 # W H A T I S the Schwarzian Derivative English
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?W H A T I S . . .the Schwarzian Derivative? Valentin Ovsienko and Sergei Tabachnikov Almost every mathematician has encountered, at some point of his or her education, the following rather intimidating expression and, most likely, tried to forget it right away: (1) S (f (x)) = f ???(x) f ?(x) ? 3 2 ( f ??(x) f ?(x) )2 . Here f (x) is a function in one (real or complex) variable and f ?(x), f ??(x), ... are its derivatives. This is the celebrated Schwarzian derivative, or the Schwarzian, for short. It was discovered by Lagrange in his treatise “Sur la construction des cartes géographiques” (1781); the Schwarzian also appeared in a paper by Kummer (1836), and it was named after Schwarz by Cayley. Expression (1) is ubiquitous and tends to ap- pear in seemingly unrelated fields ofmathematics: classical complex analysis, differential equations, and one-dimensional dynamics, as well as, more recently, Teichmüller theory, integrable systems, and conformal field theory. Leaving these numer- ous applications aside, we focus on the basic properties of the Schwarzian itself. Two examples. a) The first example is perhaps the oldest one.

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• interesting multi-dimensional

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• projective differ

• ential geometry

• corresponding local

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I S . . . the Schwarzian Derivative? Valentin Ovsienko and Sergei Tabachnikov
Almost every mathematician has encountered, at some point of his or her education, the following rather intimidating expression and, most likely, tried to forget it right away:  ! 2 ′′′ ′′ f (x)3f (x) (1)S (f (x))= −. ′ ′ f (x)2f (x)
Heref (x)is a function in one (real or complex) ′ ′′ variable andf (x), f (x), ...are its derivatives. This is the celebrated Schwarzian derivative, or theSchwarzian, for short. It was discovered by Lagrange in his treatise “Sur la construction des cartes géographiques” (1781); the Schwarzian also appeared in a paper by Kummer (1836), and it was named after Schwarz by Cayley. Expression (1) is ubiquitous and tends to ap pear in seemingly unrelated ﬁelds of mathematics: classical complex analysis, diﬀerential equations, and onedimensional dynamics, as well as, more recently, Teichmüller theory, integrable systems, and conformal ﬁeld theory. Leaving these numer ous applications aside, we focus on the basic properties of the Schwarzian itself.
Two examples.a) The ﬁrst example is perhaps the oldest one. Consider the simplest secondorder diﬀerential equation, the SturmLiouville equation, ′′ (2)ϕ (x)+u(x) ϕ(x)=0
Valentin Ovsienko is researcher of Centre National de la Recherche Scientiﬁque at Université Claude Bernard Lyon 1. His email address isovsienko@math. univlyon1.fr. Sergei Tabachnikov is professor of mathematics at Pennsylvania State University, University Park. His email address istabachni@math.psu.edu. Partially supported by an NSF grant DMS0555803.
where the potentialu(x)is a (real or complex valued) smooth function. The space of solutions is twodimensional and spanned by any two lin early independent solutions,ϕ1andϕ2. Suppose that we know the quotientf (x)=ϕ1(x)/ϕ2(x); can one reconstruct the potential? The reader can carry out the relevant computations to check 1 thatu=S(f ).The geometrical meaning of this 2 formula is as follows. The quotientt=ϕ12 1 is anaﬃne coordinateon the projective lineP 1 so thatt=f (x)is a parametrized curve inP. This curve has nonvanishing speed, i.e.,f0, since the Wronski determinant of two solutions of (2) is a nonzero constant. The Schwarzian then reconstructs a SturmLiouville equation from such a curve. b) The next example is due to C. Duval, L. Guieu, and the ﬁrst author (2000). Consider the Lorentz plane with the metric g=dxdyand a curve y=f (x). If>f (x) 0, then its Lorentz curvature ′′ ′ 3/2 can be easily computed:̺(x)=(f (x))f (x) , and the Schwarzian enters the game when one com p ′ ′ putes̺=S(f )/ f .Thus, informally speaking, the Schwarzian derivative is curvature. The following beautiful theorem of E. Ghys (1995) is a manifestation of this principle:for an ar bitrary diﬀeomorphismfof the real projective line, its Schwarzian derivativeS(f )vanishes at least at 4 distinct points.Ghys’ theorem is analogous to the classical 4 vertex theorem of Mukhopadhyaya (1909):the Euclidean curvature of a smooth closed 2 convex curve inRhas at least 4 distinct extrema. Not a function.A surprise hidden in formula (1) is that the Schwarzian is actually not a func tion. The diﬀerence between a function and a more complicated tensor ﬁeld is in its behavior
Notices of the AMS
Volume56, Number1
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