Schwarzian derivative

Template:Short description Template:Use American English 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 Template:Mvar of one complex variable Template:Mvar is defined by

$(Sf)(z)=\left(\frac{f^{″}(z)}{f^{\prime }(z)}\right)-\frac{1}{2}{\left(\frac{f^{″}(z)}{f^{\prime }(z)}\right)}^2=\frac{f^{‴}(z)}{f^{\prime }(z)}-\frac{3}{2}{\left(\frac{f^{″}(z)}{f^{\prime }(z)}\right)}^2.$

The same formula also defines the Schwarzian derivative of a [[Smoothness|Template:Math function]] of one real variable. The alternative notation

$\{f,z\}=(Sf)(z)$

is frequently used.

Properties

The Schwarzian derivative of any Möbius transformation

$g(z)=\frac{az+b}{cz+d}$

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 Template:Math is a Möbius transformation, then the composition Template:Math has the same Schwarzian derivative as Template:Math; and on the other hand, the Schwarzian derivative of Template:Math is given by the chain rule

$(S(f\circ g))(z)=(Sf)(g(z))\cdot g^{\prime }(z{)}^2.$

More generally, for any sufficiently differentiable functions Template:Math and Template:Math

$S(f\circ g)=((Sf)\circ g)\cdot (g^{\prime }{)}^2+Sg.$

When Template:Math and Template:Math 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

$F(z,w)=\log \left(\frac{f(z)-f(w)}{z-w}\right),$

its second mixed partial derivative is given by

$\frac{{\partial }^{2}F(z,w)}{\partial z\partial w}=\frac{f^{\prime }(z)f^{\prime }(w)}{(f(z)-f(w){)}^2}-\frac{1}{(z-w{)}^2},$

and the Schwarzian derivative is given by the formula:

$(Sf)(w)={6\cdot \frac{{\partial }^{2}F(z,w)}{\partial z\partial w}|}_{z=w}.$

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

$(Sw)(v)=-{\left(\frac{dw}{dv}\right)}^2(Sv)(w)$

or more explicitly, $Sf+(f^{\prime }{)}^2((S{f}^{-1})\circ f)=0$. 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 $f$ be a conformal mapping in a neighborhood of $z_0\in ℂ.$ Then there exists a unique Möbius transformation $M$ such that $M,f$ has the same 0, 1, 2-th order derivatives at $z_0.$

Now $(M^{-1}\circ f)(z-z_0)=z_0+(z-z_0)+\frac{1}{6}a(z-z_0{)}^3+\cdots .$ To explicitly solve for $a,$ it suffices to solve the case of $z_0=0.$ Let $M^{-1}(z)=$$(Az+B)/(Cz+1),$ and solve for the $A,B,C$ that make the first three coefficients of $M^{-1}\circ f$ equal to $0,1,0.$ Plugging it into the fourth coefficient, we get $a=(Sf)(z_0)$.

After a translation, rotation, and scaling of the complex plane, $(M^{-1}\circ f)(z)=$$z+z^3+O(z^4)$ in a neighborhood of zero. Up to third order this function maps the circle of radius $r$ to the parametric curve defined by $(r\cos \theta +r^3\cos 3\theta ,r\sin \theta +r^3\sin 3\theta ),$ where $\theta \in [0,2\pi ].$ This curve is, up to fourth order, an ellipse with semiaxes $r+r^3$ and Template:Nobr

$$\begin{array}{cc}\frac{(r\cos \theta +r^3\cos 3\theta {)}^2}{(r+r^3{)}^2}+\frac{(r\sin \theta +r^3\sin 3\theta {)}^2}{(r-r^3{)}^2}& =\frac{1+8r^4{\sin }^2(2\theta )+O(r^6)}{(1-r^4{)}^2}\\ & \to 1+8r^4{\sin }^2(2\theta )+O(r^6)\end{array}$$

as $r\to 0.$

Since Möbius transformations always map circles to circles or lines, the eccentricity measures the deviation of $f$ from a Möbius transform.

Differential equation

Consider the linear second-order ordinary differential equation $$x^{″}(t)+p(t)x(t)=0$$ where $x$ is a real-valued function of a real parameter $t$. Let $X$ denote the two-dimensional space of solutions. For $t\in ℝ$, let ${\mathrm{ev}}_t:X\to ℝ$ be the evaluation functional ${\mathrm{ev}}_t(x)=x(t)$. The map $t\mapsto \ker ({\mathrm{ev}}_t)$ gives, for each point $t$ of the domain of $X$, a one-dimensional linear subspace of $X$. 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 $2p(t)$ Template:Harv.

