Schwarzian derivative
In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces. It is named after the German mathematician Hermann Schwarz. Definition
The Schwarzian derivative of a holomorphic function  of one complex variable  is defined by

The same formula also defines the Schwarzian derivative of a  function of one real variable. The alternative notation

is frequently used. Properties
The Schwarzian derivative of any Möbius transformation

 

is zero. Conversely, the Möbius transformations are the only functions with this property. Thus, the Schwarzian derivative precisely measures the degree to which a function fails to be a Möbius transformation. If  is a Möbius transformation, then the composition  has the same Schwarzian derivative as ; and on the other hand, the Schwarzian derivative of  is given by the chain rule

 

More generally, for any sufficiently differentiable functions  and 

 

When  and  are smooth real-valued functions, this implies that all iterations of a function with negative (or positive) Schwarzian will remain negative (resp. positive), a fact of use in the study of one-dimensional dynamics. Introducing the function of two complex variables

its second mixed partial derivative is given by

and the Schwarzian derivative is given by the formula:

The Schwarzian derivative has a simple inversion formula, exchanging the dependent and the independent variables. One has

or more explicitly, . This follows from the chain rule above. Geometric interpretation 
William Thurston interprets the Schwarzian derivative as a measure of how much a conformal map deviates from a Möbius transformation. Let  be a conformal mapping in a neighborhood of  Then there exists a unique Möbius transformation  such that  has the same 0, 1, 2-th order derivatives at 

Now  To explicitly solve for  it suffices to solve the case of  Let  and solve for the  that make the first three coefficients of  equal to  Plugging it into the fourth coefficient, we get . After a translation, rotation, and scaling of the complex plane,  in a neighborhood of zero. Up to third order this function maps the circle of radius  to the parametric curve defined by  where  This curve is, up to fourth order, an ellipse with semiaxes  and 

as 

Since Möbius transformations always map circles to circles or lines, the eccentricity measures the deviation of  from a Möbius transform. Differential equation
Consider the linear second-order ordinary differential equation

where  is a real-valued function of a real parameter . Let  denote the two-dimensional space of solutions. For , let  be the evaluation functional . The map  gives, for each point  of the domain of , a one-dimensional linear subspace of . That is, the kernel defines a mapping from the real line to the real projective line. The Schwarzian of this mapping is well-defined, and in fact is equal to  . Owing to this interpretation of the Schwarzian, if two diffeomorphisms of a common open interval into  have the same Schwarzian, then they are (locally) related by an element of the general linear group acting on the two-dimensional vector space of solutions to the same differential equation, i.e., a fractional linear transformation of . Alternatively, consider the second-order linear ordinary differential equation in the complex plane

Let  and  be two linearly independent holomorphic solutions. Then the ratio  satisfies

over the domain on which  and   are defined, and  The converse is also true: if such a  exists, and it is holomorphic on a simply connected domain, then two solutions  and  can be found, and furthermore, these are unique up to a common scale factor. When a linear second-order ordinary differential equation can be brought into the above form, the resulting  is sometimes called the Q-value of the equation. Note that the Gaussian hypergeometric differential equation can be brought into the above form, and thus pairs of solutions to the hypergeometric equation are related in this way. Conditions for univalence
If  is a holomorphic function on the unit disc, , then W. Kraus (1932) and Nehari (1949) proved that a necessary condition for  to be univalent is

Conversely if  is a holomorphic function on  satisfying

then Nehari proved that  is univalent. In particular a sufficient condition for univalence is

Conformal mapping of circular arc polygons

The Schwarzian derivative and associated second-order ordinary differential equation can be used to determine the Riemann mapping between the upper half-plane or unit circle and any bounded polygon in the complex plane, the edges of which are circular arcs or straight lines. For polygons with straight edges, this reduces to the Schwarz–Christoffel mapping, which can be derived directly without using the Schwarzian derivative. The accessory parameters that arise as constants of integration are related to the eigenvalues of the second-order differential equation. Already in 1890 Felix Klein had studied the case of quadrilaterals in terms of the Lamé differential equation. Let  be a circular arc polygon with angles  in clockwise order. Let  be a holomorphic map extending continuously to a map between the boundaries. Let the vertices correspond to points  on the real axis. Then  is real-valued for  real and not one of the points. By the Schwarz reflection principle  extends to a rational function on the complex plane with a double pole at :

The real numbers  are called accessory parameters. They are subject to three linear constraints:

which correspond to the vanishing of the coefficients of  and  in the expansion of  around . The mapping  can then be written as

where  and  are linearly independent holomorphic solutions of the linear second-order ordinary differential equation

