First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 1/13 Classification problems in conformal geometry Introduction to conformal differential geometry Michael Eastwood Australian National University
First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 2/13 Motivation from physics GR: null geodesics are conformally invariant Maxwell s equations are conformally invariant g ab = (pseudo-)metric on M, a smooth n-manifold ĝ ab =Ω 2 g ab = conformally related metric (angles OK) g 2 : T M R dg 2 X g 2 geodesic spray ĝ 2 =Ω 2 g 2 dĝ 2 =Ω 2 dg 2 + g 2 dω 2 Xĝ2 g=0 X g 2 g=0 g ab ǫ ab de volume form (e.g.ǫ ab de ǫ ab de = n!) ĝ ab =Ω 2 g ab = ˆǫ ab de =Ω n ǫ ab de ˆǫ cd ab =ǫ cd ab when n=4 F ab F ab ǫ cd ab F cd is invariant df= 0 d F= 0
First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 3/13 Motivation from geometry S n stereographic projection R n conformal R n x 1 trough 4x x 2 +4 S n x 2 4
First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 4/13 Motivation from navigation Mercator (Cartographer) 1569 Wright (Mathematician) 1599 Jac= S 2 \{poles} stereographic R 2 \{0}=C\{0} log C u x v x u y v y = c -s s c u x = v y Cauchyv x = -u y Riemann
First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 5/13 Euclidean symmetries X=vector field X=X 1 x 1 + X 2 x 2 + +X n x n = X a a Infinitesimal Euclidean symmetry: }{{} L X δ ab = 0. Lie derivative Compute L X δ ab = X c c δ ab +δ cb a X c +δ ac b X c = a X b + b X a L X δ ab = 0 (a X b) = 0 Killing field
First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 6/13 Killing fields by prolongation Killing operator: X a (a X b) Kernel in flat space: K ab a X b is skew. Claim: a K bc = 0. a K bc = c K ba b K ca = c b X a b c X a = 0, as required. Hence, (a X b) = 0 ax b = K ab a K bc = 0 Closed! Conclusion: X a = s a + m ab x b where m ab = m ba. translations rotations
First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 7/13 Conformal symmetries trace-free part (a X b) = 0 conformal Killing field Rewrite as a X b = K ab +Λδ ab where K ab is skew. a K bc = c K ba b K ca = c b X a b c X a δ ab c Λ+δ ac b Λ so a K bc =δ ab Q c δ ac Q b where a Λ= Q c but 0 = δ ab ( d a K bc a d K bc ) = δ ab (δ ab d Q c δ ac d Q b δ db a Q c +δ dc a Q b ) = (n 2) d Q c +δ dc a Q a whence a Q b = 0 if n 3 Closed!!
First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 8/13 Conformal symmetries cont d Solve a X b = K ab +Λδ ab a K bc = δ ab Q c δ ac Q b a Λ = Q c a Q b = 0 Q b = r b Λ=λ+r b x b K bc = r b x c r c x b m bc X a = s a + m ab x b +λx a + r b x b x a 1 2 r ax b x b translation + rotation + dilation + inversion Integrate the inversions x a x a 1 2 ra x 2 1 r a x a + 1 4 r 2 x 2
First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 9/13 Conformal group SO(n+1, 1) acts on S n by conformal transformations semisimple S n = SO(n+1, 1)/P flat model of conformal differential geometry R n = (SO(n) R n )/SO(n) flat model of Riemannian differential geometry {generators} S n parabolic
First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 10/13 A simple question onr n, n 3 Question: Which linear differential operators preserve harmonic functions? Answer onr 3 : Zeroth order f constant f 1 First order 1 = / x 1 2 = / x 2 3 = / x 3 3 x 1 2 x 2 1 &c. 3 x 1 1 + x 2 2 + x 3 3 +1/2 1 (x 2 1 x 2 2 x 2 3 ) 1 + 2x 1 x 2 2 + 2x 1 x 3 3 + x 1 3 &c. Dimensions............................................... 10 [D 1,D 2 ] D 1 D 2 D 2 D 1 Lie Algebra so(4, 1)= conformal algebra NB!
Surroundings First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 11/13
First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 12/13 Next time What about higher order operators preserving harmonic functions? (Beyond first order) What about a classification of conformally invariant operators? (Beyond Maxwell) Further Reading C.P. Boyer, E.G. Kalnins, and W. Miller, Jr., Symmetry and separation of variables for the Helmholtz and Laplace equations, Nagoya Math. Jour. 60 (1976) 35 80. M.G. Eastwood, Notes on conformal differential geometry, Suppl. Rendi. Circ. Mat. Palermo 43 (1996) 57 76. R. Penrose and W. Rindler, Spinors and space-time, vols 1 and 2, Cambridge University Press 1984 and 1986.
First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 13/13 THANK YOU END OF PART ONE