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I recently learned from Jim Carlson’s blog of the passing of Harsh Pittie on January 16 this year. I never met Harsh, but he is very familiar to me through his work, in particular for his classic book Characteristic classes of foliations. I first encountered this book as a graduate student, late last millenium. The book has several features which make it very attractive. First of all, it is short — only 105 pages. My strong instinct is always to read books cover-to-cover from start to finish, and the motivated reader can go through Harsh’s book in its entirety in maybe 4 hours. Second of all, the exposition is quite lovely: an excellent balance is struck between providing enough details to meet the demands of rigor, and sustaining the arc of an idea so that the reader can get the point. Third of all, the book achieves a difficult synthesis of two “opposing” points of view, namely the (differential-)geometric and the algebraic. In fact, this achievement is noted and praised in the MathSciNet review of the book (linked above) by Dmitri Fuks.

An amenable group $G$ acting by homeomorphisms on a compact topological space $X$ preserves a probability measure on $X$; in fact, one can given a definition of amenability in such terms. For example, if $G$ is finite, it preserves an atomic measure supported on any orbit. If $G = \mathbb{Z}$, one can take a sequence of almost invariant probability measures, supported on the subset $[-n,n] \cdot p$ (where $p \in X$ is arbitrary), and any weak limit will be invariant. For a general amenable group, in place of the subsets $[-n,n] \subset \mathbb{Z}$, one works with a sequence of Folner sets; i.e. subsets with the property that the ratio of their size to the size of their boundary goes to zero (so to speak).

But if $G$ is not amenable, it is generally not true that there is any probability measure on $X$ invariant under the action of $G$. The best one can expect is a probability measure which is invariant on average. Such a measure is called a harmonic measure (or a stationary measure) for the $G$-action on $X$. To be concrete, suppose $G$ is finitely generated by a symmetric generating set $S$ (symmetric here means that if $s \in S$, then $s^{-1} \in S$). Let $M(X)$ denote the space of probability measures on $X$. One can form an operator $\Delta:M(X) \to M(X)$ defined by the formula

$\Delta(\mu) = \frac {1} {|S|} \sum_{s \in S} s_*\mu$

and then look for a probability measure $\nu$ stationary under $\Delta$, which exists for quite general reasons. This measure $\nu$ is the harmonic measure: the expectation of the $\nu$-measure of $s(A)$ under a randomly chosen $s \in S$ is equal to the $\nu$-measure of $A$. Note for any probability measure $\mu$ that $s_*\mu$ is absolutely continuous with respect to $\Delta(\mu)$; in fact, the Radon-Nikodym derivative satisfies $ds_*\mu/d\Delta(\mu) \le |S|$. Substituting $\nu$ for $\mu$ in this formula, one sees that the measure class of $\nu$ is preserved by $G$, and that for every $g \in G$, we have $dg_*\nu/d\nu \le |S|^{|g|}$, where $|g|$ denotes word length with respect to the given generating set.

The existence of harmonic measure is especially useful when $X$ is one-dimensional, e.g. in the case that $X=S^1$. In one dimension, a measure (at least one of full support without atoms) can be “integrated” to a path metric. Consequently, any finitely generated group of homeomorphisms of the circle is conjugate to a group of bilipschitz homeomorphisms (if the harmonic measure associated to the original action does not have full support, or has atoms, one can “throw in” another random generator to the group; the resulting action can be assumed to have a harmonic measure of full support without atoms, which can be integrated to give a structure with respect to which the group action is bilipschitz). In fact, Deroin-Kleptsyn-Navas showed that any countable group of homeomorphisms of the circle (or interval) is conjugate to a group of bilipschitz homeomorphisms (the hypothesis that $G$ be countable is essential; for example, the group $\mathbb{Z}^{\mathbb{Z}}$ acts in a non-bilipschitz way on the interval — see here).

