NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY y Michael Eastwoo d These notes are in no way meant to b e comprehensive neither in treatment nor in references to the extensive literature They are merely meant as an introduction to a small selection of topics in the eld They were presented as a series of four lectures at th the Winter School on Geometry and Physics SrnCzech Republic January Lecture One Recall that a Riemannian manifold M is really a pair M g consisting of a smo oth manifold M and a metric g a smo oth and everywhere p ositive denite section of J T M the symmetric tensor pro duct of the cotangent bundle Following standard practise we shall usually write g instead of g and more generally we shall adorn ab tensors with upp er and lower indices in corresp ondence with the tangent or cotangent bundle We shall also use the Einstein summation convention to denote the natural a pairing of vectors and covectors Thus V denotes a tangent vector or a vector eld a b and g V V denotes the square of its length with resp ect to g We shall often raise ab a b and lower indices without commentif V is a vector eld then V g V is the a ab corresp onding form More precisely this is Penroses abstract index notation see for details A conformal manifold M is a pair M g where g is a Riemannian metric dened J only up to scale In other words g is a section of R T M the bundle of rays J in T M such that a representative g is p ositive denite In yet other words a conformal manifold is an equivalence class of Riemannian manifolds where two b b metrics g and g are said to b e equivalent if g is a multiple of g In this case ab ab ab ab b it is convenient to write g g for some smo oth function On a conformal ab ab manifold one can measure angles b etween vectors but not lengths This pap er is in nal form and no version of it will b e submitted for publication elsewhere y ARC Senior Research Fellow University of Adelaide MICHAEL EASTWOOD For example in two dimensions an oriented conformal manifold is precisely a Riemann surface ie a onedimensional complex manifold Certainly if M is a Riemann surface with lo cal coordinate z x iy then dx dy is a Riemannian metric and if w u iv is another then the Cauchy Riemann equations u v v u x y y x imply u v v u dx dy dx dy du dv x y x y u v dx dy x x Conversely there is a theorem of Korn and Lichtenstein which says that any Rieman nian metric in two dimensions may b e written lo cally in the form dx dy for some smo oth function and suitable coordinates x y chosen compatible with the orientation Taking z x iy denes a lo cal complex coordinate and any other choice is holomorphically related Indeed the Cauchy Riemann equations are precisely that the Jacobian matrix u x u y v x v y is prop ortional to an orthogonal one Notice that a conformal manifold in two dimensions has no lo cal invariantsall Riemann surfaces are lo cally indistinguishable The only lo cal invariant of a two dimensional Riemannian manifold is its scalar curvature and this has b een eliminated by the freedom to scale the metric In higher dimensions it is reasonable to exp ect some of the rigidity of Riemannian geometry This is indeed the case and evidence in its favour can b e found as follows n Firstly let is consider rigid motions of R ie the connected comp onent of the group n of isometries of R with its usual metric Of course this is wellknown to b e generated by translations and rotations One way of seeing this is to consider an innitesimal a n motion ie a vector eld V on R with the prop erty that a V a son ie V V V ab ab b b x where square brackets around indices denote taking the skew part Consider now a V a son ie V V and V V V abc abc abc abc bc b c x x NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY where round brackets denote the symmetric part However this rst prolongation son vanishes V V V V V V V abc bac bca cba cab acb abc By integration the result concerning rigid motions follows There are some observa tions to b e made The calculation is the principal ingredient in constructing the LeviCivita connection on a Riemannian manifold n The at mo del of Riemannian geometry is R as a homogeneous space n R o SOn rigid motions n R SOn stabiliser of a p oint As an alternative pro of we could show that geo desics ie straight lines with unit sp eed parameterisation are preserved by isometries and by ring out geo desics conclude that an isometry xing a p oint to rst order is necessarily the identity Let us try the same technique for the conformal case in an attempt to identify the n conformal motions of R An innitesimal