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In the previous set of notes we introduced the notion of a complex diffeomorphism ${f: U \rightarrow V}$ between two open subsets ${U,V}$ of the complex plane ${{\bf C}}$ (or more generally, two Riemann surfaces): an invertible holomorphic map whose inverse was also holomorphic. (Actually, the last part is automatic, thanks to Exercise 40 of Notes 4.) Such maps are also known as biholomorphic maps or conformal maps (although in some literature the notion of “conformal map” is expanded to permit maps such as the complex conjugation map ${z \mapsto \overline{z}}$ that are angle-preserving but not orientation-preserving, as well as maps such as the exponential map ${z \mapsto \exp(z)}$ from ${{\bf C}}$ to ${{\bf C} \backslash \{0\}}$ that are only locally injective rather than globally injective). Such complex diffeomorphisms can be used in complex analysis (or in the analysis of harmonic functions) to change the underlying domain ${U}$ to a domain that may be more convenient for calculations, thanks to the following basic lemma:

Lemma 1 (Holomorphicity and harmonicity are conformal invariants) Let ${\phi: U \rightarrow V}$ be a complex diffeomorphism between two Riemann surfaces ${U,V}$.

• (i) If ${f: V \rightarrow W}$ is a function to another Riemann surface ${W}$, then ${f}$ is holomorphic if and only if ${f \circ \phi: U \rightarrow W}$ is holomorphic.
• (ii) If ${U,V}$ are open subsets of ${{\bf C}}$ and ${u: V \rightarrow {\bf R}}$ is a function, then ${u}$ is harmonic if and only if ${u \circ \phi: U \rightarrow {\bf R}}$ is harmonic.

Proof: Part (i) is immediate since the composition of two holomorphic functions is holomorphic. For part (ii), observe that if ${u: V \rightarrow {\bf R}}$ is harmonic then on any ball ${B(z_0,r)}$ in ${V}$, ${u}$ is the real part of some holomorphic function ${f: B(z_0,r) \rightarrow {\bf C}}$ thanks to Exercise 62 of Notes 3. By part (i), ${f \circ \phi: B(z_0,r) \rightarrow {\bf C}}$ is also holomorphic. Taking real parts we see that ${u \circ \phi}$ is harmonic on each ball ${B(z_0,r)}$ in ${V}$, and hence harmonic on all of ${V}$, giving one direction of (ii); the other direction is proven similarly. $\Box$

Exercise 2 Establish Lemma 1(ii) by direct calculation, avoiding the use of holomorphic functions. (Hint: the calculations are cleanest if one uses Wirtinger derivatives, as per Exercise 27 of Notes 1.)

Exercise 3 Let ${\phi: U \rightarrow V}$ be a complex diffeomorphism between two open subsets ${U,V}$ of ${{\bf C}}$, let ${z_0}$ be a point in ${U}$, let ${m}$ be a natural number, and let ${f: V \rightarrow {\bf C} \cup \{\infty\}}$ be holomorphic. Show that ${f: V \rightarrow {\bf C} \cup \{\infty\}}$ has a zero (resp. a pole) of order ${m}$ at ${\phi(z_0)}$ if and only if ${f \circ \phi: U \rightarrow {\bf C} \cup \{\infty\}}$ has a zero (resp. a pole) of order ${m}$ at ${z_0}$.

From Lemma 1(ii) we can now define the notion of a harmonic function ${u: M \rightarrow {\bf R}}$ on a Riemann surface ${M}$; such a function ${u}$ is harmonic if, for every coordinate chart ${\phi_\alpha: U_\alpha \rightarrow V_\alpha}$ in some atlas, the map ${u \circ \phi_\alpha^{-1}: V_\alpha \rightarrow {\bf R}}$ is harmonic. Lemma 1(ii) ensures that this definition of harmonicity does not depend on the choice of atlas. Similarly, using Exercise 3 one can define what it means for a holomorphic map ${f: M \rightarrow {\bf C} \cup \{\infty\}}$ on a Riemann surface ${M}$ to have a pole or zero of a given order at a point ${p_0 \in M}$, with the definition being independent of the choice of atlas.

In view of Lemma 1, it is thus natural to ask which Riemann surfaces are complex diffeomorphic to each other, and more generally to understand the space of holomorphic maps from one given Riemann surface to another. We will initially focus attention on three important model Riemann surfaces:

• (i) (Elliptic model) The Riemann sphere ${{\bf C} \cup \{\infty\}}$;
• (ii) (Parabolic model) The complex plane ${{\bf C}}$; and
• (iii) (Hyperbolic model) The unit disk ${D(0,1)}$.

The designation of these model Riemann surfaces as elliptic, parabolic, and hyperbolic comes from Riemannian geometry, where it is natural to endow each of these surfaces with a constant curvature Riemannian metric which is positive, zero, or negative in the elliptic, parabolic, and hyperbolic cases respectively. However, we will not discuss Riemannian geometry further here.

All three model Riemann surfaces are simply connected, but none of them are complex diffeomorphic to any other; indeed, there are no non-constant holomorphic maps from the Riemann sphere to the plane or the disk, nor are there any non-constant holomorphic maps from the plane to the disk (although there are plenty of holomorphic maps going in the opposite directions). The complex automorphisms (that is, the complex diffeomorphisms from a surface to itself) of each of the three surfaces can be classified explicitly. The automorphisms of the Riemann sphere turn out to be the Möbius transformations ${z \mapsto \frac{az+b}{cz+d}}$ with ${ad-bc \neq 0}$, also known as fractional linear transformations. The automorphisms of the complex plane are the linear transformations ${z \mapsto az+b}$ with ${a \neq 0}$, and the automorphisms of the disk are the fractional linear transformations of the form ${z \mapsto e^{i\theta} \frac{\alpha - z}{1 - \overline{\alpha} z}}$ for ${\theta \in {\bf R}}$ and ${\alpha \in D(0,1)}$. Holomorphic maps ${f: D(0,1) \rightarrow D(0,1)}$ from the disk ${D(0,1)}$ to itself that fix the origin obey a basic but incredibly important estimate known as the Schwarz lemma: they are “dominated” by the identity function ${z \mapsto z}$ in the sense that ${|f(z)| \leq |z|}$ for all ${z \in D(0,1)}$. Among other things, this lemma gives guidance to determine when a given Riemann surface is complex diffeomorphic to a disk; we shall discuss this point further below.

It is a beautiful and fundamental fact in complex analysis that these three model Riemann surfaces are in fact an exhaustive list of the simply connected Riemann surfaces, up to complex diffeomorphism. More precisely, we have the Riemann mapping theorem and the uniformisation theorem:

Theorem 4 (Riemann mapping theorem) Let ${U}$ be a simply connected open subset of ${{\bf C}}$ that is not all of ${{\bf C}}$. Then ${U}$ is complex diffeomorphic to ${D(0,1)}$.

