 µ F 1,2 (C 4 ) ν Gr 2 (C 4 ) NB RP 3 CP 3 O( 2) RP3 = E( 2) H 1 (CP + 3, O( 2)) P

4 The 34th Czech Winter School in Geometry and Physics, Srní p. 4/13 Machinery for the Penrose transform F 1,2 (C 4 ) µ ν V = holomorphic vector bundle on CP 3 CP Gr 2 (C 4 3 ) 0 µ 1 O(V ) Ω µ µ V E p,q 1 = ν q (Ω p µ V ) = H p+q (O(V )) Easy to use (when V is irreducible homogeneous) Ω µ O O A (1)[ 1] O(2)[ 3] on F 1,2 (C 4 ) O( 3) O( 3) ν 1 O A [ 1] O A ( 2)[ 1] ν 1 D O A [ 2] O( 1)[ 3] Dirac operator! on M = Gr 2 (C 4 )

5 The 34th Czech Winter School in Geometry and Physics, Srní p. 5/13 Examples of the Penrose transform O( 2) O( 2) O A ( 1)[ 1] O[ 3] ν 1 O[ 1] O Ω 0 µ Ω 1 µ Ω 2 µ ν 0 ν 0 ν 0 Ω Ω 0 d Ω 1 d + Ω ν 1 ν 0 ν 0 Ω 0 Ω 2 d Ω 3 ν 0 O[ 3] BGG Th m EPW H 1 (CP +, O) Γ(M ++, Max ) exact sequence 0 H 1 (CP + 3, O) d H 1 (CP 3, Ω 1 ) Γ(M ++, Ω 0 ) 2 Γ(M ++, Ω 4 )

6 The 34th Czech Winter School in Geometry and Physics, Srní p. 6/13 Geometry of the Penrose/X-ray transform CP 3 F 1,2 (C 4 ) Gr 2 (C 4 ) SL(2, H)-orbits CP 3 ν 1 (S 4 ) τ S 4 SL(4, R)-orbits RP 3 F 1,2 (R 4 ) projective geometry Gr 2 (R 4 ) another parabolic geometry! conformal geometry SL(4, R) = Spin(3, 3) Plücker: Gr 2 (R 4 ) RP 5 cf. Charles Frances talks

7 The 34th Czech Winter School in Geometry and Physics, Srní p. 7/13 Machinery for the X-ray transform Complex analysis comes into play in two ways: constructing a spectral sequence, computing with the spectral sequence. More generally: F 1,2 (C n+1 ) µ ν CP n Gr 2 (C n+1 ) a correspondence dim R = 2n but not a double fibration ν 1 (Gr 2 (R n+1 )) CP n RP n η F 1,2 (R n+1 ) τ Gr 2 (R n+1 ) F Gr 2 (R n+1 )

8 The 34th Czech Winter School in Geometry and Physics, Srní p. 8/13 Real blow up F = η CP n = L Cn+1 is a complex line (L,P) s.t. P R n+1 is a real plane R(L) P (generic equality) { L s.t. L C n+1 is a complex line } F F 1,2 (R n+1 ) η CP n RP n Real blow up of CP n along RP n!

9 The 34th Czech Winter School in Geometry and Physics, Srní p. 9/13 Involutive structure complex manifold Y Ω J : TΩ TΩ s.t. J 2 = Id... { T 0,1 CTΩ s.t. [T 0,1,T 0,1 ] T 0,1 Λ 0,0 Λ 0,1 Λ 0,2 s.t. 2 = 0 totally real submanifold M Ω dim R M = dim C Ω and TM JTM = 0 Y involutive structure Ω Σ η Ω M real blow-up Involutive cohomology H r ( Ω) (cf. Dolbeault, b,... )

10 The 34th Czech Winter School in Geometry and Physics, Srní p. 10/13 The X-ray machine η RP n CP n Gr 2 (R n+1 ) F Pull-back to F τ V = holomorphic vector bundle on CP n 0 Γ(CP n, O(V )) Γ(RP n, E(V )) H 1 (F,Ṽ ) H1 (CP n, O(V )) 0 Example Γ(RP n, E( 2)) H 1 (F, Œ( 2)) Push-down to Gr 2 (R n+1 ) E p,q 1 = Γ(Gr 2 (R n+1 ),τ q Œ p η(ṽ )) = H (F,Ṽ ) Just like the Penrose transform!!

11 The 34th Czech Winter School in Geometry and Physics, Srní p. 11/13 Examples of the X-ray transform E( 2) Th m J 0 Γ(RP 3, E( 2)) X Γ(Gr 2 (R 4 ), Ẽ[ 1]) Γ(Gr 2 (R 4 ), Ẽ[ 3]) E( 3) 0 Γ(RP 3, E( 3)) X Γ(Gr 2 (R 4 ), ẼA [ 1]) D Γ(Gr 2 (R 4 ), ẼA[ 2]) Λ 1 0 R Γ(RP 3, Λ 0 ) Unique conformally covariant operator: Λ 0 Λ 4 cf. Jean-Louis Clerc s talks d Γ(RP 3, Λ 1 ) X Γ(Gr 2 (R 4 ), Λ 0 ) 2 Γ(Gr 2 (R 4 ), Λ 4 )

12 The 34th Czech Winter School in Geometry and Physics, Srní p. 12/13 X-ray kernels (Bailey-E 1997) Theorem The X-ray transform is injective on Γ(RP n, Λ 0 ( 2)) and on various other tensor fields: 0 R Γ(RP n, Λ 0 ) d Γ(RP n, Λ 1 ) X R. Michel (1978) Γ(RPn, Λ 1 (2)) Γ(RP n, 2 Λ 1 (2)) X R. Michel (1973) Killing operator Γ(RPn, 2 Λ 1 (4)) Γ(RP n, 3 Λ 1 (4)) X P. Estezet (1988) first BGG operator a b c a b c 0 Γ(RPn, ) a+1 a 2 a + b + 1 c X Γ(RP n, ) first BGG operator

13 The 34th Czech Winter School in Geometry and Physics, Srní p. 13/13 END OF PART TWO THANK YOU