Owing to this interpretation of the Schwarzian, if two diffeomorphisms of a common open interval into ${ℝ\mathbb{P}}^1$ 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 ${ℝ\mathbb{P}}^1$.

Alternatively, consider the second-order linear ordinary differential equation in the complex plane

$\frac{d^{2}f}{d{z}^2}+Q(z)f(z)=0.$

Let $f_1(z)$ and $f_2(z)$ be two linearly independent holomorphic solutions. Then the ratio $g(z)=f_1(z)/f_2(z)$ satisfies

$(Sg)(z)=2Q(z)$

over the domain on which $f_1(z)$ and $f_2(z)$ are defined, and $f_2(z)\ne 0.$ The converse is also true: if such a Template:Math exists, and it is holomorphic on a simply connected domain, then two solutions $f_1$ and $f_2$ 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 Template:Math 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 Template:Math is a holomorphic function on the unit disc, Template:Math, then W. Kraus (1932) and Nehari (1949) proved that a necessary condition for Template:Math to be univalent is

$|S(f)|\le 6(1-|z{|}^2{)}^{-2}.$

Conversely if Template:Math is a holomorphic function on Template:Math satisfying

$|S(f)(z)|\le 2(1-|z{|}^2{)}^{-2},$

then Nehari proved that Template:Math is univalent.

In particular a sufficient condition for univalence is

$|S(f)|\le 2.$

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 Template:Math be a circular arc polygon with angles $\pi {\alpha }_1,\dots ,\pi {\alpha }_n$ in clockwise order. Let Template:Math be a holomorphic map extending continuously to a map between the boundaries. Let the vertices correspond to points $a_1,\dots ,a_n$ on the real axis. Then Template:Math is real-valued when Template:Math is real and different from all the points Template:Math. By the Schwarz reflection principle Template:Math extends to a rational function on the complex plane with a double pole at Template:Math:

$p(z)=\sum _{i=1}^n\frac{(1-{\alpha }_i^2)}{2(z-a_i{)}^2}+\frac{{\beta }_i}{z-a_i}.$

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

$\sum {\beta }_i=0$

$\sum 2a_i{\beta }_i+(1-{\alpha }_i^2)=0$

$\sum a_i^2{\beta }_i+a_i(1-{\alpha }_i^2)=0$

which correspond to the vanishing of the coefficients of $z^{-1},z^{-2}$ and $z^{-3}$ in the expansion of Template:Math around Template:Math. The mapping Template:Math can then be written as

$f(z)=\frac{u_1(z)}{u_2(z)},$

where $u_1(z)$ and $u_2(z)$ are linearly independent holomorphic solutions of the linear second-order ordinary differential equation

$u^{\prime \prime }(z)+\frac{1}{2}p(z)u(z)=0.$

There are Template:Math linearly independent accessory parameters, which can be difficult to determine in practise.

For a triangle, when Template:Math, there are no accessory parameters. The ordinary differential equation is equivalent to the hypergeometric differential equation and Template:Math 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 Template:Math. Writing Template:Math for a suitable choice of Template:Math, the ordinary differential equation takes the form

$a(z)U^{\prime \prime }(z)+b(z)U^{\prime }(z)+(c(z)+\lambda )U(z)=0.$

Thus $q(z)u_i(z)$ are eigenfunctions of a Sturm–Liouville equation on the interval $[a_i,a_{i+1}]$. By the Sturm separation theorem, the non-vanishing of $u_2(z)$ forces Template:Math 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 Template:Math, or equivalently the upper half-plane Template:Math, 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 Template:Math with the lower hemisphere of the Riemann sphere, any quasiconformal self-map Template:Math of the lower hemisphere corresponds naturally to a conformal mapping of the upper hemisphere $\tilde{f}$ onto itself. In fact $\tilde{f}$ is determined as the restriction to the upper hemisphere of the solution of the Beltrami differential equation

$\frac{\partial F}{\partial \overline{z}}=\mu (z)\frac{\partial F}{\partial z},$

where μ is the bounded measurable function defined by

$\mu (z)=\frac{\partial f}{\partial \overline{z}}/\frac{\partial f}{\partial z}$

on the lower hemisphere, extended to 0 on the upper hemisphere.

