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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.

Quadratic forms (i.e. homogeneous polynomials of degree two) are fundamental mathematical objects. For the ancient Greeks, quadratic forms manifested in the geometry of conic sections, and in Pythagoras’ theorem. Riemann recognized the importance of studying abstract smooth manifolds equipped with a field of infinitesimal quadratic forms (i.e. a Riemannian metric), giving rise to the theory of Riemannian manifolds. In contrast to more general norms, an inner product on a vector space enjoys a big group of symmetries; thus infinitesimal Riemannian geometry inherits all the richness of the representation theory of orthogonal groups, which organizes the various curvature tensors and Weitzenbock formulae. It is natural that quadratic forms should come up in so many distinct ways in differential geometry: one uses calculus to approximate a smooth object near some point by a linear object, and the “difference” is a second-order term, which can often be interpreted as a quadratic form. For example:

1. If $M$ is a Riemannian manifold, at any point $p$ one can choose an orthonormal frame for $T_p M$, and exponentiate to obtain geodesic normal co-ordinates. In such local co-ordinates, the metric tensor $g_{ij}$ satisfies $g_{ij}(p)=\delta_{ij}$ and $\partial_kg_{ij}(p) = 0$. The second order derivatives can be expressed in terms of the Riemann curvature tensor at $p$.
2. If $S$ is an immersed submanifold of Euclidean space, at every point $p \in S$ there is a unique linear subspace that is tangent to $S$ at $p$. The second order difference between these two spaces is measured by the second fundamental form of $S$, a quadratic form (with coefficients in the normal bundle) whose eigenvectors are the directions of (extrinsic) principal curvature. If $S$ has codimension one, the second fundamental form is easily described in terms of the Gauss map $g: S \to S^{n-1}$ taking each point on $S$ to the unique unit normal to $S$ at that point, and using the flatness of the ambient Euclidean space to identify the normal spheres at different points with “the” standard sphere. The second fundamental form is then defined by the formula $II(v,w) = \langle dg(v),w \rangle$. For higher codimension, one considers Gauss maps with values in an appropriate Grassmannian.
3. If $f$ is a smooth function on a manifold $M$, a critical point $p$ of $f$ is a point at which $df=0$ (i.e. at which all the partial derivatives of $f$ in some local coordinates vanish). At such a point, one defines the Hessian $Hf$, which is a quadratic form on $T_pM$, determined by the second partial derivatives of $f$ at such a point. If $\nabla$ is a Levi-Civita connection on $T^*M$ (determined by an Riemannian metric on $M$ compatible with the smooth structure) then $Hf = \nabla df$. The condition that the Levi-Civita connection is torsion-free translates into the fact that the antisymmetric part of $\nabla \theta$ is equal to $d\theta$ for any $1$-form $\theta$; in this context, this means that the antisymmetric part of the Hessian vanishes — i.e. that it is symmetric (and therefore a quadratic form). If $\nabla'$ is a different connection, then $\nabla' df = \nabla df + \alpha \wedge df$ for some $1$-form $\alpha$, and therefore their values at $p$ agree, and $Hf$ is well-defined, independent of a choice of metric.

By contrast, cubic forms are less often encountered, either in geometry or in other parts of mathematics; their appearance is often indicative of unusual richness. For example: Lie groups arise as the subgroups of automorphisms of vector spaces preserving certain structure. Orthogonal and symplectic groups are those that preserve certain (symmetric or alternating) quadratic forms. The exceptional Lie group $G_2$ is the group of automorphisms of $\mathbb{R}^7$ that preserves a generic (i.e. nondegenerate) alternating $3$-form. One expects to encounter cubic forms most often in flavors of geometry in which the local transformation pseudogroups are bigger than the orthogonal group.

One example is that of $1$-dimensional complex projective geometry. If $U$ is a domain in the Riemann sphere, one can think of $U$ as a geometric space in at least two natural ways: by considering the local pseudogroup of all holomorphic self-maps between open subsets of the Riemann sphere, restricted to $U$ (i.e. all holomorphic functions), or by considering only those holomorphic maps that extend to the entire Riemann sphere (i.e. the projective transformations: $z \to \frac {az+b} {cz+d}$). The difference between these two geometric structures is measured by a third-order term, called the Schwarzian derivative. If $U$ is homeomorphic to a disk, then we can think of $U$ as the image of the round unit disk $D$ under a uniformizing map $f$. At every point $p \in D$ there is a unique projective transformation $f_p$ that osculates to $f$ to second order at $p$ (i.e. has the same value, first derivative, and second derivative as $f$ at the point $p$); the (scaled) third derivative is the Schwarzian of $f$ at $p$. In local co-ordinates, $Sf = f'''/f' - \frac {3} {2} \left( f''/f'\right)^2$. Actually, although the Schwarzian is sensitive to third-order information, it should really be thought of as a quadratic form on the (one-dimensional) complex tangent space to $p$.

Real projective geometry gives rise to similar invariants. Consider an immersed curve in the (real projective) plane. At every point, there is a unique osculating conic, that agrees with the immersed curve to second order. The projective curvature (really a cubic form) measures the third order deviation between these two immersed submanifolds at this point. See e.g. the book by Ovsienko and Tabachnikov for more details.