Theorem 5 (Uniformisation theorem) Let ${M}$ be a simply connected Riemann surface. Then ${M}$ is complex diffeomorphic to ${{\bf C} \cup \{\infty\}}$, ${{\bf C}}$, or ${D(0,1)}$.

As we shall see, every connected Riemann surface can be viewed as the quotient of its simply connected universal cover by a discrete group of automorphisms known as deck transformations. This in principle gives a complete classification of Riemann surfaces up to complex diffeomorphism, although the situation is still somewhat complicated in the hyperbolic case because of the wide variety of discrete groups of automorphisms available in that case.

We will prove the Riemann mapping theorem in these notes, using the elegant argument of Koebe that is based on the Schwarz lemma and Montel’s theorem (Exercise 57 of Notes 4). The uniformisation theorem is however more difficult to establish; we discuss some components of a proof (based on the Perron method of subharmonic functions) here, but stop short of providing a complete proof.

The above theorems show that it is in principle possible to conformally map various domains into model domains such as the unit disk, but the proofs of these theorems do not readily produce explicit conformal maps for this purpose. For some domains we can just write down a suitable such map. For instance:

Exercise 6 (Cayley transform) Let ${{\bf H} := \{ z \in {\bf C}: \mathrm{Im} z > 0 \}}$ be the upper half-plane. Show that the Cayley transform ${\phi: {\bf H} \rightarrow D(0,1)}$, defined by

$\displaystyle \phi(z) := \frac{z-i}{z+i},$

is a complex diffeomorphism from the upper half-plane ${{\bf H}}$ to the disk ${D(0,1)}$, with inverse map ${\phi^{-1}: D(0,1) \rightarrow {\bf H}}$ given by

$\displaystyle \phi^{-1}(w) := i \frac{1+w}{1-w}.$

Exercise 7 Show that for any real numbers ${a, the strip ${\{ z \in {\bf C}: a < \mathrm{Re}(z) < b \}}$ is complex diffeomorphic to the disk ${D(0,1)}$. (Hint: use the complex exponential and a linear transformation to map the strip onto the half-plane ${{\bf H}}$.)

We will discuss some other explicit conformal maps in this set of notes, such as the Schwarz-Christoffel maps that transform the upper half-plane ${{\bf H}}$ to polygonal regions. Further examples of conformal mapping can be found in the text of Stein-Shakarchi.

In the previous set of notes we saw that functions ${f: U \rightarrow {\bf C}}$ that were holomorphic on an open set ${U}$ enjoyed a large number of useful properties, particularly if the domain ${U}$ was simply connected. In many situations, though, we need to consider functions ${f}$ that are only holomorphic (or even well-defined) on most of a domain ${U}$, thus they are actually functions ${f: U \backslash S \rightarrow {\bf C}}$ outside of some small singular set ${S}$ inside ${U}$. (In this set of notes we only consider interior singularities; one can also discuss singular behaviour at the boundary of ${U}$, but this is a whole separate topic and will not be pursued here.) Since we have only defined the notion of holomorphicity on open sets, we will require the singular sets ${S}$ to be closed, so that the domain ${U \backslash S}$ on which ${f}$ remains holomorphic is still open. A typical class of examples are the functions of the form ${\frac{f(z)}{z-z_0}}$ that were already encountered in the Cauchy integral formula; if ${f: U \rightarrow {\bf C}}$ is holomorphic and ${z_0 \in U}$, such a function would be holomorphic save for a singularity at ${z_0}$. Another basic class of examples are the rational functions ${P(z)/Q(z)}$, which are holomorphic outside of the zeroes of the denominator ${Q}$.

Singularities come in varying levels of “badness” in complex analysis. The least harmful type of singularity is the removable singularity – a point ${z_0}$ which is an isolated singularity (i.e., an isolated point of the singular set ${S}$) where the function ${f}$ is undefined, but for which one can extend the function across the singularity in such a fashion that the function becomes holomorphic in a neighbourhood of the singularity. A typical example is that of the complex sinc function ${\frac{\sin(z)}{z}}$, which has a removable singularity at the origin ${0}$, which can be removed by declaring the sinc function to equal ${1}$ at ${0}$. The detection of isolated removable singularities can be accomplished by Riemann’s theorem on removable singularities (Exercise 35 from Notes 3): if a holomorphic function ${f: U \backslash S \rightarrow {\bf C}}$ is bounded near an isolated singularity ${z_0 \in S}$, then the singularity at ${z_0}$ may be removed.

After removable singularities, the mildest form of singularity one can encounter is that of a pole – an isolated singularity ${z_0}$ such that ${f(z)}$ can be factored as ${f(z) = \frac{g(z)}{(z-z_0)^m}}$ for some ${m \geq 1}$ (known as the order of the pole), where ${g}$ has a removable singularity at ${z_0}$ (and is non-zero at ${z_0}$ once the singularity is removed). Such functions have already made a frequent appearance in previous notes, particularly the case of simple poles when ${m=1}$. The behaviour near ${z_0}$ of function ${f}$ with a pole of order ${m}$ is well understood: for instance, ${|f(z)|}$ goes to infinity as ${z}$ approaches ${z_0}$ (at a rate comparable to ${|z-z_0|^{-m}}$). These singularities are not, strictly speaking, removable; but if one compactifies the range ${{\bf C}}$ of the holomorphic function ${f: U \backslash S \rightarrow {\bf C}}$ to a slightly larger space ${{\bf C} \cup \{\infty\}}$ known as the Riemann sphere, then the singularity can be removed. In particular, functions ${f: U \backslash S \rightarrow {\bf C}}$ which only have isolated singularities that are either poles or removable can be extended to holomorphic functions ${f: U \rightarrow {\bf C} \cup \{\infty\}}$ to the Riemann sphere. Such functions are known as meromorphic functions, and are nearly as well-behaved as holomorphic functions in many ways. In fact, in one key respect, the family of meromorphic functions is better: the meromorphic functions on ${U}$ turn out to form a field, in particular the quotient of two meromorphic functions is again meromorphic (if the denominator is not identically zero).

