WHAT IS... ? the Schwarzian Derivative? Valentin Ovsienko and Sergei Tabachnikov

Almost every mathematician has encountered, at where the potential u(x) is a (real or complex some point of his or her education, the following valued) smooth function. The space of solutions rather intimidating expression and, most likely, is two-dimensional and spanned by any two lin-

tried to forget it right away: early independent solutions, ϕ1 and ϕ2. Suppose 2 that we know the quotient f (x) = ϕ1(x)/ϕ2(x); f ′′′(x) 3 f ′′(x) (1) S (f (x)) = − . can one reconstruct the potential? The reader f ′(x) 2 f ′(x) ! can carry out the relevant computations to check that u = 1 S(f ). The geometrical meaning of this Here f (x) is a function in one (real or complex) 2 variable and f ′(x), f ′′(x), ... are its derivatives. formula is as follows. The quotient t = ϕ1/ϕ2 P1 This is the celebrated Schwarzian derivative, or is an affine coordinate on the projective line P1 the Schwarzian, for short. It was discovered by so that t = f (x) is a parametrized curve in . ′ ≠ Lagrange in his treatise “Sur la construction des This curve has non-vanishing speed, i.e., f 0, cartes géographiques” (1781); the Schwarzian also since the Wronski determinant of two solutions appeared in a paper by Kummer (1836), and it was of (2) is a non-zero constant. The Schwarzian then named after Schwarz by Cayley. reconstructs a Sturm-Liouville equation from such Expression (1) is ubiquitous and tends to ap- a curve. pear in seemingly unrelated fields of mathematics: b) The next example is due to C. Duval, L. Guieu, classical complex analysis, differential equations, and the first author (2000). Consider the Lorentz and one-dimensional dynamics, as well as, more plane with the metric g = dxdy and a curve ′ recently, Teichmüller theory, integrable systems, y = f (x). If f (x) > 0, then its Lorentz curvature ′′ ′ −3/2 and conformal field theory. Leaving these numer- can be easily computed: ̺(x) = f (x) (f (x)) , ous applications aside, we focus on the basic and the Schwarzian enters the game when one com- ′ properties of the Schwarzian itself. putes ̺ = S(f )/ f ′. Thus, informally speaking, the Schwarzian derivative is curvature. Two examples. a) The first example is perhaps p The following beautiful theorem of E. Ghys the oldest one. Consider the simplestsecond-order (1995) is a manifestation of this principle: for an ar- differential equation, the Sturm-Liouville equation, bitrary diffeomorphism f of the real projective line, (2) ϕ′′(x) + u(x) ϕ(x) = 0 its Schwarzian derivative S(f ) vanishes at least at 4 distinct points. Ghys’ theorem is analogous to Valentin Ovsienko is researcher of Centre National the classical 4 vertex theorem of Mukhopadhyaya de la Recherche Scientifique at Université Claude (1909): the Euclidean curvature of a smooth closed Bernard Lyon 1. His email address is ovsienko@math. convex curve in R2 has at least 4 distinct extrema. univ-lyon1.fr. Not a function. A surprise hidden in formula Sergei Tabachnikov is professor of mathematics at Pennsylvania State University, University Park. His email (1) is that the Schwarzian is actually not a func- address is [email protected]. tion. The difference between a function and a Partially supported by an NSF grant DMS-0555803. more complicated tensor field is in its behavior