motion is a vector eld satisfying a V a con ie V g V ab ab b b x Thus a V con ie V g and V V bc abc c ab abc abc This rst prolongation con is no longer zero In fact V g g g a abc c ab b ac a bc n identies con with R However a con ie V g and V V V bcd cd ab abcd abcd abcd and this second prolongation con vanishes if the dimension n is greater than ab ab ab n g V g V g V V cd abcd acbd cabd acbd ab ab g g V g V bd ac cdab cd cdab So cd cd ab cd cd ng g g g g ng cd cd ab cd cd and is tracefree Since n then cd ab n g V cd cdab MICHAEL EASTWOOD implies that the right hand side is symmetric in cd in which case ab ab n g V g g cd ab cd cdab which implies that is pure trace Now V and we nd that V is zero by cd abcd abcd applying on the rst three indices n We may conclude that a conformal motion of R for n which xes the origin to second order is necessarily the identity More precisely we may identify the group P of conformal motions xing the identity as matrices of the form a a n a a for m SOn r R r m b b a a r r r m a a b a n acting on x R by a b c a m x x x r b c a x a b c b r m x x x r r a b c b It is a go o d exercise to check that this really is conformal ie that its derivative is everywhere prop ortional to an orthogonal matrix Notice that the denominator b b c a vanishes when x m r r r On the other hand this formula is forced by the c a prolongation argument In order to allow nonzero r we are therefore obliged to n n n compactify R with a single p oint to obtain the sphere S R fg Further reasoning along these lines identies the full group of conformal motions of the round n n sphere S conformally containing Euclidean R via stereographic pro jection as the identity connected comp onent G of SOn The sphere is realised as the space of future p ointing null rays e n space S of generators n in R The form of P as ab ove is obtained by taking G to preserve the form x x x x x n n n and the basep oint of S to b e represented by the null vector e NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY In summary the at mo del of conformal geometry is n S GP for G SO n and P a suitable parabolic subgroup The other two observations in the Riemannian case will turn out to have analogues in the conformal case We shall nd an invariant connection but not on the tangent bundle In fact we shall nd a somewhat b etter dierential op erator originally found by TY Thomas We shall also nd analogues of geo desics in the conformal case They are known as n conformal circles b ecause in the at case they coincide with the round circles on S equipp ed with their standard pro jective parameterisations Certainly these circles are preserved by G Lecture Two So much for the at case To pro ceed in general the nave approach is to work with a metric in the conformal class and then see how things change when the metric b is scaled If g is replaced by g g then the LeviCivita connection r is ab ab ab a c replaced by the connection r acting on forms according to a c c r r g a b a b a b b a c ab where r Indeed this formula surely denes a torsionfree connection a a and induces d d c r r g g a bc a bc a bc b ac c ba dc ab bd ac b on contravariant tensors clearly annihilating g Corresp ondingly the new ab ab connection on vector elds is b b b b c b c r V r V V V V a a a a c a b where is the Kronecker delta It is convenient to introduce a line bundle E on a M as follows If a metric g in the conformal class is chosen then E is identied ab with the trivial bundle E Equivalently a lo cal section of E may b e regarded as a b b function say f If however g is replaced by g g then the function f repres ab ab ab th b b enting with resp ect to the metric g is given by f f The w p ower of E will ab b e denoted by E w and its sections called conformally weighted functions of weight w w b Such a section may b e represented by a function scaling according to f f We a shall write E and E for the tangent and cotangent bundle resp ectively Other tensor a bundles will b e denoted by adorning E with the corresp onding indices The conformal metric may b e regarded as an invariantly dened section of E Raising and lower ab a ing indices denes a canonical isomorphism of E w with E w for any weight w a Cho osing a metric induces a connection on E w which transforms according to c r r w a a a MICHAEL EASTWOOD under scaling of the metric The Riemann curvature is dened by c c d r r r r V R V a b b a ab d and transforms by b R R g g g g abcd abcd ac bd bc ad bd ac ad bc under scaling of the metric where c r g ab a b a b c ab In particular the variation is entirely through traces The totally