Unfortunately, there are isolated singularities that are neither removable or poles, and are known as essential singularities. A typical example is the function ${f(z) = e^{1/z}}$, which turns out to have an essential singularity at ${z=0}$. The behaviour of such essential singularities is quite wild; we will show here the Casorati-Weierstrass theorem, which shows that the image of ${f}$ near the essential singularity is dense in the complex plane, as well as the more difficult great Picard theorem which asserts that in fact the image can omit at most one point in the complex plane. Nevertheless, around any isolated singularity (even the essential ones) ${z_0}$, it is possible to expand ${f}$ as a variant of a Taylor series known as a Laurent series ${\sum_{n=-\infty}^\infty a_n (z-z_0)^n}$. The ${\frac{1}{z-z_0}}$ coefficient ${a_{-1}}$ of this series is particularly important for contour integration purposes, and is known as the residue of ${f}$ at the isolated singularity ${z_0}$. These residues play a central role in a common generalisation of Cauchy’s theorem and the Cauchy integral formula known as the residue theorem, which is a particularly useful tool for computing (or at least transforming) contour integrals of meromorphic functions, and has proven to be a particularly popular technique to use in analytic number theory. Within complex analysis, one important consequence of the residue theorem is the argument principle, which gives a topological (and analytical) way to control the zeroes and poles of a meromorphic function.

Finally, there are the non-isolated singularities. Little can be said about these singularities in general (for instance, the residue theorem does not directly apply in the presence of such singularities), but certain types of non-isolated singularities are still relatively easy to understand. One particularly common example of such non-isolated singularity arises when trying to invert a non-injective function, such as the complex exponential ${z \mapsto \exp(z)}$ or a power function ${z \mapsto z^n}$, leading to branches of multivalued functions such as the complex logarithm ${z \mapsto \log(z)}$ or the ${n^{th}}$ root function ${z \mapsto z^{1/n}}$ respectively. Such branches will typically have a non-isolated singularity along a branch cut; this branch cut can be moved around the complex domain by switching from one branch to another, but usually cannot be eliminated entirely, unless one is willing to lift up the domain ${U}$ to a more general type of domain known as a Riemann surface. As such, one can view branch cuts as being an “artificial” form of singularity, being an artefact of a choice of local coordinates of a Riemann surface, rather than reflecting any intrinsic singularity of the function itself. The further study of Riemann surfaces is an important topic in complex analysis (as well as the related fields of complex geometry and algebraic geometry), but unfortunately this topic will probably be postponed to the next course in this sequence (which I will not be teaching).

Throughout this post we shall always work in the smooth category, thus all manifolds, maps, coordinate charts, and functions are assumed to be smooth unless explicitly stated otherwise.

A (real) manifold ${M}$ can be defined in at least two ways. On one hand, one can define the manifold extrinsically, as a subset of some standard space such as a Euclidean space ${{\bf R}^d}$. On the other hand, one can define the manifold intrinsically, as a topological space equipped with an atlas of coordinate charts. The fundamental embedding theorems show that, under reasonable assumptions, the intrinsic and extrinsic approaches give the same classes of manifolds (up to isomorphism in various categories). For instance, we have the following (special case of) the Whitney embedding theorem:

Theorem 1 (Whitney embedding theorem) Let ${M}$ be a compact manifold. Then there exists an embedding ${u: M \rightarrow {\bf R}^d}$ from ${M}$ to a Euclidean space ${{\bf R}^d}$.

In fact, if ${M}$ is ${n}$-dimensional, one can take ${d}$ to equal ${2n}$, which is often best possible (easy examples include the circle ${{\bf R}/{\bf Z}}$ which embeds into ${{\bf R}^2}$ but not ${{\bf R}^1}$, or the Klein bottle that embeds into ${{\bf R}^4}$ but not ${{\bf R}^3}$). One can also relax the compactness hypothesis on ${M}$ to second countability, but we will not pursue this extension here. We give a “cheap” proof of this theorem below the fold which allows one to take ${d}$ equal to ${2n+1}$.

A significant strengthening of the Whitney embedding theorem is (a special case of) the Nash embedding theorem:

Theorem 2 (Nash embedding theorem) Let ${(M,g)}$ be a compact Riemannian manifold. Then there exists a isometric embedding ${u: M \rightarrow {\bf R}^d}$ from ${M}$ to a Euclidean space ${{\bf R}^d}$.

In order to obtain the isometric embedding, the dimension ${d}$ has to be a bit larger than what is needed for the Whitney embedding theorem; in this article of Gunther the bound

$\displaystyle d = \max( n(n+5)/2, n(n+3)/2 + 5) \ \ \ \ \ (1)$

is attained, which I believe is still the record for large ${n}$. (In the converse direction, one cannot do better than ${d = \frac{n(n+1)}{2}}$, basically because this is the number of degrees of freedom in the Riemannian metric ${g}$.) Nash’s original proof of theorem used what is now known as Nash-Moser inverse function theorem, but a subsequent simplification of Gunther allowed one to proceed using just the ordinary inverse function theorem (in Banach spaces).

I recently had the need to invoke the Nash embedding theorem to establish a blowup result for a nonlinear wave equation, which motivated me to go through the proof of the theorem more carefully. Below the fold I give a proof of the theorem that does not attempt to give an optimal value of ${d}$, but which hopefully isolates the main ideas of the argument (as simplified by Gunther). One advantage of not optimising in ${d}$ is that it allows one to freely exploit the very useful tool of pairing together two maps ${u_1: M \rightarrow {\bf R}^{d_1}}$, ${u_2: M \rightarrow {\bf R}^{d_2}}$ to form a combined map ${(u_1,u_2): M \rightarrow {\bf R}^{d_1+d_2}}$ that can be closer to an embedding or an isometric embedding than the original maps ${u_1,u_2}$. This lets one perform a “divide and conquer” strategy in which one first starts with the simpler problem of constructing some “partial” embeddings of ${M}$ and then pairs them together to form a “better” embedding.

In preparing these notes, I found the articles of Deane Yang and of Siyuan Lu to be helpful.

In addition to the Fields medallists mentioned in the previous post, the IMU also awarded the Nevanlinna prize to Subhash Khot, the Gauss prize to Stan Osher (my colleague here at UCLA!), and the Chern medal to Phillip Griffiths. Like I did in 2010, I’ll try to briefly discuss one result of each of the prize winners, though the fields of mathematics here are even further from my expertise than those discussed in the previous post (and all the caveats from that post apply here also).

Subhash Khot is best known for his Unique Games Conjecture, a problem in complexity theory that is perhaps second in importance only to the ${P \neq NP}$ problem for the purposes of demarcating the mysterious line between “easy” and “hard” problems (if one follows standard practice and uses “polynomial time” as the definition of “easy”). The ${P \neq NP}$ problem can be viewed as an assertion that it is difficult to find exact solutions to certain standard theoretical computer science problems (such as ${k}$-SAT); thanks to the NP-completeness phenomenon, it turns out that the precise problem posed here is not of critical importance, and ${k}$-SAT may be substituted with one of the many other problems known to be NP-complete. The unique games conjecture is similarly an assertion about the difficulty of finding even approximate solutions to certain standard problems, in particular “unique games” problems in which one needs to colour the vertices of a graph in such a way that the colour of one vertex of an edge is determined uniquely (via a specified matching) by the colour of the other vertex. This is an easy problem to solve if one insists on exact solutions (in which 100% of the edges have a colouring compatible with the specified matching), but becomes extremely difficult if one permits approximate solutions, with no exact solution available. In analogy with the NP-completeness phenomenon, the threshold for approximate satisfiability of many other problems (such as the MAX-CUT problem) is closely connected with the truth of the unique games conjecture; remarkably, the truth of the unique games conjecture would imply asymptotically sharp thresholds for many of these problems. This has implications for many theoretical computer science constructions which rely on hardness of approximation, such as probabilistically checkable proofs. For a more detailed survey of the unique games conjecture and its implications, see this Bulletin article of Trevisan.