There are  linearly independent accessory parameters, which can be difficult to determine in practise. For a triangle, when , there are no accessory parameters. The ordinary differential equation is equivalent to the hypergeometric differential equation and  is the Schwarz triangle function, which can be written in terms of hypergeometric functions. For a quadrilateral the accessory parameters depend on one independent variable . Writing  for a suitable choice of , the ordinary differential equation takes the form

Thus  are eigenfunctions of a Sturm–Liouville equation on the interval . By the Sturm separation theorem, the non-vanishing of  forces  to be the lowest eigenvalue. Complex structure on Teichmüller space
Universal Teichmüller space is defined to be the space of real analytic quasiconformal mappings of the unit disc , or equivalently the upper half-plane , onto itself, with two mappings considered to be equivalent if on the boundary one is obtained from the other by composition with a Möbius transformation. Identifying  with the lower hemisphere of the Riemann sphere, any quasiconformal self-map  of the lower hemisphere corresponds naturally to a conformal mapping of the upper hemisphere  onto itself. In fact  is determined as the restriction to the upper hemisphere of the solution of the Beltrami differential equation

where μ is the bounded measurable function defined by

on the lower hemisphere, extended to 0 on the upper hemisphere. Identifying the upper hemisphere with , Lipman Bers used the Schwarzian derivative to define a mapping

which embeds universal Teichmüller space into an open subset  of the space of bounded holomorphic functions  on  with the uniform norm. Frederick Gehring showed in 1977 that  is the interior of the closed subset of Schwarzian derivatives of univalent functions. For a compact Riemann surface  of genus greater than 1, its universal covering space is the unit disc  on which its fundamental group  acts by Möbius transformations. The Teichmüller space of  can be identified with the subspace of the universal Teichmüller space invariant under . The holomorphic functions  have the property that

is invariant under , so determine quadratic differentials on . In this way, the Teichmüller space of  is realized as an open subspace of the finite-dimensional complex vector space of quadratic differentials on . Diffeomorphism group of the circle

Crossed homomorphisms
The transformation property

 

allows the Schwarzian derivative to be interpreted as a continuous 1-cocycle or crossed homomorphism of the diffeomorphism group of the circle with coefficients in the module of densities of degree 2 on the circle. Let  be the space of tensor densities of degree  on . The group of orientation-preserving diffeomorphisms of , acts on  via pushforwards. If  is an element of  then consider the mapping

In the language of group cohomology the chain-like rule above says that this mapping is a 1-cocycle on  with coefficients in . In fact

and the 1-cocycle generating the cohomology is . The computation of 1-cohomology is a particular case of the more general result

Note that if  is a group and  a -module, then the identity defining a crossed homomorphism   of  into  can be expressed in terms of standard homomorphisms of groups: it is encoded in a homomorphism  of  into the semidirect product  such that the composition of  with the projection  onto  is the identity map; the correspondence is by the map . The crossed homomorphisms form a vector space and containing as a subspace the coboundary crossed homomorphisms  for  in . A simple averaging argument shows that, if  is a compact group and  a topological vector space on which K acts continuously, then the higher cohomology groups vanish  for . In particular for 1-cocycles χ with

averaging over , using left invariant of the Haar measure on  gives

 
with

Thus by averaging it may be assumed that  satisfies the normalisation condition  for  in . Note that if any element  in  satisfies  then . But then, since  is a homomorphism,
, so that  satisfies the equivariance condition . Thus it may be assumed that the cocycle satisfies these normalisation conditions for . The Schwarzian derivative in fact vanishes whenever  is a Möbius transformation corresponding to . The other two 1-cycles discussed below vanish only on . There is an infinitesimal version of this result giving a 1-cocycle for , the Lie algebra of smooth vector fields, and hence for the Witt algebra, the subalgebra of trigonometric polynomial vector fields. Indeed, when  is  a Lie group and the action of  on  is smooth, there is a Lie algebraic version of crossed homomorphism obtained by taking the corresponding homomorphisms of the Lie algebras (the derivatives of the homomorphisms at the identity). This also makes sense for  and leads to the 1-cocycle

which satisfies the identity

In the Lie algebra case, the coboundary maps have the form  for  in . In both cases the 1-cohomology is defined as the space of crossed homomorphisms modulo coboundaries. The natural correspondence between group homomorphisms and Lie algebra homomorphisms leads to the "van Est inclusion map"

In this way the calculation can be reduced to that of Lie algebra cohomology. By continuity this reduces to the computation of crossed homomorphisms  of the Witt algebra into . The normalisations conditions on the group crossed homomorphism imply the following additional conditions for :

for  in . Following the conventions of , a basis of the Witt algebra is given by

so that . A basis for  the complexification of  is given by

so that

for  in . This forces  for suitable coefficients . The crossed homomorphism condition
 gives a recurrence relation for the :