Suppose now that $G = \pi_1(M)$ for some manifold $M$. The action of $G$ on $S^1$ determines a foliated circle bundle $S^1 \to E \to M$; i.e. a circle bundle, together with a codimension one foliation transverse to the circle fibers. To see this, first form the product $\widetilde{M} \times S^1$ with its product foliation by leaves $\widetilde{M} \times \text{point}$, where $\widetilde{M}$ denotes the universal cover of $M$. The group $G = \pi_1(M)$ acts on $\widetilde{M}$ as the deck group of the covering, and on $S^1$ by the given action; the quotient of this diagonal action on the product is the desired circle bundle $E$. The foliation makes $E$ into a “flat” circle bundle with structure group $\text{Homeo}^+(S^1)$. The foliation allows us to associate to each path $\gamma$ in $M$ a homeomorphism from the fiber over $\gamma(0)$ to the fiber over $\gamma(1)$; integrability (or flatness) implies that this homeomorphism only depends on the relative homotopy class of $\gamma$ in $M$. This identification of fibers is called the holonomy of the foliation along the path $\gamma$. If $M$ is a Riemannian manifold, there is another kind of harmonic measure on the circle bundle; in other words, a probability measure on each circle with the property that the holonomy associated to an infinitesimal random walk on $M$ preserves the expected value of the measure. This is (very closely related to) a special case of a construction due to Lucy Garnett which associates a harmonic transverse measure to any foliation $\mathcal{F}$ of a manifold $N$, by finding a fixed point of the leafwise heat flow on the space of probability measures on $N$, and disintegrating this measure into the product of the leafwise area measure, and a “harmonic” transverse measure.

In any case, we normalize our foliated circle bundle so that each circle has length $2\pi$ in its harmonic measure. Let $X$ be the vector field on the circle bundle that rotates each circle at unit speed, and let $\alpha$ be the $1$-form on $E$ whose kernel is tangent to the leaves of the foliation. We scale $\alpha$ so that $\alpha(X)=1$ everywhere. The integrability condition for a foliation is expressed in terms of the $1$-form as the identity $\alpha \wedge d\alpha = 0$, and we can write $d\alpha = -\beta \wedge \alpha$ where $\beta(X)=0$. More intrinsically, $\beta$ descends to a $1$-form on the leaves of the foliation which measures the logarithm of the rate at which the transverse measure expands under holonomy in a given direction (the leafwise form $\beta$ is sometimes called the Godbillon class, since it is “half” of the Godbillon-Vey class associated to a codimension one foliation; see e.g. Candel-Conlon volume 2, Chapter 7). Identifying the universal cover of each leaf with $\widetilde{M}$ by projection, the fact that our measure is harmonic means that $\beta$ “is” the gradient of the logarithm of a positive harmonic function on $\widetilde{M}$. As observed by Thurston, the geometry of $M$ then puts constraints on the size of $\beta$. The following discussion is taken largely from Thurston’s paper “Three-manifolds, foliations and circles II” (unfortunately this mostly unwritten paper is not publicly available; some details can be found in my foliations book, example 4.6).

An orthogonal connection on $E$ can be obtained by averaging $\alpha$ under the flow of $X$; i.e. if $\phi_t$ is the diffeomorphism of $E$ which rotates each circle through angle $t$, then

$\omega = \frac {1} {2\pi} \int_{0}^{2\pi} \phi_t^* \alpha$

is an $X$-invariant $1$-form on $E$, which therefore descends to a $1$-form on $M$, which can be thought of as a connection form for an $\text{SO}(2)$-structure on the bundle $E$. The curvature of the connection (in the usual sense) is the $2$-form $d\omega$, and we have a formula

$d\omega = \frac {1} {2\pi} \int_{0}^{2\pi} \phi_t^*(d\alpha) = \frac {1} {2\pi} \int_{0}^{2\pi} \phi_t^*(-\beta \wedge \alpha)$

The action of the $1$-parameter group $\phi_t$ trivializes the cotangent bundle to $E$ over each fiber. After choosing such a trivialization, we can think of the values of $\alpha$ at each point on a fiber as sweeping out a circle $\gamma$ in a fixed vector space $V$. The tangent to this circle is found by taking the Lie derivative

$\mathcal{L}_X(\alpha) = \iota_X d\alpha + d\iota_X \alpha = \alpha(X)\beta = \beta$

In other words, $\beta$ is identified with $d\gamma$ under the identification of $\alpha$ with $\gamma$, and $\int \phi_t^*(-\beta \wedge \alpha) = \int \gamma \wedge d\gamma$; i.e. the absolute value of the curvature of the connection is equal to $1/\pi$ times the area enclosed by $\gamma$.