2 Notices of the AMS Volume 56, Number 1 under coordinate changes. Choose another coor- homogeneous of degree 2 in v, and therefore S(f ) dinate y. What is the formula for S(f )(y)? For is indeed a quadratic differential. a function the answer is simply f (y) = f (x(y)), Schwarzian as a cocycle. Let G be a group but for its derivative, due to the Chain Rule, it is acting on a vector space V , i.e., there is a homo- ′ ′ ′ different: f (y) = f (x(y)) x (y); this is why the morphism ρ : G → End(V). A map c : G → V is a invariant (geometric) quantity is not the derivative 1-cocycle on G with coefficients in V if but the differential df = f ′(x) dx—a distinction ρ : (v, λ) ֏ (ρ v + λ c(g), λ) that tortures countless calculus students. The g g geometric quantity corresponding to (1) is the is again a G-action on V ⊕ R. The cocycle c is non- quadratic differential: S(f ) = S (f (x)) (dx)2. In trivial if thise action is not isomorphic to that with other words, S(f ) is a quadratic function on the c = 0. In this case, c defines a class of cohomology tangent space T R, just like a metric but without the of G, the notion that plays a fundamental role in 1 non-vanishing condition. We denote by Q2(RP ) geometry, algebra, and topology. the space of the quadratic differentials on the real Diffeomorphisms of RP1 form an infinite- projective line RP1. dimensional group, Diff(RP1), which acts on all 1 Main properties. 1. S(f ) = S(g) if and only tensor fields on RP . Property 2 means precisely if g(x) = (af (x) + b)/(cf (x) + d), where a,b,c,d that the Schwarzian is a 1-cocycle with coef- are (real or complex) constants with ad − bc ≠ 0. ficients in the space of quadratic differentials RP1 In particular, S(f ) = 0 if and only if f is a Q2( ). Moreover, one can prove uniqueness: linear-fractional transformation: the Schwarzian is the only projectively invariant RP1 ax + b 1-cocycle on Diff( ). This serves as a good (3) f (x) = . intrinsic definition. cx + d The algebra of vector fields, Vect(RP1), is the Note that f (−d/c) = ∞, but f (x) is well-defined Lie algebra of the group Diff(RP1). Every differen- for x = ∞. We can understand f as a diffeomor- tiable map on Diff(RP1) corresponds to a map on RP1 phism of . The transformations (3) with real Vect(RP1), its infinitesimal version. The infinites- coefficients form a group of projective symmetries imal version of the Schwarzian is easy to compute RP1 R of , which is SL(2, )/{±1}. It follows that the (substitute f (x) = x + ε X(x) in (1) and differen- Schwarzian is a projective invariant. tiate with respect to ε at ε = 0): s(X(x) d/dx) = RP1 2. Given two diffeomorphisms f , g of , one X′′′(x) (dx)2. This is a projectively invariant 1- has: S g ◦ f = S(g)◦f +S(f ), where the first sum- 1 1 cocycle on Vect(RP ) with coefficients in Q2(RP ). mand is the
action of f on a quadratic differential, ′ 2 Moreover, the invariant pairing between quadratic (u ◦ f )(x) = u(f (x)) (f (x)) . differentials and vector fields yields a 2-cocycle on From discrete projective invariants to differ- Vect(RP1) with trivial coefficients: ential ones. The reader may be familiar with d d projective invariants; see F. Labourie’s article [1]. ω X(x) , Y (x) = X′′′(x) Y (x) dx. dx dx RP1 Recall that a quadruple of points in P1 has a nu-   Z merical invariant. Choose an affine coordinate that This is the Gelfand-Fuchs cocycle (1967); it defines a central extension of Vect(RP1) called the Vira- represents the points by four numbers t1,t2,t3, and soro algebra, perhaps the most famous infinite- t4; the cross-ratio dimensional Lie algebra, defined on the space (t1 − t3)(t2 − t4) RP1 R [t1,t2,t3,t4] = Vect( ) ⊕ by the commutator (t1 − t2)(t3 − t4) [(X, α), (Y , β)] = ([X, Y ], ω(X, Y )) , is invariant under the projective transformations of the projective line. What is the relation of this where [X, Y ] is the commutator of vector fields. discrete invariant to the Schwarzian? The Schwarzian contains complete information Consider a diffeomorphism f of RP1. The about the Gelfand-Fuchs cocycle; for instance, the Schwarzian measures how f changes the cross- 2-cocycle condition follows from Property 2. The ratio of infinitesimally close points. Let t be a point relations between the Schwarzian and the Vira- in RP1 and v a tangent vector to RP1 at t. Extend soro algebra were discovered, independently, by v to a vector field in a vicinity of t and denote A. Kirillov and G. Segal (1980). by φs the corresponding local flow. Consider four Multi-dimensional versions of the Schwarzian. points: t,t1 = φε(t), t2 = φ2ε(t), t3 = φ3ε(t). The Here is a “universal method” of discovering a cross-ratio does not change in the first order in ε: multidimensional Schwarzian: a) choose a group of diffeomorphisms and a [f (t), f (t1), f (t2), f (t3)] = subgroup G that has a nice geometrical meaning, 2 3 [t,t1,t2,t3] − 2ε S(f )(t) + O(ε ). b) find a G-invariant 1-cocycle on the group of The coefficient of ε2 depends on the diffeomor- diffeomorphisms, phism f , the point t, and the tangent vector v, c) (the most important step) check that no one did but not on its extension to a vector field. It is it before.