My colleague Stan Osher has worked in many areas of applied mathematics, ranging from image processing to modeling fluids for major animation studies such as Pixar or Dreamworks, but today I would like to talk about one of his contributions that is close to an area of my own expertise, namely compressed sensing. One of the basic reconstruction problem in compressed sensing is the basis pursuit problem of finding the vector ${x \in {\bf R}^n}$ in an affine space ${\{ x \in {\bf R}^n: Ax = b \}}$ (where ${b \in {\bf R}^m}$ and ${A \in {\bf R}^{m\times n}}$ are given, and ${m}$ is typically somewhat smaller than ${n}$) which minimises the ${\ell^1}$-norm ${\|x\|_{\ell^1} := \sum_{i=1}^n |x_i|}$ of the vector ${x}$. This is a convex optimisation problem, and thus solvable in principle (it is a polynomial time problem, and thus “easy” in the above theoretical computer science sense). However, once ${n}$ and ${m}$ get moderately large (e.g. of the order of ${10^6}$), standard linear optimisation routines begin to become computationally expensive; also, it is difficult for off-the-shelf methods to exploit any additional structure (e.g. sparsity) in the measurement matrix ${A}$. Much of the problem comes from the fact that the functional ${x \mapsto \|x\|_1}$ is only barely convex. One way to speed up the optimisation problem is to relax it by replacing the constraint ${Ax=b}$ with a convex penalty term ${\frac{1}{2 \epsilon} \|Ax-b\|_{\ell^2}^2}$, thus one is now trying to minimise the unconstrained functional

$\displaystyle \|x\|_1 + \frac{1}{2\epsilon} \|Ax-b\|_{\ell^2}^2.$

This functional is more convex, and is over a computationally simpler domain ${{\bf R}^n}$ than the affine space ${\{x \in {\bf R}^n: Ax=b\}}$, so is easier (though still not entirely trivial) to optimise over. However, the minimiser ${x^\epsilon}$ to this problem need not match the minimiser ${x^0}$ to the original problem, particularly if the (sub-)gradient ${\partial \|x\|_1}$ of the original functional ${\|x\|_1}$ is large at ${x^0}$, and if ${\epsilon}$ is not set to be small. (And setting ${\epsilon}$ too small will cause other difficulties with numerically solving the optimisation problem, due to the need to divide by very small denominators.) However, if one modifies the objective function by an additional linear term

$\displaystyle \|x\|_1 - \langle p, x \rangle + \frac{1}{2 \epsilon} \|Ax-b\|_{\ell^2}^2$

then some simple convexity considerations reveal that the minimiser to this new problem will match the minimiser ${x^0}$ to the original problem, provided that ${p}$ is (or more precisely, lies in) the (sub-)gradient ${\partial \|x\|_1}$ of ${\|x\|_1}$ at ${x^0}$ – even if ${\epsilon}$ is not small. But, one does not know in advance what the correct value of ${p}$ should be, because one does not know what the minimiser ${x^0}$ is.

With Yin, Goldfarb and Darbon, Osher introduced a Bregman iteration method in which one solves for ${x}$ and ${p}$ simultaneously; given an initial guess ${x^k, p^k}$ for both ${x^k}$ and ${p^k}$, one first updates ${x^k}$ to the minimiser ${x^{k+1} \in {\bf R}^n}$ of the convex functional

$\displaystyle \|x\|_1 - \langle p^k, x \rangle + \frac{1}{2 \epsilon} \|Ax-b\|_{\ell^2}^2 \ \ \ \ \ (1)$

and then updates ${p^{k+1}}$ to the natural value of the subgradient ${\partial \|x\|_1}$ at ${x^{k+1}}$, namely

$\displaystyle p^{k+1} := p^k - \nabla \frac{1}{2 \epsilon} \|Ax-b\|_{\ell^2}^2|_{x=x^{k+1}} = p_k - \frac{1}{\epsilon} (Ax^k - b)$

(note upon taking the first variation of (1) that ${p^{k+1}}$ is indeed in the subgradient). This procedure converges remarkably quickly (both in theory and in practice) to the true minimiser ${x^0}$ even for non-small values of ${\epsilon}$, and also has some ability to be parallelised, and has led to actual performance improvements of an order of magnitude or more in certain compressed sensing problems (such as reconstructing an MRI image).

Phillip Griffiths has made many contributions to complex, algebraic and differential geometry, and I am not qualified to describe most of these; my primary exposure to his work is through his text on algebraic geometry with Harris, but as excellent though that text is, it is not really representative of his research. But I thought I would mention one cute result of his related to the famous Nash embedding theorem. Suppose that one has a smooth ${n}$-dimensional Riemannian manifold that one wants to embed locally into a Euclidean space ${{\bf R}^m}$. The Nash embedding theorem guarantees that one can do this if ${m}$ is large enough depending on ${n}$, but what is the minimal value of ${m}$ one can get away with? Many years ago, my colleague Robert Greene showed that ${m = \frac{n(n+1)}{2} + n}$ sufficed (a simplified proof was subsequently given by Gunther). However, this is not believed to be sharp; if one replaces “smooth” with “real analytic” then a standard Cauchy-Kovalevski argument shows that ${m = \frac{n(n+1)}{2}}$ is possible, and no better. So this suggests that ${m = \frac{n(n+1)}{2}}$ is the threshold for the smooth problem also, but this remains open in general. The cases ${n=1}$ is trivial, and the ${n=2}$ case is not too difficult (if the curvature is non-zero) as the codimension ${m-n}$ is one in this case, and the problem reduces to that of solving a Monge-Ampere equation. With Bryant and Yang, Griffiths settled the ${n=3}$ case, under a non-degeneracy condition on the Einstein tensor. This is quite a serious paper – over 100 pages combining differential geometry, PDE methods (e.g. Nash-Moser iteration), and even some harmonic analysis (e.g. they rely at one key juncture on an extension theorem of my advisor, Elias Stein). The main difficulty is that that the relevant PDE degenerates along a certain characteristic submanifold of the cotangent bundle, which then requires an extremely delicate analysis to handle.