Now suppose $M$ is a hyperbolic $n$-manifold, i.e. a manifold of dimension $n$ with constant curvature $-1$ everywhere. Equivalently, think of $M$ as a quotient of hyperbolic space $\mathbb{H}^n$ by a discrete group of isometries. A positive harmonic function on $\mathbb{H}^n$ has a logarithmic derivative which is bounded pointwise by $(n-1)$; identifying positive harmonic functions on hyperbolic space with distributions on the sphere at infinity, one sees that the  “worst case” is the harmonic extension of an atomic measure concentrated at a single point at infinity, since every other positive harmonic function is the weighted average of such examples. As one moves towards or away from a blob at infinity concentrated near this point, the radius of the blob expands like $e^t$; since the sphere at infinity has dimension $n-1$, the conclusion follows. But this means that the speed of $\gamma$ (i.e. the size of $d\gamma$) is pointwise bounded by $(n-1)$, and the length of the $\gamma$ circle is at most $2\pi(n-1)$. A circle of length $2\pi(n-1)$ can enclose a disk of area at most $\pi (n-1)^2$, so the curvature of the connection has absolute value pointwise bounded by $(n-1)^2$.

One corollary is a new proof of the Milnor-Wood inequality, which says that a foliated circle bundle $E$ over a closed oriented surface $S$ of genus at least $2$ satisfies $|e(E)| \le -\chi(S)$, where $e(E)$ is the Euler number of the bundle (a topological invariant). For, the surface $S$ can be given a hyperbolic metric, and the bundle a harmonic connection whose average is an orthogonal connection with pointwise curvature of absolute value at most $1$. The Euler class of the bundle evaluated on the fundamental class of $S$ is the Euler number $e(E)$; we have

$|e(E)| = \frac {1} {2\pi} |\int_S \omega| \le \text{area}(S)/2\pi = -\chi(S)$

where the first equality is the Chern-Weil formula for the Euler class of a bundle in terms of the curvature of a connection, and the last equality is the Gauss-Bonnet theorem for a hyperbolic surface. Another corollary gives lower bounds on the area of an incompressible surface in a hyperbolic manifold. Suppose $S \to M$ is an immersion which is injective on $\pi_1$. There is a cover $\widehat{M}$ of $M$ for which the immersion lifts to a homotopy equivalence, and we get an action of $\pi_1(\widehat{M})$ on the circle at infinity of $S$, and hence a foliated circle bundle as above with $e(E) = -\chi(S)$. Integrating as above over the image of $S$ in $\widehat{M}$, and using the fact that the curvature of $\omega$ is pointwise bounded by $(n-1)^2$, we deduce that the area of $S$ is at least $-2\pi\chi(S)/(n-1)^2$. If $M$ is a $3$-manifold, we obtain $\text{area}(S) \ge -2\pi\chi(S)/4$.

(A somewhat more subtle argument allows one to get better bounds, e.g. replacing $4$ by $(\pi/2)^2$ for $n=3$, and better estimates for higher $n$.)

A geometric structure on a manifold is an atlas of charts with values in some kind of “model space”, and transformation functions taken from some pseudogroup of transformations on the model space. If $X$ is the model space, and $G$ is the pseudo-group, one talks about a $(G,X)$-structure on a manifold $M$. One usually (but not always) wants $X$ to be homogeneous with respect to $G$. So, for instance, one talks about smooth structures, conformal structures, projective structures, bilipschitz structures, piecewise linear structures, symplectic structures, and so on, and so on. Riemannian geometry does not easily fit into this picture, because there are so few (germs of) isometries of a typical Riemannian metric, and so many local invariants; but Riemannian metrics modeled on a locally symmetric space, with $G$ a Lie group of symmetries of $X$, are a very significant example.

Sometimes the abstract details of a theory are hard to grasp before looking at some fundamental examples. The case of geometric structures on $1$-manifolds is a nice example, which is surprisingly rich in some ways.