Identifying the upper hemisphere with Template:Math, Lipman Bers used the Schwarzian derivative to define a mapping

$g=S(\tilde{f}),$

which embeds universal Teichmüller space into an open subset Template:Math of the space of bounded holomorphic functions Template:Math on Template:Math with the uniform norm. Frederick Gehring showed in 1977 that Template:Math is the interior of the closed subset of Schwarzian derivatives of univalent functions.

For a compact Riemann surface Template:Math of genus greater than 1, its universal covering space is the unit disc Template:Math on which its fundamental group Template:Math acts by Möbius transformations. The Teichmüller space of Template:Math can be identified with the subspace of the universal Teichmüller space invariant under Template:Math. The holomorphic functions Template:Math have the property that

$g(z)d{z}^2$

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

Diffeomorphism group of the circle

Crossed homomorphisms

The transformation property

$S(f\circ g)=(S(f)\circ g)\cdot (g^{\prime }{)}^2+S(g).$

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 Template:Math be the space of tensor densities of degree Template:Math on Template:Math. The group of orientation-preserving diffeomorphisms of Template:Math, acts on Template:Math via pushforwards. If $f\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(S^1)$ then consider the mapping

$f\to S(f^{-1}).$

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

$H^1(\text{Diff}({𝐒}^1);F_2({𝐒}^1))=𝐑$

and the 1-cocycle generating the cohomology is $f\mapsto S(f^{-1})$. The computation of 1-cohomology is a particular case of the more general result

$H^1(\text{Diff}({𝐒}^1);F_{\lambda }({𝐒}^1))=𝐑\text{for }\lambda =0,1,2\mathrm{a}\mathrm{n}\mathrm{d}(0)\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{.}$

Note that if Template:Math is a group and Template:Math a Template:Math-module, then the identity defining a crossed homomorphism Template:Math of Template:Math into Template:Math can be expressed in terms of standard homomorphisms of groups: it is encoded in a homomorphism Template:Phi of Template:Math into the semidirect product $M⋊G$ such that the composition of Template:Phi with the projection $M⋊G$ onto Template:Math is the identity map; the correspondence is by the map

$C(g)=(c(g),g).$

The crossed homomorphisms form a vector space and containing as a subspace the coboundary crossed homomorphisms

$b(g)=g\cdot m-m$

in Template:Math. A simple averaging argument shows that, if Template:Math is a compact group and Template:Math a topological vector space on which K acts continuously, then the higher cohomology groups vanish

$H^m(V,K)=(0)\text{for }m>0.$

In particular for 1-cocycles χ with

$\chi (xy)=\chi (x)+x\cdot \chi (y),$

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

$\chi (x)=m-x\cdot m,$

with

$m=\underset{K}{\int }\chi (y)dy.$

Thus by averaging it may be assumed that Template:Math satisfies the normalisation condition $c(x)=0$ for $x\in \mathrm{R}\mathrm{o}\mathrm{t}(S^1)$. Note that if any element Template:Math in Template:Math satisfies Template:Math then

$C(x)=(0,x).$

But then, since Template:Math is a homomorphism,

$C(xg{x}^{-1})=C(x)C(g)C(x{)}^{-1},$

so that Template:Math satisfies the equivariance condition

$c(xg{x}^{-1})=x\cdot c(g).$

Thus it may be assumed that the cocycle satisfies these normalisation conditions for Template:Math. The Schwarzian derivative in fact vanishes whenever Template:Math is a Möbius transformation corresponding to Template:Math. The other two 1-cycles discussed below vanish only on Template:Math.

There is an infinitesimal version of this result giving a 1-cocycle for Template:Math, the Lie algebra of smooth vector fields, and hence for the Witt algebra, the subalgebra of trigonometric polynomial vector fields. Indeed, when Template:Math is a Lie group and the action of Template:Math on Template:Math 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 Template:Math and leads to the 1-cocycle

$s\left(f\frac{d}{d\theta }\right)=\frac{d^{3}f}{d{\theta }^3}(d\theta {)}^2$

which satisfies the identity

$s([X,Y])=X\cdot s(Y)-Y\cdot s(X).$

In the Lie algebra case, the coboundary maps have the form

$b(X)=X\cdot m,m\in M$

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"

$H^1(\mathrm{Diff}({𝐒}^1);F_{\lambda }({𝐒}^1))\hookrightarrow H^1(\mathrm{Vect}({𝐒}^1);F_{\lambda }({𝐒}^1)),$

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

$\begin{array}{cc}\phi (\mathrm{Ad}(x)X)& =x\cdot \phi (X)\\ \phi \left(\frac{d}{d\theta }\right)& =0\end{array}$

for Template:Math in Template:Math.