Let ${f: {\bf R}^3 \rightarrow {\bf R}}$ be an irreducible polynomial in three variables. As ${{\bf R}}$ is not algebraically closed, the zero set ${Z_{\bf R}(f) = \{ x \in{\bf R}^3: f(x)=0\}}$ can split into various components of dimension between ${0}$ and ${2}$. For instance, if ${f(x_1,x_2,x_3) = x_1^2+x_2^2}$, the zero set ${Z_{\bf R}(f)}$ is a line; more interestingly, if ${f(x_1,x_2,x_3) = x_3^2 + x_2^2 - x_2^3}$, then ${Z_{\bf R}(f)}$ is the union of a line and a surface (or the product of an acnodal cubic curve with a line). We will assume that the ${2}$-dimensional component ${Z_{{\bf R},2}(f)}$ is non-empty, thus defining a real surface in ${{\bf R}^3}$. In particular, this hypothesis implies that ${f}$ is not just irreducible over ${{\bf R}}$, but is in fact absolutely irreducible (i.e. irreducible over ${{\bf C}}$), since otherwise one could use the complex factorisation of ${f}$ to contain ${Z_{\bf R}(f)}$ inside the intersection ${{\bf Z}_{\bf C}(g) \cap {\bf Z}_{\bf C}(\bar{g})}$ of the complex zero locus of complex polynomial ${g}$ and its complex conjugate, with ${g,\bar{g}}$ having no common factor, forcing ${Z_{\bf R}(f)}$ to be at most one-dimensional. (For instance, in the case ${f(x_1,x_2,x_3)=x_1^2+x_2^2}$, one can take ${g(z_1,z_2,z_3) = z_1 + i z_2}$.) Among other things, this makes ${{\bf Z}_{{\bf R},2}(f)}$ a Zariski-dense subset of ${{\bf Z}_{\bf C}(f)}$, thus any polynomial identity which holds true at every point of ${{\bf Z}_{{\bf R},2}(f)}$, also holds true on all of ${{\bf Z}_{\bf C}(f)}$. This allows us to easily use tools from algebraic geometry in this real setting, even though the reals are not quite algebraically closed.

The surface ${Z_{{\bf R},2}(f)}$ is said to be ruled if, for a Zariski open dense set of points ${x \in Z_{{\bf R},2}(f)}$, there exists a line ${l_x = \{ x+tv_x: t \in {\bf R} \}}$ through ${x}$ for some non-zero ${v_x \in {\bf R}^3}$ which is completely contained in ${Z_{{\bf R},2}(f)}$, thus

$\displaystyle f(x+tv_x)=0$

for all ${t \in {\bf R}}$. Also, a point ${x \in {\bf Z}_{{\bf R},2}(f)}$ is said to be a flecnode if there exists a line ${l_x = \{ x+tv_x: t \in {\bf R}\}}$ through ${x}$ for some non-zero ${v_x \in {\bf R}^3}$ which is tangent to ${Z_{{\bf R},2}(f)}$ to third order, in the sense that

$\displaystyle f(x+tv_x)=O(t^4)$

as ${t \rightarrow 0}$, or equivalently that

$\displaystyle \frac{d^j}{dt^j} f(x+tv_x)|_{t=0} = 0 \ \ \ \ \ (1)$

for ${j=0,1,2,3}$. Clearly, if ${Z_{{\bf R},2}(f)}$ is a ruled surface, then a Zariski open dense set of points on ${Z_{{\bf R},2}}$ are a flecnode. We then have the remarkable theorem (discovered first by Monge, and then later by Cayley and Salmon) asserting the converse:

Theorem 1 (Monge-Cayley-Salmon theorem) Let ${f: {\bf R}^3 \rightarrow {\bf R}}$ be an irreducible polynomial with ${{\bf Z}_{{\bf R},2}}$ non-empty. Suppose that a Zariski dense set of points in ${Z_{{\bf R},2}(f)}$ are flecnodes. Then ${Z_{{\bf R},2}(f)}$ is a ruled surface.

Among other things, this theorem was used in the celebrated result of Guth and Katz that almost solved the Erdos distance problem in two dimensions, as discussed in this previous blog post. Vanishing to third order is necessary: observe that in a surface of negative curvature, such as the saddle ${\{ (x_1,x_2,x_3): x_3 = x_1^2 - x_2^2 \}}$, every point on the surface is tangent to second order to a line (the line in the direction for which the second fundamental form vanishes). This surface happens to be ruled, but a generic perturbation of this surface (e.g. ${x_3 = x_1^2 - x_2^2 + x_2^4}$) will no longer be ruled, although it is still negative curvature near the origin.

The original proof of the Monge-Cayley-Salmon theorem is not easily accessible and not written in modern language. A modern proof of this theorem (together with substantial generalisations, for instance to higher dimensions) is given by Landsberg; the proof uses the machinery of modern algebraic geometry. The purpose of this post is to record an alternate proof of the Monge-Cayley-Salmon theorem based on classical differential geometry (in particular, the notion of torsion of a curve) and basic ODE methods (in particular, Gronwall’s inequality and the Picard existence theorem). The idea is to “integrate” the lines ${l_x}$ indicated by the flecnode to produce smooth curves ${\gamma}$ on the surface ${{\bf Z}_{{\bf R},2}}$; one then uses the vanishing (1) and some basic calculus to conclude that these curves have zero torsion and are thus planar curves. Some further manipulation using (1) (now just to second order instead of third) then shows that these curves are in fact straight lines, giving the ruling on the surface.

Update: Janos Kollar has informed me that the above theorem was essentially known to Monge in 1809; see his recent arXiv note for more details.

I thank Larry Guth and Micha Sharir for conversations leading to this post.

In this set of notes, we describe the basic analytic structure theory of Lie groups, by relating them to the simpler concept of a Lie algebra. Roughly speaking, the Lie algebra encodes the “infinitesimal” structure of a Lie group, but is a simpler object, being a vector space rather than a nonlinear manifold. Nevertheless, thanks to the fundamental theorems of Lie, the Lie algebra can be used to reconstruct the Lie group (at a local level, at least), by means of the exponential map and the Baker-Campbell-Hausdorff formula. As such, the local theory of Lie groups is completely described (in principle, at least) by the theory of Lie algebras, which leads to a number of useful consequences, such as the following:

• (Local Lie implies Lie) A topological group ${G}$ is Lie (i.e. it is isomorphic to a Lie group) if and only if it is locally Lie (i.e. the group operations are smooth near the origin).
• (Uniqueness of Lie structure) A topological group has at most one smooth structure on it that makes it Lie.
• (Weak regularity implies strong regularity, I) Lie groups are automatically real analytic. (In fact one only needs a “local ${C^{1,1}}$” regularity on the group structure to obtain real analyticity.)
• (Weak regularity implies strong regularity, II) A continuous homomorphism from one Lie group to another is automatically smooth (and real analytic).