One of the most important ways in which geometric structures arise is in the theory of ODE’s. Consider a first order ODE in one variable, e.g. an equation like $y' = f(y,t)$. If we fix an “initial” value $y(t_0)=y_0$, then we are guaranteed short time existence and uniqueness of a solution (providing the function $f$ is nice enough). But if we do not fix an initial value, we can instead think of an ODE as a $1$-parameter family of (perhaps partially defined) maps from $\mathbb{R}$ to itself. For each fixed $t$, the function $f(y,t)$ defines a vector field on $\mathbb{R}$. We can think of the ODE as specifying a path in the Lie algebra of vector fields on $\mathbb{R}$; solving the ODE amounts to finding a path in the Lie group of diffeomorphisms of $\mathbb{R}$ (or some partially defined Lie pseudogroup of diffeomorphisms on some restricted subdomain) which is tangent to the given family of vector fields. It makes sense therefore to study special classes of equations, and ask when this family of maps is conjugate into an interesting pseudogroup; equivalently, that the evolution of the solutions preserves an interesting geometric structure on $\mathbb{R}$. We consider some examples in turn.

1. Indefinite integral $y' = a(t)$. The group in this case is $\mathbb{R}$, acting on $\mathbb{R}$ by translation. The equation is solved by integrating: $y=\int a(t)dt + C$.
2. Linear homogeneous ODE $y' = a(t)y$. The group in this case is $\mathbb{R}^+$, acting on $\mathbb{R}$ by multiplication (notice that this group action is not transitive; the point $0 \in \mathbb{R}$ is preserved; this corresponds to the fact that $y = 0$ is always a solution of a homogeneous linear ODE). The Lie algebra is $\mathbb{R}$, and the ODE is “solved” by exponentiating the vector field, and integrating. Hence $y = C e^{\int a(t)dt}$ is the general solution. In fact, in the previous example, the Lie algebra of the group of translations is also identified with $\mathbb{R}$, and “exponentiating” is the identity map.
3. Linear inhomogeneous ODE $y' = a(t)y + b(t)$. The group in this case is the affine group $\mathbb{R}^+ \ltimes \mathbb{R}$ where the first factor acts by dilations and the second by translation. The affine group is not abelian, so one cannot “integrate” a vector field directly, but it is solvable: there is a short exact sequence $\mathbb{R} \to \mathbb{R}^+ \ltimes \mathbb{R} \to \mathbb{R}^+$. The image in the Lie algebra of the group of dilations is the term $a(t)y$, which can be integrated as before to give an integrating factor $e^{\int a(t)dt}$. Setting $z = ye^{-\int a(t)dt}$ gives $z' = y'e^{-\int a(t)dt} - a(t)ye^{-\int a(t)dt} = b(t)e^{-\int a(t)dt}$ which is an indefinite integral, and can be solved by a further integration. In other words, we do one integration to change the structure group from $\mathbb{R}^+ \ltimes \mathbb{R}$ to $\mathbb{R}$ (“integrating out” the group of dilations) and then what is left is an abelian structure group, in which we can do “ordinary” integration. This procedure works whenever the structure group is solvable; i.e. whenever there is a finite sequence $G=G_0,\cdots,G_n=0$ where each $G_i$ surjects onto an abelian group, with kernel $G_{i-1}$, and after finitely many steps, the last kernel is trivial.
4. Ricatti equation $y' = a(t)y^2 + b(t)y + c(t)$. In this case, it is well-known that the equation can blow up in finite time, and one does not obtain a group of transformations of $\mathbb{R}$, but rather a group of transformations of the projective line $\mathbb{RP}^1 = \mathbb{R} \cup \infty$; another point of view says that one obtains a pseudogroup of transformations of subsets of $\mathbb{R}$. The group in this case is the projective group $\text{PSL}(2,\mathbb{R})$, acting by projective linear transformations. Let $A(t)$ be a $1$-parameter family of matrices in $\text{PSL}(2,\mathbb{R})$, say $A(t)=\left( \begin{smallmatrix} u(t) & v(t) \\ w(t) & x(t) \end{smallmatrix} \right)$, with $A(0)=\text{id}$. Matrices act on $\mathbb{R}$ by fractional linear maps; that is, $Az = (uz + v)/(wz+x)$ for $z \in \mathbb{R}$. Differentiating $A(t)z$ at $t=0$ one obtains $(Az)'(0) = (u'z+v')-z(w'z+x') = w'z^2 + (u'-x')z + v'$ which is the general form of the Ricatti equation. Since the group $\text{PSL}(2,\mathbb{R})$ is not solvable, the Ricatti equation cannot be solved in terms of elementary functions and integrals. However, if one knows one solution $y=z(t)$, one can find all other solutions as follows. Do a change of co-ordinates, by sending the solution $z(t)$ “to infinity”; i.e. define $x = 1/(y-z)$. Then as a function of $x$, the Ricatti equation reduces to a linear inhomogeneous ODE. In other words, the structure group reduces to the subgroup of $\text{PSL}(2,\mathbb{R})$ fixing the point at infinity (i.e. the solution $z(t)$), which is the affine group $\mathbb{R}^+ \ltimes \mathbb{R}$. One can therefore solve for $x$, and by substituting back, for $y$.