Following the conventions of Template:Harvtxt, a basis of the Witt algebra is given by

$d_n=i{e}^{in\theta }\frac{d}{d\theta }$

so that

$[d_m,d_n]=(m-n)d_{m+n}.$

A basis for the complexification of Template:Math is given by

$v_n=e^{in\theta }(d\theta {)}^{\lambda },$

so that

$\begin{array}{cc}{d}_m\cdot v_n& =-(n+\lambda m)v_{n+m}\\ g_{\zeta }\cdot v_n& ={\zeta }^n{v}_n,\end{array}$

for $g_{\zeta }\in \mathrm{R}\mathrm{o}\mathrm{t}(S^1)=𝐓$. This forces

$\phi (d_n)=a_n\cdot v_n$

for suitable coefficients Template:Math. The crossed homomorphism condition

$\phi ([X,Y])=X\phi (Y)-Y\phi (X)$

gives a recurrence relation for the $a_n$:

$(m-n)a_{m+n}=(m+\lambda n)a_m-(n+\lambda m)a_n.$

The condition

$\phi \left(\frac{d}{d\theta }\right)=0$

implies that $a_0=0$. From this condition and the recurrence relation, it follows that up to scalar multiples, this has a unique non-zero solution when $\lambda =0,1,2$ and only the zero solution otherwise. The solution for $\lambda =1$ corresponds to the group 1-cocycle

${\phi }_1(f)=\frac{f^{\prime \prime }}{f^{\prime }}d\theta .$

The solution for $\lambda =0$ corresponds to the group 1-cocycle

${\phi }_0(f)=\log f^{\prime }.$

The corresponding Lie algebra 1-cocycles for Template:Math are given up to a scalar multiple by

${\phi }_{\lambda }\left(F\frac{d}{d\theta }\right)=\frac{d^{\lambda +1}F}{d{\theta }^{\lambda +1}}(d\theta {)}^{\lambda }.$

Central extensions

The crossed homomorphisms in turn give rise to the central extension of Template:Math and of its Lie algebra Template:Math, the so-called Virasoro algebra.

Coadjoint action

The group Template:Math and its central extension also appear naturally in the context of Teichmüller theory and string theory. In fact the homeomorphisms of Template:Math induced by quasiconformal self-maps of Template:Math are precisely the quasisymmetric homeomorphisms of Template:Math; 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 Template:Math by the subgroup of Möbius transformations Template:Math. (It can also be realized naturally as the space of quasicircles in Template:Math.) Since

$\mathrm{Moeb}({𝐒}^1)\subset \mathrm{Diff}({𝐒}^1)\subset \text{QS}({𝐒}^1)$

the homogeneous space Template:Math 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 Template:Math can be identified with the space of Hill's operators on Template:Math

$\frac{d^2}{d{\theta }^2}+q(\theta ),$

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

$\frac{d^2}{d{\theta }^2}+f^{\prime }(\theta {)}^{2}q\circ f(\theta )+\frac{1}{2}S(f)(\theta ).$

Pseudogroups and connections

The Schwarzian derivative and the other 1-cocycle defined on Template:Math 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 Template:Math on Template:Math consists of a collection of biholomorphisms Template:Math between open sets Template:Math and Template:Math in Template:Math which contains the identity maps for each open Template:Math, 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 Template:Math, then it too is in Template:Math. The pseudogroup is said to be transitive if, given Template:Math and Template:Math in Template:Math, there is a biholomorphism Template:Math in Template:Math such that Template:Math. A particular case of transitive pseudogroups are those which are flat, i.e. contain all complex translations Template:Math. Let Template:Math be the group, under composition, of formal power series transformations Template:Math with Template:Math. A holomorphic pseudogroup Template:Math defines a subgroup Template:Math of Template:Math, namely the subgroup defined by the Taylor series expansion about 0 (or "jet") of elements Template:Math of Template:Math with Template:Math. Conversely if Template:Math is flat it is uniquely determined by Template:Math: a biholomorphism Template:Math on Template:Math is contained in Template:Math in if and only if the power series of Template:Math lies in Template:Math for every Template:Math in Template:Math: in other words the formal power series for Template:Math at Template:Math is given by an element of Template:Math with Template:Math replaced by Template:Math; or more briefly all the jets of Template:Math lie in Template:Math.