January 2009 Notices of the AMS 3 One of the most interesting multi-dimensional Among other generalizations, let us mention the Schwarzians is that of Osgood and Stowe (1992). “Lagrangian Schwarzian” modeled on symmetric Consider a Riemannian surface (M, g) and the matrices, Ovsienko (1989); a more general non- group Diffc (M) of all conformal transformations commutative Schwarzian of Retakh and Shander of M. The Riemann uniformization theorem im- (1993); and various generalized Schwarzians with plies that M is conformally flat. One can (locally) coefficients in the space of differential operators. ∗ express the metric as g = (1/F)ψ g0, where ψ is Last but not least, the Schwarzian derivative a conformal diffeomorphism of M, F is a smooth plays a key role in Teichmüller theory, namely, function, and g0 is a metric of constant curvature. the Bers embedding of the Teichmüller space of The conformal Schwarzian is given by a Riemann surface into an appropriate complex space. However, this vast topic deserves a separate ∇dF 3 dF ⊗ dF 1 g−1(dF,dF) g treatment. S(ψ) = − + , F 2 F 2 4 F 2 Further Reading where ∇ is the covariant derivative corresponding [1] F. Labourie, What is...a cross-ratio?, Notices 55 to the Levi-Civita connection. This is a 1-cocycle (2008), No 10. (November), 1234–1235. on Diffc (M), invariant with respect to the (local) [2] V. Ovsienko and S. Tabachnikov, Projective Differ- Möbius subgroup SO(3, 1) associated to the metric ential Geometry Old and New. From the Schwarzian g0. The construction also makes sense if dim M > Derivative to the Cohomology of Diffeomorphism 2, but the conformal group is finite-dimensional Groups, Cambridge University Press, Cambridge, in this case. 2005.

Lagrange and the Schwarzian Derivative He studies the maps given by a conformal map- The name “Schwarzian de- ping from a spherical region to the plane that rivative” was coined by takes all the meridians and all the parallels to Cayley, but he points arcs of circles (as does stereographic projection). out that Schwarz him- This is equivalent to describing local conformal self says that it occurs mappings z ֏ f (z) for which the image of each already, at least implicit- horizontal and each vertical line is a circular arc. ly, in Lagrange’s essay on He proves that the conditions on horizontals and cartes géographiques. Fe- verticals are equivalent, and that both are equiv- lix Klein learned about alent, in contemporary notation, to the equation Lagrange’s work through ImS(f ) = 0. This implies in turn that S(f ) = const. a private communication He solves this equation and explicitly describes its from Schwarz (noted in solutions. The Schwarzian derivative S(f ) appears §III.5 of Lectures on the in his paper as φ′′/φ where φ = 1/ f ′), which Icosahedron). It is not quite excuses Sylvester to some extent for missingp it. straightforward, however, All this was found again much later, but appar- in reading Lagrange, to see ently quite independently of Lagrange’s original what he is doing, and it is work. What is sometimes called Arnold’s Law as- not clear to what extent lat- serts, “Discoveries are rarely attributed to the er mathematicians went to correct person.” One might add to this (Michael) the original work. Joseph Sylvester, in “Method Berry’s Law, prompted by the observation that the of reciprocants” records that he tried to track sequence of antecedents under the previous law down Lagrange’s use of the Schwarzian, but then seems endless: “Nothing is ever discovered for the only concludes that “There are two papers by first time.” Lagrange … but I have not been able to discover Lagrange’s article on cartes géographiques is in the Schwarzian derivative in either one of them.” Volume IV of his collected works. This is not, un- Even in modern times Lagrange’s role has been fortunately, available at http:gallica.bnf.fr missed—for example, George Heine in an article as are Volumes II and VIII, but we have made a in the recent book Euler at 300: An Appreciation scan of it available at says that Lagrange’s work had little influence on http:www.math.ubc.ca/˜cass/cartes.pdf. either mathematics or cartography. It is this volume, incidentally, that contains as Schwarz was, however, correct—Lagrange did frontispiece the well known portrait of Lagrange introduce some version of the Schwarzian de- reproduced here. rivative S(f ), and for an interesting purpose. He considers the Earth as a general body of revolu- —Bill Casselman, Valentin Ovsienko, tion, taking into account the known non-sphericity. and Sergei Tabachnikov

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