The connection between Lie groups and Lie algebras also highlights the role of one-parameter subgroups of a topological group, which will play a central role in the solution of Hilbert’s fifth problem.

We note that there is also a very important algebraic structure theory of Lie groups and Lie algebras, in which the Lie algebra is split into solvable and semisimple components, with the latter being decomposed further into simple components, which can then be completely classified using Dynkin diagrams. This classification is of fundamental importance in many areas of mathematics (e.g. representation theory, arithmetic geometry, and group theory), and many of the deeper facts about Lie groups and Lie algebras are proven via this classification (although in such cases it can be of interest to also find alternate proofs that avoid the classification). However, it turns out that we will not need this theory in this course, and so we will not discuss it further here (though it can of course be found in any graduate text on Lie groups and Lie algebras).

Over the past few months or so, I have been brushing up on my Lie group theory, as part of my project to fully understand the theory surrounding Hilbert’s fifth problem. Every so often, I encounter a basic fact in Lie theory which requires a slightly non-trivial “trick” to prove; I am recording two of them here, so that I can find these tricks again when I need to.

The first fact concerns the exponential map ${\exp: {\mathfrak g} \rightarrow G}$ from a Lie algebra ${{\mathfrak g}}$ of a Lie group ${G}$ to that group. (For this discussion we will only consider finite-dimensional Lie groups and Lie algebras over the reals ${{\bf R}}$.) A basic fact in the subject is that the exponential map is locally a homeomorphism: there is a neighbourhood of the origin in ${{\mathfrak g}}$ that is mapped homeomorphically by the exponential map to a neighbourhood of the identity in ${G}$. This local homeomorphism property is the foundation of an important dictionary between Lie groups and Lie algebras.

It is natural to ask whether the exponential map is globally a homeomorphism, and not just locally: in particular, whether the exponential map remains both injective and surjective. For instance, this is the case for connected, simply connected, nilpotent Lie groups (as can be seen from the Baker-Campbell-Hausdorff formula.)

The circle group ${S^1}$, which has ${{\bf R}}$ as its Lie algebra, already shows that global injectivity fails for any group that contains a circle subgroup, which is a huge class of examples (including, for instance, the positive dimensional compact Lie groups, or non-simply-connected Lie groups). Surjectivity also obviously fails for disconnected groups, since the Lie algebra is necessarily connected, and so the image under the exponential map must be connected also. However, even for connected Lie groups, surjectivity can fail. To see this, first observe that if the exponential map was surjective, then every group element ${g \in G}$ has a square root (i.e. an element ${h \in G}$ with ${h^2 = g}$), since ${\exp(x)}$ has ${\exp(x/2)}$ as a square root for any ${x \in {\mathfrak g}}$. However, there exist elements in connected Lie groups without square roots. A simple example is provided by the matrix

$\displaystyle g = \begin{pmatrix} -4 & 0 \\ 0 & -1/4 \end{pmatrix}$

in the connected Lie group ${SL_2({\bf R})}$. This matrix has eigenvalues ${-4}$, ${-1/4}$. Thus, if ${h \in SL_2({\bf R})}$ is a square root of ${g}$, we see (from the Jordan normal form) that it must have at least one eigenvalue in ${\{-2i,+2i\}}$, and at least one eigenvalue in ${\{-i/2,i/2\}}$. On the other hand, as ${h}$ has real coefficients, the complex eigenvalues must come in conjugate pairs ${\{ a+bi, a-bi\}}$. Since ${h}$ can only have at most ${2}$ eigenvalues, we obtain a contradiction.

However, there is an important case where surjectivity is recovered:

Proposition 1 If ${G}$ is a compact connected Lie group, then the exponential map is surjective.

Proof: The idea here is to relate the exponential map in Lie theory to the exponential map in Riemannian geometry. We first observe that every compact Lie group ${G}$ can be given the structure of a Riemannian manifold with a bi-invariant metric. This can be seen in one of two ways. Firstly, one can put an arbitrary positive definite inner product on ${{\mathfrak g}}$ and average it against the adjoint action of ${G}$ using Haar probability measure (which is available since ${G}$ is compact); this gives an ad-invariant positive-definite inner product on ${{\mathfrak g}}$ that one can then translate by either left or right translation to give a bi-invariant Riemannian structure on ${G}$. Alternatively, one can use the Peter-Weyl theorem to embed ${G}$ in a unitary group ${U(n)}$, at which point one can induce a bi-invariant metric on ${G}$ from the one on the space ${M_n({\bf C}) \equiv {\bf C}^{n^2}}$ of ${n \times n}$ complex matrices.

As ${G}$ is connected and compact and thus complete, we can apply the Hopf-Rinow theorem and conclude that any two points are connected by at least one geodesic, so that the Riemannian exponential map from ${{\mathfrak g}}$ to ${G}$ formed by following geodesics from the origin is surjective. But one can check that the Lie exponential map and Riemannian exponential map agree; for instance, this can be seen by noting that the group structure naturally defines a connection on the tangent bundle which is both torsion-free and preserves the bi-invariant metric, and must therefore agree with the Levi-Civita metric. (Alternatively, one can embed into a unitary group ${U(n)}$ and observe that ${G}$ is totally geodesic inside ${U(n)}$, because the geodesics in ${U(n)}$ can be described explicitly in terms of one-parameter subgroups.) The claim follows. $\Box$

Remark 1 While it is quite nice to see Riemannian geometry come in to prove this proposition, I am curious to know if there is any other proof of surjectivity for compact connected Lie groups that does not require explicit introduction of Riemannian geometry concepts.

The other basic fact I learned recently concerns the algebraic nature of Lie groups and Lie algebras. An important family of examples of Lie groups are the algebraic groups – algebraic varieties with a group law given by algebraic maps. Given that one can always automatically upgrade the smooth structure on a Lie group to analytic structure (by using the Baker-Campbell-Hausdorff formula), it is natural to ask whether one can upgrade the structure further to an algebraic structure. Unfortunately, this is not always the case. A prototypical example of this is given by the one-parameter subgroup

$\displaystyle G := \{ \begin{pmatrix} t & 0 \\ 0 & t^\alpha \end{pmatrix}: t \in {\bf R}^+ \} \ \ \ \ \ (1)$

of ${GL_2({\bf R})}$. This is a Lie group for any exponent ${\alpha \in {\bf R}}$, but if ${\alpha}$ is irrational, then the curve that ${G}$ traces out is not an algebraic subset of ${GL_2({\bf R})}$ (as one can see by playing around with Puiseux series).