The group Template:Math has a natural homomorphisms onto the group Template:Math of Template:Math-jets obtained by taking the truncated power series taken up to the term z^k. This group acts faithfully on the space of polynomials of degree Template:Math (truncating terms of order higher than k). Truncations similarly define homomorphisms of Template:Math onto Template:Math; the kernel consists of maps f with Template:Math, so is Abelian. Thus the group G_k is solvable, a fact also clear from the fact that it is in triangular form for the basis of monomials.

A flat pseudogroup Template:Math is said to be "defined by differential equations" if there is a finite integer Template:Math such that homomorphism of Template:Math into Template:Math is faithful and the image is a closed subgroup. The smallest such Template:Math is said to be the order of Template:Math. There is a complete classification of all subgroups Template:Math that arise in this way which satisfy the additional assumptions that the image of Template:Math in Template:Math is a complex subgroup and that Template:Math equals Template:Math: this implies that the pseudogroup also contains the scaling transformations Template:Math for Template:Math, i.e. contains Template:Math contains every polynomial Template:Math with Template:Math.

The only possibilities in this case are that Template:Math and Template:Math}; or that Template:Math and Template:Math. The former is the pseudogroup defined by affine subgroup of the complex Möbius group (the Template:Math transformations fixing Template:Math); 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 $𝔤$ of Template:Math consists of formal vector fields Template:Math with F a formal power series. It contains the polynomial vectors fields with basis Template:Math, which is a subalgebra of the Witt algebra. The Lie brackets are given by Template:Math. Again these act on the space of polynomials of degree Template:Math by differentiation—it can be identified with Template:Math—and the images of Template:Math give a basis of the Lie algebra of Template:Math. Note that Template:Math. Let $𝔞$ denote the Lie algebra of Template:Math: it is isomorphic to a subalgebra of the Lie algebra of Template:Math. It contains Template:Math and is invariant under Template:Math. Since $𝔞$ is a Lie subalgebra of the Witt algebra, the only possibility is that it has basis Template:Math or basis Template:Math for some Template:Math. There are corresponding group elements of the form Template:Math. Composing this with translations yields Template:Math with Template:Math. Unless Template:Math, this contradicts the form of subgroup Template:Math; so Template:Math.

The Schwarzian derivative is related to the pseudogroup for the complex Möbius group. In fact if Template:Math is a biholomorphism defined on Template:Math then Template:Math is a quadratic differential on Template:Math. If Template:Math is a bihomolorphism defined on Template:Math and Template:Math and Template:Math are quadratic differentials on Template:Math; moreover Template:Math is a quadratic differential on Template:Math, so that Template:Math is also a quadratic differential on Template:Math. The identity

$S(f\circ g)=g_{*}S(f)+S(g)$

is thus the analogue of a 1-cocycle for the pseudogroup of biholomorphisms with coefficients in holomorphic quadratic differentials. Similarly ${\phi }_0(f)=\log f^{\prime }$ and ${\phi }_1(f)=f^{\prime \prime }/f^{\prime }$ 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

$\phi (f\circ g)=g_{*}\phi (f)+\phi (g).$

Applying the above identity to inclusion maps Template:Math, it follows that Template:Math; and hence that if Template:Math is the restriction of Template:Math, so that Template:Math, then Template:Math. 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 Template:Phi 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 Template:Math:

$\phi ([X,Y])=X\phi (Y)-Y\phi (X).$

Restricting to the Lie algebra of polynomial vector fields with basis Template:Math, 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 Template:Math, whereas here it is just for a subalgebra acting on holomorphic (or polynomial) differentials of order Template:Math. Again, assuming that Template:Phi vanishes on rotations of Template:Math, 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

${\phi }_k(p(z)\frac{d}{dz})=p^{(k+1)}(z)(dz{)}^k,$

where Template:Math is a polynomial.

The 1-cocycles define the three pseudogroups by Template:Math: this gives the scaling group (Template:Math); the affine group (Template:Math); and the whole complex Möbius group (Template:Math). 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 Template:Math is a pseudogroup of smooth mappings on Template:Math, a topological space Template:Math is said to have a Template:Math-structure if it has a collection of charts Template:Math that are homeomorphisms from open sets Template:Math in Template:Math to open sets Template:Math in Template:Math such that, for every non-empty intersection, the natural map from Template:Math to Template:Math lies in Template:Math. This defines the structure of a smooth Template:Math-manifold if Template:Math consists of local diffeomorphisms and a Riemann surface if Template:Math—so that Template:Math—and Template:Math consists of biholomorphisms. If Template:Math is the affine pseudogroup, Template:Math is said to have an affine structure; and if Template:Math is the Möbius pseudogroup, Template:Math is said to have a projective structure. Thus a genus one surface given as Template:Math for some lattice Template:Math has an affine structure; and a genus Template:Math 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 Template:Math, the existence of a projective connection, defined using the Schwarzian derivative Template:Phi₂ 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 Template:Math).