The Ricatti equation is important for the solution of second order linear equations, since any second order linear equation $y'' = a(t)y' + b(t)y + c(t)$ can be transformed into a system of two first order linear equations in the variables $y$ and $y'$. A system of first order ODEs in $n$ variables can be described in terms of pseudogroups of transformations of (subsets of) $\mathbb{R}^n$. A system of linear equations corresponds to the structure group $\text{GL}(n,\mathbb{R})$, hence in the case of a $2\times 2$ system, to $\text{GL}(2,\mathbb{R})$. The determinant map is a homomorphism from $\text{GL}(2,\mathbb{R})$ to $\mathbb{R}^*$ with kernel $\text{SL}(2,\mathbb{R})$; hence, after  multiplication by a suitable integrating factor, one can reduce to a system which is (equivalent to) the Ricatti equation.

Having seen these examples, one naturally wonders whether there are any other interesting families of equations and corresponding Lie groups acting on $1$-manifolds. In fact, there are (essentially) no other examples: if one insists on (finite dimensional) simple Lie groups, then $\text{SL}(2,\mathbb{R})$ is more or less the only example. Perhaps this is one of the reasons why the theory of ODEs tends to appear to undergraduates (and others) as an unstructured collection of rules and tricks. Nevertheless, recasting the theory in terms of geometric structures has the effect of clearing the air to some extent.

Geometric structures on $1$-manifolds arise also in the theory of foliations, which may be seen as a geometric abstraction of certain kinds of PDE. Suppose $M$ is a manifold, and $\mathcal{F}$ is a codimension one foliation. The foliation determines local charts on the manifold in which the leaves of the foliation intersect the chart in the level sets of a co-ordinate function. In the overlap of two such local charts, the transitions between the local co-ordinate functions take values in some pseudogroup. For certain kinds of foliations, this pseudogroup might be analytically quite rigid. For example, if $\mathcal{F}$ is tangent to the kernel of a nonsingular $1$-form $\alpha$ on $M$, then integrating $\alpha$ determines a metric on the leaf space which is preserved by the co-ordinate transformations, and the pseudogroup is conjugate into the group of translations. There are also some interesting examples where the pseudogroup has no interesting local structure, but where structure emerges on a macroscopic scale, because of some special features of the topology of $M$ and $\mathcal{F}$. For example, suppose $M$ is a $3$-manifold, and $\mathcal{F}$ is a foliation in which every leaf is dense. One knows for topological reasons (i.e. theorems of Novikov and Palmeira) that the universal cover $\tilde{M}$ is homeomorphic to $\mathbb{R}^3$ in such a way that the pulled-back foliation $\tilde{\mathcal{F}}$ is topologically a foliation by planes. One important special case is when any two leaves of $\tilde{\mathcal{F}}$ are a finite Hausdorff distance apart in $\tilde{M}$. In this case, the foliation $\tilde{\mathcal{F}}$ is topologically conjugate to a product foliation, and $\pi_1(M)$ acts on the leaf space (which is $\mathbb{R}$) by a group of homeomorphisms. The condition that pairs of leaves are a finite Hausdorff distance away implies that there are intervals $I$ in the leaf space whose translates do not nest; i.e. with the property that there is no $g \in \pi_1(M)$ for which $g(I)$ is properly contained in $I$. Let $I^\pm$ denote the two endpoints of the interval $I$. One defines a function $Z:\mathbb{R} \to \mathbb{R}$ by defining $Z(p)$ to be the supremum of the set of values $g(I^+)$ over all $g \in \pi_1(M)$ for which $g(I^-) \le p$. The non-nesting property, and the fact that every leaf of $\mathcal{F}$ is dense, together imply that $Z$ is a strictly increasing (i.e. fixed-point free) homeomorphism of $\mathbb{R}$ which commutes with the action of $\pi_1(M)$. In particular, the action of $\pi_1(M)$ is conjugate into the subgroup $\text{Homeo}^+(\mathbb{R})^{\mathbb{Z}}$ of homeomorphisms that commute with integer translation. One says in this case that the manifold $M$ slithers over a circle; it is possible to deduce a lot about the geometry and topology of $M$ and $\mathcal{F}$ from this structure. See for example Thurston’s paper, or my book.