This is not a true counterexample to the claim that every Lie group can be given the structure of an algebraic group, because one can give ${G}$ a different algebraic structure than one inherited from the ambient group ${GL_2({\bf R})}$. Indeed, ${G}$ is clearly isomorphic to the additive group ${{\bf R}}$, which is of course an algebraic group. However, a modification of the above construction works:

Proposition 2 There exists a Lie group ${G}$ that cannot be given the structure of an algebraic group.

Proof: We use an example from the text of Tauvel and Yu (that I found via this MathOverflow posting). We consider the subgroup

$\displaystyle G := \{ \begin{pmatrix} 1 & 0 & 0 \\ x & t & 0 \\ y & 0 & t^\alpha \end{pmatrix}: x, y \in {\bf R}; t \in {\bf R}^+ \}$

of ${GL_3({\bf R})}$, with ${\alpha}$ an irrational number. This is a three-dimensional (metabelian) Lie group, whose Lie algebra ${{\mathfrak g} \subset {\mathfrak gl}_3({\bf R})}$ is spanned by the elements

$\displaystyle X := \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \alpha \end{pmatrix}$

$\displaystyle Y := \begin{pmatrix} 0 & 0 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$

$\displaystyle Z := \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ -\alpha & 0 & 0 \end{pmatrix}$

with the Lie bracket given by

$\displaystyle [Y,X] = -Y; [Z,X] = -\alpha Z; [Y,Z] = 0.$

As such, we see that if we use the basis ${X, Y, Z}$ to identify ${{\mathfrak g}}$ to ${{\bf R}^3}$, then adjoint representation of ${G}$ is the identity map.

If ${G}$ is an algebraic group, it is easy to see that the adjoint representation ${\hbox{Ad}: G \rightarrow GL({\mathfrak g})}$ is also algebraic, and so ${\hbox{Ad}(G) = G}$ is algebraic in ${GL({\mathfrak g})}$. Specialising to our specific example, in which adjoint representation is the identity, we conclude that if ${G}$ has any algebraic structure, then it must also be an algebraic subgroup of ${GL_3({\bf R})}$; but ${G}$ projects to the group (1) which is not algebraic, a contradiction. $\Box$

A slight modification of the same argument also shows that not every Lie algebra is algebraic, in the sense that it is isomorphic to a Lie algebra of an algebraic group. (However, there are important classes of Lie algebras that are automatically algebraic, such as nilpotent or semisimple Lie algebras.)

Hilbert’s fifth problem asks to clarify the extent that the assumption on a differentiable or smooth structure is actually needed in the theory of Lie groups and their actions. While this question is not precisely formulated and is thus open to some interpretation, the following result of Gleason and Montgomery-Zippin answers at least one aspect of this question:

Theorem 1 (Hilbert’s fifth problem) Let ${G}$ be a topological group which is locally Euclidean (i.e. it is a topological manifold). Then ${G}$ is isomorphic to a Lie group.

Theorem 1 can be viewed as an application of the more general structural theory of locally compact groups. In particular, Theorem 1 can be deduced from the following structural theorem of Gleason and Yamabe:

Theorem 2 (Gleason-Yamabe theorem) Let ${G}$ be a locally compact group, and let ${U}$ be an open neighbourhood of the identity in ${G}$. Then there exists an open subgroup ${G'}$ of ${G}$, and a compact subgroup ${N}$ of ${G'}$ contained in ${U}$, such that ${G'/N}$ is isomorphic to a Lie group.

The deduction of Theorem 1 from Theorem 2 proceeds using the Brouwer invariance of domain theorem and is discussed in this previous post. In this post, I would like to discuss the proof of Theorem 2. We can split this proof into three parts, by introducing two additional concepts. The first is the property of having no small subgroups:

Definition 3 (NSS) A topological group ${G}$ is said to have no small subgroups, or is NSS for short, if there is an open neighbourhood ${U}$ of the identity in ${G}$ that contains no subgroups of ${G}$ other than the trivial subgroup ${\{ \hbox{id}\}}$.

An equivalent definition of an NSS group is one which has an open neighbourhood ${U}$ of the identity that every non-identity element ${g \in G \backslash \{\hbox{id}\}}$ escapes in finite time, in the sense that ${g^n \not \in U}$ for some positive integer ${n}$. It is easy to see that all Lie groups are NSS; we shall shortly see that the converse statement (in the locally compact case) is also true, though significantly harder to prove.

Another useful property is that of having what I will call a Gleason metric:

Definition 4 Let ${G}$ be a topological group. A Gleason metric on ${G}$ is a left-invariant metric ${d: G \times G \rightarrow {\bf R}^+}$ which generates the topology on ${G}$ and obeys the following properties for some constant ${C>0}$, writing ${\|g\|}$ for ${d(g,\hbox{id})}$:

• (Escape property) If ${g \in G}$ and ${n \geq 1}$ is such that ${n \|g\| \leq \frac{1}{C}}$, then ${\|g^n\| \geq \frac{1}{C} n \|g\|}$.
• (Commutator estimate) If ${g, h \in G}$ are such that ${\|g\|, \|h\| \leq \frac{1}{C}}$, then

$\displaystyle \|[g,h]\| \leq C \|g\| \|h\|, \ \ \ \ \ (1)$

where ${[g,h] := g^{-1}h^{-1}gh}$ is the commutator of ${g}$ and ${h}$.

For instance, the unitary group ${U(n)}$ with the operator norm metric ${d(g,h) := \|g-h\|_{op}}$ can easily verified to be a Gleason metric, with the commutator estimate (1) coming from the inequality

$\displaystyle \| [g,h] - 1 \|_{op} = \| gh - hg \|_{op}$

$\displaystyle = \| (g-1) (h-1) - (h-1) (g-1) \|_{op}$

$\displaystyle \leq 2 \|g-1\|_{op} \|g-1\|_{op}.$

Similarly, any left-invariant Riemannian metric on a (connected) Lie group can be verified to be a Gleason metric. From the escape property one easily sees that all groups with Gleason metrics are NSS; again, we shall see that there is a partial converse.

Remark 1 The escape and commutator properties are meant to capture “Euclidean-like” structure of the group. Other metrics, such as Carnot-Carathéodory metrics on Carnot Lie groups such as the Heisenberg group, usually fail one or both of these properties.

The proof of Theorem 2 can then be split into three subtheorems:

Theorem 5 (Reduction to the NSS case) Let ${G}$ be a locally compact group, and let ${U}$ be an open neighbourhood of the identity in ${G}$. Then there exists an open subgroup ${G'}$ of ${G}$, and a compact subgroup ${N}$ of ${G'}$ contained in ${U}$, such that ${G'/N}$ is NSS, locally compact, and metrisable.

Theorem 6 (Gleason’s lemma) Let ${G}$ be a locally compact metrisable NSS group. Then ${G}$ has a Gleason metric.