Generalizations

Template:Harvtxt describe a generalization that is applicable for mappings of conformal manifolds, in which the Schwarzian derivative becomes a symmetric tensor on the manifold. Let $M$ be a smooth manifold of dimension $n$ with a smooth metric tensor $g$. A smooth diffeomorphism $F:M\to M$ is conformal if $F^{*}g=e^{2\phi }g$ for some smooth function $\phi$. The Schwarzian is defined by $$S_g(\phi )={\mathrm{\nabla }}^2\phi -d\phi \otimes d\phi -\frac{1}{n}(\mathrm{\Delta }\phi -g(\mathrm{\nabla }\phi ,\mathrm{\nabla }\phi ))$$ where $\mathrm{\nabla }$ is the Levi-Civita connection of $g$, ${\mathrm{\nabla }}^2$ denotes the Hessian with respect to the connection, $\mathrm{\Delta }\phi$ is the Laplace–Beltrami operator (defined as the trace of the Hessian with respect to $g$).

The Schwarzian satisfies the cocycle law $$S_g(\phi +\psi )=S_g(\phi )+S_{e^{2\phi }g}(\psi ).$$ A Möbius transformation is a conformal diffeomorphism, whose conformal factor has vanishing Schwarzian. The collection of Möbius transformations of $M$ is a closed Lie subgroup of the conformal group of $M$. The solutions to $S_g(\phi )=0$ on Euclidean space, with $g$ the Euclidean metric, are precisely when $\phi$ is constant, the conformal factor giving the spherical metric $\log [(1+|𝐱{|}^2{)}^{-1}]$, or else a conformal factor for a hyperbolic Poincaré metric on the ball or half-space $\log |(1-|𝐱{|}^2{)}^{-1}|$ or $\log |x_n^{-1}|$ (respectively).

Another generalization applies to positive curves in a Lagrangian Grassmannian Template:Harv. Suppose that $(X,\omega )$ is a symplectic vector space, of dimension $2n$ over $ℝ$. Fix a pair of complementary Lagrangian subspaces $A,B\subset X$. The set of Lagrangian subspaces that are complemenary to $A$ is parameterized by the space of mappings $H:A\to B$ that are symmetric with respect to $\omega$ ($\omega (a,H(a))=\omega (H(a),a)$ for all $a\in A$). Any Lagrangian subspace complementary to $A$ is given by $\{a+H(a)|a\in A\}$ for some such tensor $H$. A curve is thus specified locally by a one-parameter family $H(u)$ of symmetric tensors. A curve is positive if $H^{\prime }(u)$ is positive definite. The Lagrangian Schwarzian is then defined as $$S(H)=(H^{\prime }{)}^{-1/2}(H^{‴}-\frac{3}{2}{H}^{″}(H^{\prime }{)}^{-1}{H}^{″})(H^{\prime }{)}^{-1/2}.$$ This has the property that $S(H)=S(G)$ if and only if there is a symplectic transformation relating the curves $H(u)$ and $G(u)$.

The Lagrangian Schwarzian is related to a second order differential equation $$\frac{d^{2}x}{d{t}^2}+Q(t)x=0,$$ where $Q(t)$ is a symmetric tensor, depending on a real variable $t$ and $x$ is a curve in ${ℝ}^n$. Let $X$ be the $2n$-dimensional space of solutions of the differential equation. Since $Q$ is symmetric, the form on $X$ given by $\omega (x,y)=x(t)\cdot y^{\prime }(t)-x^{\prime }(t)\cdot y(t)$ is independent of $t$, and so gives $X$ a symplectic structure. Let ${\mathrm{ev}}_t:X\to ℝ$ the evaluation functional. Then for any $t$ in the domain of $X$, the kernel of ${\mathrm{ev}}_t$ is a Lagrangian subspace of $X$, and so the kernel defines a curve in the Lagrangian Grassmannian of $(X,\omega )$. The Lagrangian Schwarzian of this curve is then $2Q(t)$.

See also

• Riccati equation

References

• Template:Citation, Chapter 6, "Teichmüller Spaces"
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• Template:Citation, Chapter 10, "The Schwarzian".
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• Template:Citation, Section 12, "Mapping of polygons with circular arcs".
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