A third significant way in which geometric structures arise on circles is in the theory of conformal welding. Let $\gamma:S^1 \to \mathbb{CP}^1$ be a Jordan curve in the Riemann sphere. The image of the curve decomposes the sphere into two regions homeomorphic to disks. Each open disk region can be uniformized by a holomorphic map from the open unit disk, which extends continuously to the boundary circle. These uniformizing maps are well-defined up to composition with an element of the Möbius group $\text{PSL}(2,\mathbb{R})$, and their difference is therefore a coset in $\text{Homeo}^+(S^1)/\text{PSL}(2,\mathbb{R})$ called the welding homeomorphism. Conversely, given a homeomorphism of the circle, one can ask when it arises from a Jordan curve in the Riemann sphere as above, and if it does, whether the curve is unique (up to conformal self-maps of the Riemann sphere). Neither existence nor uniqueness hold in great generality. For example, if the image $\gamma(S^1)$ has positive (Hausdorff) measure, any quasiconformal deformation of the complex structure on the Riemann sphere supported on the image of the curve will deform the curve but not the welding homeomorphism. One significant special case in which existence and uniqueness is assured is the case that $\gamma(S^1)$ is a quasicircle. This means that there is a constant $K$ with the property that if two points $p,q$ are contained in the quasicircle, and the spherical distance between the two points is $d(p,q)$, then at least one arc of the quasicircle joining $p$ to $q$ has spherical diameter at most $Kd(p,q)$. In other words, there are no bottlenecks where two points on the quasicircle come very close in the sphere without being close in the curve. Welding maps corresponding to quasicircles are precisely the quasisymmetric homeomorphisms. A homeomorphism is quasisymmetric if for every sufficiently small interval in the circle, the image of the midpoint of the interval under the homeomorphism is not too far from being the midpoint of the image of the interval; i.e. it divides the image of the interval into two pieces whose lengths have a ratio which is bounded below and above by some fixed constant. Other classes of geometric structures can be detected by welding: smooth Jordan circles correspond to smooth welding maps, real analytic circles correspond to real analytic welding maps, round circles correspond to welding maps in $\text{PSL}(2,\mathbb{R})$, and so on. Recent work of  Eero Saksman and his collaborators has sought to find the correct idea of a “random” welding, which corresponds to the kinds of Jordan curves generated by stochastic processes such as SLE. In general, the precise correspondence between the analytic quality of $\gamma$ and of the welding map is given by the Hilbert transform.

This list of examples of geometric structures on $1$-manifolds is by no means exhaustive. There are many very special features of $1$-dimensional geometry: oriented $1$-manifolds have a natural causal structure, which may be seen as a special case of contact/symplectic geometry; (nonatomic) measures on $1$-manifolds can be integrated to metrics; connections on $1$-manifolds are automatically flat, and correspond to representations. It would be interesting to hear other examples, and how they arise in various mathematical fields.