AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 1/18 Representation theory and the X-ray transform Differential geometry on real and complex projective space Michael Eastwood Australian National University
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 2/18 Topics Connections Affine connections Levi-Civita connection Round sphere and real projective space Complex projective space Fubini-Study curvature Model embeddings Kähler form and model pullback Representation theory Tensors under model pullback Pullback of curvature et cetera
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 3/18 Connections E= smooth vector bundle (real or complex) Connection : E Λ 1 E Coupled de Rham E s.t. ( fσ)= f σ+d f σ Λ 1 E Λ 2 } {{ E} Λ n E 0 ω σ dω σ ω σ Curvature 2 : E Λ 2 E 2 ( fσ)= f 2 σ κ Γ(Λ 2 End(E)) Bianchi identity: 3 = 0 or κ=0
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 4/18 Induced connections If E and F are equipped with connections, then there are induced connections on the following bundles. E E E E E E E End(E)= E E E F Hom(E, F)= E F and so on... Leibniz rule, e.g. (σ τ)=( σ) τ+σ ( τ)
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 5/18 Existence and freedom, ˆ connections on E h + (1 h) ˆ also a connection. Therefore, partition of unity existence. Freedom:, ˆ connections on E ˆ = +Γ forγ : E Λ 1 E a homomorphism.
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 6/18 Affine connections Affine connection onλ 1 (or the tangent bundle) Problem: :Λ 1 Λ 1 Λ 1 Λ 2 may not agree with the exterior derivative! Solution: T d :Λ 1 Λ 2 Λ 1 Λ 1 is a homomorphism torsion Γ(Λ 1 End(Λ 1 )). ˆ T is a torsion free connection onλ 1. Consequence:Λ 1 Λ 1 ˆ Λ 2 Λ 1 is unambiguous.
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 7/18 Indices Covariant tensorsφ a,ψ ab,.... (Anti)-symmetrisation: φ [ab] 1 2 (φ ab φ ba ) φ (ab)c 1 2 (φ abc+φ bac ) Contravariant tensors, e.g. X a = vector field. E.g. torsion tensor T ab c s.t. T ab c = T [ab] c or T (ab) c = 0 Einstein summation convention: X ω X a ω a Curvature of a torsion-free affine connection ( a b b a )X c c = R ab d X d c ( a b b a )ω d = R ab d ω c
Levi-Civita connection Given g ab a metric,! affine a characterised by a is torsion-free a g bc = 0. Proof Choose any ˆ a torsion-free. Consider a φ b = ˆ a φ b Γ ab c φ c. Want: Γ abc =Γ (ab)c andγ a(bc) = 1 2 ˆ a g bc but Λ 1 Λ 2 Λ 2 Λ 1 NB K abc K [ab]c QED! K a[bc] AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 8/18
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 9/18 Round sphere Riemannian curvature tensor ( a b b a )ω d = R ab c d ω c R abcd = R [ab][cd] R [abc]d = 0 ( R abcd = R cdab ) = SO(n+1)/SO(n) R abcd = g ac g bd g bc g ad R ab R ca c b = (n 1)g ab constant curvature R R a a = n(n 1)
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 10/18 Real projective space antipodal identification = SO(n+1)/S(O(1) O(n)) R abcd = g ac g bd g bc g ad constant curvature i.e. ( b c c b )ω a = g ab ω c g ac ω b RP n ={linear L R n+1 s.t. dim L=1}
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 11/18 Complex projective space CP n RP n = {linear L C n+1 s.t. dim L=1} = SU(n+1)/S(U(1) U(n)) Fubini-Study metric g ab complex span CP n totally geodesic embedding J b a s.t. J b a J c c b = δ a complex structure (orthogonal) J ab ( J c a g bc ) Kähler form (skew) a J bc = 0 Fubini-Study curvature R abcd = g ac g bd g bc g ad + J ac J bd J bc J ad + 2J ab J cd NB: R ab c d J ce = 2J a(d g e)b 2J b(d g e)a + 2J ab g de
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 12/18 Model embeddings ι Recall totally geodesic embeddingrp n CP n Recall SU(n+1) acting oncp n by isometries Model embeddings:rp n µ CP n. J is type (1, 1) ι J= 0. Thereforeµ J= 0, i.e. model embeddings are Lagrangian. Conversely, Model embeddings though p CP n Linear Algebra Lagrangian subspaces of T p CP n ( Lagrangian Grassmannian U(n)/O(n) E.g. Helgason Geometric Analysis..., Exercise I.A.4(ii) )
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 13/18 Kähler form a J bc = 0 dj=0 J ab is non-degenerate } CP n is a symplectic manifold Letψbe a 2-form, i.e. with indicesψ ab =ψ [ab]. Then ψ ab = ψ ab 1 2n Jcd ψ cd J ab + 1 2n Jcd ψ cd J ab } {{ } J-trace-free ψ ab = ψ ab +θj ab ψ = ψ +θj Λ 2 = Λ 2 Λ 0 J Symplectic decomposition
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 14/18 Representation theory Λ 2 =Λ 2 Λ0 J (on any symplectic manifold) Let J be a non-degenerate skew 2n 2n real matrix. Sp(2n,R) {A=2n 2n matrix s.t. AJA t = J} Defining representation :R 2n Λ 2 R 2n =Λ 2 R 2n R 0 1 0 0 0 0 = 0 1 0 0 0 0 0 0 0 0 2n 1 nodes n nodes n nodes Branching for SL(2n, R) Sp(2n, R). (Induced vector bundles from co-frame bundle.)
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 15/18 Model pullback Recall model embeddingsµ :RP n CP n. Supposeψis a two-form oncp n. Lemma: µ ψ=0 µ ψ=θj ψ = 0. Proof: Fix attention on p CP n. ψ(p) Λ 2 p s.t.ψ(p) L = 0 Lagrangian L T p CP n NB: invariant under Sp(2n,R) acting on T p CP n. Recallµ J= 0. ψ s.t.µ ψ 0. Schur! Generalises to suitable tensors, using a b c d 0 0 0 = a b c d (case n=4).
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 16/18 Curvature pullback 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
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 17/18 Summary 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
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 18/18 THANK YOU END OF PART ONE