Another example is the so-called symplectic curvature. Let $X$ be a flat symplectic space; this could be ordinary Euclidean space $\mathbb{R}^{2n}$ with its standard symplectic form, or a quotient of such a space by a discrete group of translations. A linear subspace $\pi$ of $\mathbb{R}^{2n}$ through the origin is a Lagrangian subspace if it has (maximal) dimension $n$, and the restriction of the symplectic form to $\pi$ is identically zero. A smooth submanifold $L$ of dimension $n$ is Lagrangian if its tangent space at every point is a Lagrangian submanifold. A Lagrangian submanifold of a flat symplectic space inherits a natural cubic form on the tangent space at every point, which can be defined in any of the following equivalent ways:

1. If $W$ is a symplectic manifold and $L$ is a Lagrangian submanifold, then near any point $p$ one can find a neighborhood $U$ and choose symplectic coordinates so that $U$ is symplectomorphic to a neighborhood of some point in $T^*L$. Moreover, every other Lagrangian submanifold $L'$ sufficiently close (in $C^1$) to $L$ can be taken in some possibly smaller neighborhood to be of the form $df$, where $f$ is a smooth function on $L$ (well-defined up to a constant), thought of as a section of $T^*L$. In the context above, choose local symplectic coordinates (by a linear symplectic transformation) for which the flat space looks locally like $T^*\pi$ and $L$ looks locally like $df$. The condition that $\pi$ and $L$ are tangent at the origin means that the $2$-jet of $f$ vanishes. The first nonvanishing term are the third partial derivatives of $f$, which can be thought of as the coefficients of a (symmetric) cubic form on $\pi$.
2. If we choose a Euclidean metric on $X$ compatible with the flat symplectic structure, the second fundamental form of $L$ at some point is a quadratic form on $\pi$ with coefficients in the normal bundle to $\pi$. The symplectic form identifies the normal $\pi^\perp$ to $\pi$ with the dual $\pi^*$, so by contracting indices, one obtains a cubic form on $\pi$. This form does not depend on the choice of Euclidean metric, since a different metric skews the normal bundle $\pi^\perp$ replacing it with $\pi^\perp + \alpha\pi$. But since $\pi$ is Lagrangian, the identification of this normal bundle with $\pi^*$ is insensitive to the skewed term, and therefore independent of the choices.
3. The space of all Lagrangian subspaces $\Lambda$ of $\mathbb{R}^{2n}$ is a symmetric space, homeomorphic to $U(n)/O(n)$, sometimes called the Shilov boundary of the Siegel upper half-space. If $\pi \in \Lambda$ and $\pi'_0$ is a tangent vector to $\pi$ in $\Lambda$, then one obtains a symmetric quadratic form on $\pi$ in the following way. If $\sigma$ is a transverse Lagrangian to $\pi$, and $\pi_t$ is a $1$-parameter family of Lagrangians starting at $\pi$, then for small  $t$ the Lagrangians $\pi_t$ and $\sigma$ are transverse, and span $\mathbb{R}^{2n}$. For any $v \in \mathbb{R}^{2n}$ there is a unique decomposition $v = v(\pi_t) + v(\sigma)$. Define $q_t(v,w) = \omega(v(\pi_t),w(\sigma))$. Then $q'_0$ is a symmetric bilinear form that vanishes on $\sigma$, and therefore descends to a form on $\pi$ that depends only on $\pi'_0$. A Lagrangian submanifold $L$ maps to $\Lambda$ by the Gauss map $g$. One obtains a cubic form on $\pi$ associated to $L$ as follows: if $u,v,w \in \pi$ then $dg(u)$ is a tangent vector to $\pi$ in $\Lambda$, and therefore determines a quadratic form on $\pi$; this form is then evaluated on the vectors $v,w$.

One application of symplectic curvature is to homological mirror symmetry, where the symplectic curvature associated to a Lagrangian family of Calabi-Yau $3$-folds $Y$ in $H^3(Y)$ determines the so-called “Yukawa 3-differential”, whose expression in a certain local coordinate gives the generating function for the number of rational curves of degree $d$ in a generic quintic hypersurface in $\mathbb{CP}^4$. This geometric picture is described explicitly in the work of Givental (e.g. here). In another more recent paper, Givental shows how the topological recursion relations, the string equation and the dilaton equation in Gromov-Witten theory can be reformulated in terms of the geometry of a certain Lagrangian cone in a formal loop space (the geometric property of this cone is that it is overruled — i.e. each tangent space $L$ is tangent to the cone exactly along $zL$, where $z$ is a formal variable). This geometric condition translates into properties of the symplectic curvature of the Lagrangian cone, from which one can read off the “gravitational descendents” in the theory (let me add that this subject is quite far from my area of expertise, and that I come to this material as an interested outsider).

Cubic forms occur naturally in other “special” geometric contexts, e.g. holomorphic symplectic geometry (Rozansky-Witten invariants), affine differential geometry (related to the discussion of the Schwarzian above), etc. Each of these contexts is the start of a long story, which is best kept for another post.