The condition , implies that . From this condition and the recurrence relation, it follows that up to scalar multiples, this has a unique non-zero solution when  equals 0, 1 or 2 and only the zero solution otherwise. The solution for  corresponds to the group 1-cocycle . The solution for  corresponds to the group 1-cocycle . The corresponding Lie algebra 1-cocycles for  are given up to a scalar multiple by

Central extensions
The crossed homomorphisms in turn give rise to the central extension of  and of its Lie algebra , the so-called Virasoro algebra. Coadjoint action
The group  and its central extension also appear naturally in the context of Teichmüller theory and string theory. In fact the homeomorphisms of  induced by quasiconformal self-maps of  are precisely the quasisymmetric homeomorphisms of ; these are exactly homeomorphisms which do not send four points with cross ratio 1/2 to points with cross ratio near 1 or 0. Taking boundary values, universal Teichmüller can be identified with the quotient of the group of quasisymmetric homeomorphisms  by the subgroup of Möbius transformations . (It can also be realized naturally as the space of quasicircles in .) Since

the homogeneous space  is naturally a subspace of universal Teichmüller space. It is also naturally a complex manifold and this and other natural geometric structures are compatible with those on Teichmüller space. The dual of the Lie algebra of  can be identified with the space of Hill's operators on 

and the coadjoint action of  invokes the Schwarzian derivative. The inverse of the diffeomorphism  sends the Hill's operator to

Pseudogroups and connections
The Schwarzian derivative and the other 1-cocycle defined on  can be extended to biholomorphic between open sets in the complex plane. In this case the local description leads to the theory of analytic pseudogroups, formalizing the theory of infinite-dimensional groups and Lie algebras first studied by Élie Cartan in the 1910s. This is related to affine and projective structures on Riemann surfaces as well as the theory of Schwarzian or projective connections, discussed by Gunning, Schiffer and Hawley. A holomorphic pseudogroup  on  consists of a collection of biholomorphisms  between open sets  and  in  which contains the identity maps for each open , which is closed under restricting to opens, which is closed under composition (when possible), which is closed under taking inverses and such that if a biholomorphisms is locally in , then it too is in . The pseudogroup is said to be transitive if, given  and  in , there is a biholomorphism  in  such that . A particular case of transitive pseudogroups are those which are flat, i.e. contain all complex translations . Let  be the group, under composition, of formal power series transformations   with . A holomorphic pseudogroup  defines a subgroup  of , namely the subgroup defined by the Taylor series expansion about 0  (or "jet") of elements  of  with . Conversely if  is flat it is uniquely determined by : a biholomorphism  on  is contained in  in if and only if the power series  of  lies in  for every  in : in other words the formal power series for  at  is given by an element of  with  replaced by ; or more briefly all the jets of  lie in . The group  has a natural homomorphisms onto the  group  of -jets obtained by taking the truncated power series taken up to the term zk. This group acts faithfully on the space of polynomials of degree  (truncating terms of order higher than k). Truncations similarly define homomorphisms of  onto ; the kernel consists of maps f with , so is Abelian. Thus the group Gk is solvable, a fact also clear from the fact that it is in triangular form for the basis of monomials. A flat pseudogroup  is said to be "defined by differential equations" if there is a finite integer  such that homomorphism of  into  is faithful and the image is a closed subgroup. The smallest such  is said to be the order of . There is a complete classification of all subgroups  that arise in this way which satisfy the additional assumptions that the image of  in  is a complex subgroup and that  equals : this implies that the pseudogroup also contains the scaling transformations  for , i.e. contains  contains every polynomial  with . The only possibilities in this case are that  and }; or that  and . The former is the pseudogroup defined by affine subgroup of the complex Möbius group (the  transformations fixing ); the latter is the pseudogroup defined by the whole complex Möbius group. This classification can easily be reduced to a Lie algebraic problem since the formal Lie algebra  of  consists of formal vector fields  with F a formal power series. It contains the polynomial vectors fields with basis , which is a subalgebra of the Witt algebra. The Lie brackets are given by . Again these act on the space of polynomials of degree  by differentiation—it can be identified with —and the images of  give a basis of the Lie algebra of . Note that . Let  denote the Lie algebra of : it is isomorphic to a subalgebra of the Lie algebra of . It contains  and is invariant under . Since  is a Lie subalgebra of the Witt algebra, the only possibility is that it has basis  or basis  for some . There are corresponding group elements of the form . Composing this with translations yields  with . Unless , this contradicts the form of subgroup ; so . The Schwarzian derivative is related to the pseudogroup for the complex Möbius group. In fact if  is a biholomorphism defined on  then  is a quadratic differential on . If  is a bihomolorphism defined on  and  and  are quadratic differentials on ; moreover  is a quadratic differential on , so that  is also a quadratic differential on . The identity

is thus the analogue of a 1-cocycle for the pseudogroup of biholomorphisms with coefficients in holomorphic quadratic differentials. Similarly  and  are 1-cocycles for the same pseudogroup with values in holomorphic functions and holomorphic differentials. In general 1-cocycle can be defined for holomorphic differentials of any order so that

