Conformal differential geometry

and its interaction with representation theory

Introduction to conformaldifferential geometry

Michael Eastwood

Australian National University

Spring Lecture One at the University of Arkansas – p. 1/13

Motivation from physics• GR: null geodesics are conformally invariant• Maxwell’s equations are conformally invariant

gab = (pseudo-)metric onM, a smoothn-manifoldgab = Ω

2gab = conformally related metric(angles OK)

g2 : T ∗M → R dg2 Xg2 geodesic spray

g2 = Ω2g2 dg2 = Ω2dg2 + g2dΩ2

∴ Xg2|g=0 ∝ Xg2|g=0

gab ǫab···de volume form (e.g.ǫab···deǫab···de = n!)∴ gab = Ω

2gab =⇒ ǫab···de = Ωnǫab···de

∴ ǫabcd = ǫab

cd whenn = 4∴ Fab 7→ ∗Fab ≡ ǫab

cdFcd is invariantdF = 0d∗F = 0

Spring Lecture One at the University of Arkansas – p. 2/13

Motivation from geometry

stereographic projectionS n

Rn

SS

SS

SS

SS

SS•

•

BB

BBB

conformal

QQ

QQ

QQ

QQ

QQ

QQ

Q

•

•

trough

Rn ∋ x 7→ 1

‖x‖2+4

4x‖x‖2 − 4

∈ S n

Spring Lecture One at the University of Arkansas – p. 3/13

Motivation from navigation•Mercator(Cartographer) 1569

•Wright (Mathematician) 1599

S 2 \ polesstereographic−−−−−−−−−→ R2 \ 0 = C \ 0

log−−→ C

Jac=

ux uy

vx vy

=

c -s

s c

⇐⇒

ux = vy

vx = -uy

Cauchy-Riemann

Spring Lecture One at the University of Arkansas – p. 4/13

Euclidean symmetriesX =vector field

X = X1 ∂

∂x1+ X2 ∂

∂x2+ · · · + Xn ∂

∂xn= Xa∇a

Infinitesimal Euclidean symmetry: LXδab︸︷︷︸

Lie derivative

= 0.

Compute

LXδab = Xc∇cδab + δcb∇aXc + δac∇bXc

= ∇aXb + ∇bXa

∴ LXδab = 0 ⇐⇒ ∇(aXb) = 0 Killing field

Spring Lecture One at the University of Arkansas – p. 5/13

Killing fields by prolongationKilling operator: Xa 7→ ∇(aXb)

Kernelin flat space:Kab ≡ ∇aXb is skew.

Claim: ∇aKbc = 0. ∇aKbc = ∇cKba − ∇bKca

= ∇c∇bXa − ∇b∇cXa

= 0, as required.

Hence,∇(aXb) = 0 ⇐⇒∇aXb = Kab

∇aKbc = 0Closed!

Conclusion: Xa = sa + mabxb wheremab = −mba.

translations

6

rotations

6

Spring Lecture One at the University of Arkansas – p. 6/13

Conformal symmetries

trace-free part∇(aXb) = 0 conformal Killing field

Rewrite as ∇aXb = Kab + Λδab whereKab is skew.

∇aKbc = ∇cKba − ∇bKca

= ∇c∇bXa − ∇b∇cXa − δab∇cΛ + δac∇bΛ so

∇aKbc = δabQc − δacQb where ∇aΛ = −Qc but

0 = δab(∇d∇aKbc − ∇a∇dKbc)= δab(δab∇dQc − δac∇dQb − δdb∇aQc + δdc∇aQb)= (n − 2)∇dQc + δdc∇

aQa whence

∇aQb = 0 if n ≥ 3 Closed!!

Spring Lecture One at the University of Arkansas – p. 7/13

Conformal symmetries cont’d

Solve ∇aXb = Kab + Λδab

∇aKbc = δabQc − δacQb

∇aΛ = −Qc

∇aQb = 0

Qb = −rb Λ = λ + rbxb Kbc = rbxc − rcxb − mbc

Xa = sa + mabxb + λxa + rbxbxa −12raxbxb

translation+ rotation+ dilation+ inversion

Integrate the inversions

xa 7−→xa − 1

2ra‖x‖2

1− raxa + 14‖r‖

2‖x‖2

Spring Lecture One at the University of Arkansas – p. 8/13

Conformal group

SO(n + 1,1) acts onS n

by conformal transformations

AA

AA

AA

AA

AA S n

CCCCCCCCCCCO generators

S n = SO(n + 1,1)/P

flat model of conformal differential geometry

Rn = (SO(n) ⋉ Rn)/SO(n)

flat model of Riemannian differential geometry

semisimpleHHj parabolic

Spring Lecture One at the University of Arkansas – p. 9/13

A simple question onRn, n ≥ 3Question: Which linear differential operators preserveharmonic functions? Answer onR3:–Zeroth order f 7→ constant× fFirst order∇1 = ∂/∂x1 ∇2 = ∂/∂x2 ∇3 = ∂/∂x3

x1∇2 − x2∇1 &c.x1∇1 + x2∇2 + x3∇3 +1/2

(x12 − x2

2 − x32)∇1 + 2x1x2∇2 + 2x1x3∇3 + x1

&c.Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3313

10

1

[D1,D2] ≡ D1D2 −D2D1

Lie Algebra so(4,1) = conformal algebra← NB!

Spring Lecture One at the University of Arkansas – p. 10/13

Surroundings

Spring Lecture One at the University of Arkansas – p. 11/13

Next four talks• What about conformally invariant operators?

(Beyond Maxwell)• What about higher order operators preserving

harmonic functions? (Beyond first order)• Bateman’s formula and twistor theory• The X-ray transform

Further Reading• M.G. Eastwood, Notes on conformal differential geometry,

Suppl. Rendi. Circ. Mat. Palermo43 (1996) 57–76.

• R. Penrose and W. Rindler, Spinors and space-time, vols 1

and 2, Cambridge University Press 1984 and 1986.

Spring Lecture One at the University of Arkansas – p. 12/13

THANK YOU

END OF PART ONE

Spring Lecture One at the University of Arkansas – p. 13/13