Applying the above identity to inclusion maps , it follows that ; and hence that if  is the restriction of , so that , then . On the other hand, taking the local holomororphic flow defined by holomorphic vector fields—the exponential of the vector fields—the holomorphic pseudogroup of local biholomorphisms is generated by holomorphic vector fields. If the 1-cocycle  satisfies suitable continuity or analyticity conditions, it induces a 1-cocycle of holomorphic vector fields, also compatible with restriction. Accordingly, it defines a 1-cocycle on holomorphic vector fields on :

Restricting to the Lie algebra of polynomial vector fields with basis , these can be determined using the same methods of Lie algebra cohomology (as in the previous section on crossed homomorphisms). There the calculation was for the whole Witt algebra acting on densities of order , whereas here it is just for a subalgebra acting on holomorphic (or polynomial) differentials of order . Again, assuming that  vanishes on rotations of , there are non-zero 1-cocycles, unique up to scalar multiples. only for differentials of degree 0, 1 and 2 given by the same derivative formula

where  is a polynomial. The 1-cocycles define the three pseudogroups by : this gives the scaling group (); the affine group (); and the whole complex Möbius group (). So these 1-cocycles are the special ordinary differential equations defining the pseudogroup. More significantly they can be used to define corresponding affine or projective structures and connections on Riemann surfaces. If  is a pseudogroup of smooth mappings on , a topological space  is said to have a -structure if it has a collection of charts  that are homeomorphisms from open sets  in  to open sets  in  such that, for every non-empty intersection, the natural map from  to  lies in  . This defines the structure of a smooth -manifold if  consists of local diffeomorphims and a Riemann surface if —so that —and  consists of biholomorphisms. If  is the affine pseudogroup,  is said to have an affine structure; and if  is the Möbius pseudogroup,  is said to have a projective structure. Thus a genus one surface given as  for some lattice  has an affine structure; and a genus  surface given as the quotient of the upper half plane or unit disk by a Fuchsian group has a projective structure. Gunning in 1966 describes how this process can be reversed: for genus , the existence of a projective connection, defined using the Schwarzian derivative 2 and proved using standard results on cohomology, can be used to identify the universal covering surface with the upper half plane or unit disk (a similar result holds for genus 1, using affine connections and ). Generalizations 
 describe a generalization that is applicable for mappings of conformal manifolds, in which the Schwarzian derivative becomes a symmetric tensor on the manifold. Let  be a smooth manifold of dimension  with a smooth metric tensor . A smooth diffeomorphism  is conformal if  for some smooth function . The Schwarzian is defined by

where  is the Levi-Civita connection of ,  denotes the Hessian with respect to the connection,  is the Laplace–Beltrami operator (defined as the trace of the Hessian with respect to ). The Schwarzian satisfies the cocycle law

A Möbius transformation is a conformal diffeomorphism, whose conformal factor has vanishing Schwarzian. The collection of Möbius transformations of  is a closed Lie subgroup of the conformal group of . The solutions to  on Euclidean space, with  the Euclidean metric, are precisely when  is constant, the conformal factor giving the spherical metric , or else a conformal factor for a hyperbolic Poincaré metric on the ball or half-space  or  (respectively). Another generalization applies to positive curves in a Lagrangian Grassmannian . Suppose that  is a symplectic vector space, of dimension  over . Fix a pair of complementary Lagrangian subspaces . The set of Lagrangian subspaces that are complemenary to  is parameterized by the space of mappings  that are symmetric with respect to  ( for all ). Any Lagrangian subspace complementary to  is given by  for some such tensor . A curve is thus specified locally by a one-parameter family  of symmetric tensors. A curve is positive if  is positive definite. The Lagrangian Schwarzian is then defined as

This has the property that  if and only if there is a symplectic transformation relating the curves  and . The Lagrangian Schwarzian is related to a second order differential equation

where  is a symmetric tensor, depending on a real variable  and  is a curve in . Let  be the -dimensional space of solutions of the differential equation. Since  is symmetric, the form on  given by  is independent of , and so gives  a symplectic structure. Let  the evaluation functional. Then for any  in the domain of , the kernel of  is a Lagrangian subspace of , and so the kernel defines a curve in the Lagrangian Grassmannian of . The Lagrangian Schwarzian of this curve is then . See also 
 Riccati equation

Notes

References
, Chapter 6, "Teichmüller Spaces"
]

, Chapter 10,  "The Schwarzian". , Section 12, "Mapping of polygons with circular arcs". , "On the theory of generalized Lamé functions". . Projective geometry
Modular forms
Ordinary differential equations
Complex analysis
Conformal mappings