Theorem 7 (Building a Lie structure) Let ${G}$ be a locally compact group with a Gleason metric. Then ${G}$ is isomorphic to a Lie group.

Clearly, by combining Theorem 5, Theorem 6, and Theorem 7 one obtains Theorem 2 (and hence Theorem 1).

Theorem 5 and Theorem 6 proceed by some elementary combinatorial analysis, together with the use of Haar measure (to build convolutions, and thence to build “smooth” bump functions with which to create a metric, in a variant of the analysis used to prove the Birkhoff-Kakutani theorem); Theorem 5 also requires Peter-Weyl theorem (to dispose of certain compact subgroups that arise en route to the reduction to the NSS case), which was discussed previously on this blog.

In this post I would like to detail the final component to the proof of Theorem 2, namely Theorem 7. (I plan to discuss the other two steps, Theorem 5 and Theorem 6, in a separate post.) The strategy is similar to that used to prove von Neumann’s theorem, as discussed in this previous post (and von Neumann’s theorem is also used in the proof), but with the Gleason metric serving as a substitute for the faithful linear representation. Namely, one first gives the space ${L(G)}$ of one-parameter subgroups of ${G}$ enough of a structure that it can serve as a proxy for the “Lie algebra” of ${G}$; specifically, it needs to be a vector space, and the “exponential map” needs to cover an open neighbourhood of the identity. This is enough to set up an “adjoint” representation of ${G}$, whose image is a Lie group by von Neumann’s theorem; the kernel is essentially the centre of ${G}$, which is abelian and can also be shown to be a Lie group by a similar analysis. To finish the job one needs to use arguments of Kuranishi and of Gleason, as discussed in this previous post.

The arguments here can be phrased either in the standard analysis setting (using sequences, and passing to subsequences often) or in the nonstandard analysis setting (selecting an ultrafilter, and then working with infinitesimals). In my view, the two approaches have roughly the same level of complexity in this case, and I have elected for the standard analysis approach.

Remark 2 From Theorem 7 we see that a Gleason metric structure is a good enough substitute for smooth structure that it can actually be used to reconstruct the entire smooth structure; roughly speaking, the commutator estimate (1) allows for enough “Taylor expansion” of expressions such as ${g^n h^n}$ that one can simulate the fundamentals of Lie theory (in particular, construction of the Lie algebra and the exponential map, and its basic properties. The advantage of working with a Gleason metric rather than a smoother structure, though, is that it is relatively undemanding with regards to regularity; in particular, the commutator estimate (1) is roughly comparable to the imposition ${C^{1,1}}$ structure on the group ${G}$, as this is the minimal regularity to get the type of Taylor approximation (with quadratic errors) that would be needed to obtain a bound of the form (1). We will return to this point in a later post.

A (smooth) Riemannian manifold is a smooth manifold ${M}$ without boundary, equipped with a Riemannian metric ${{\rm g}}$, which assigns a length ${|v|_{{\rm g}(x)} \in {\bf R}^+}$ to every tangent vector ${v \in T_x M}$ at a point ${x \in M}$, and more generally assigns an inner product

$\displaystyle \langle v, w \rangle_{{\rm g}(x)} \in {\bf R}$

to every pair of tangent vectors ${v, w \in T_x M}$ at a point ${x \in M}$. (We use Roman font for ${g}$ here, as we will need to use ${g}$ to denote group elements later in this post.) This inner product is assumed to symmetric, positive definite, and smoothly varying in ${x}$, and the length is then given in terms of the inner product by the formula

$\displaystyle |v|_{{\rm g}(x)}^2 := \langle v, v \rangle_{{\rm g}(x)}.$

In coordinates (and also using abstract index notation), the metric ${{\rm g}}$ can be viewed as an invertible symmetric rank ${(0,2)}$ tensor ${{\rm g}_{ij}(x)}$, with

$\displaystyle \langle v, w \rangle_{{\rm g}(x)} = {\rm g}_{ij}(x) v^i w^j.$

One can also view the Riemannian metric as providing a (self-adjoint) identification between the tangent bundle ${TM}$ of the manifold and the cotangent bundle ${T^* M}$; indeed, every tangent vector ${v \in T_x M}$ is then identified with the cotangent vector ${\iota_{TM \rightarrow T^* M}(v) \in T_x^* M}$, defined by the formula

$\displaystyle \iota_{TM \rightarrow T^* M}(v)(w) := \langle v, w \rangle_{{\rm g}(x)}.$

In coordinates, ${\iota_{TM \rightarrow T^* M}(v)_i = {\rm g}_{ij} v^j}$.

A fundamental dynamical system on the tangent bundle (or equivalently, the cotangent bundle, using the above identification) of a Riemannian manifold is that of geodesic flow. Recall that geodesics are smooth curves ${\gamma: [a,b] \rightarrow M}$ that minimise the length

$\displaystyle |\gamma| := \int_a^b |\gamma'(t)|_{{\rm g}(\gamma(t))}\ dt.$

There is some degeneracy in this definition, because one can reparameterise the curve ${\gamma}$ without affecting the length. In order to fix this degeneracy (and also because the square of the speed is a more tractable quantity analytically than the speed itself), it is better if one replaces the length with the energy

$\displaystyle E(\gamma) := \frac{1}{2} \int_a^b |\gamma'(t)|_{{\rm g}(\gamma(t))}^2\ dt.$

Minimising the energy of a parameterised curve ${\gamma}$ turns out to be the same as minimising the length, together with an additional requirement that the speed ${|\gamma'(t)|_{{\rm g}(\gamma(t))}}$ stay constant in time. Minimisers (and more generally, critical points) of the energy functional (holding the endpoints fixed) are known as geodesic flows. From a physical perspective, geodesic flow governs the motion of a particle that is subject to no external forces and thus moves freely, save for the constraint that it must always lie on the manifold ${M}$.

One can also view geodesic flows as a dynamical system on the tangent bundle (with the state at any time ${t}$ given by the position ${\gamma(t) \in M}$ and the velocity ${\gamma'(t) \in T_{\gamma(t)} M}$) or on the cotangent bundle (with the state then given by the position ${\gamma(t) \in M}$ and the momentum ${\iota_{TM \rightarrow T^* M}( \gamma'(t) ) \in T_{\gamma(t)}^* M}$). With the latter perspective (sometimes referred to as cogeodesic flow), geodesic flow becomes a Hamiltonian flow, with Hamiltonian ${H: T^* M \rightarrow {\bf R}}$ given as

$\displaystyle H( x, p ) := \frac{1}{2} \langle p, p \rangle_{{\rm g}(x)^{-1}} = \frac{1}{2} {\rm g}^{ij}(x) p_i p_j$

where ${\langle ,\rangle_{{\rm g}(x)^{-1}}: T^*_x M \times T^*_x M \rightarrow {\bf R}}$ is the inverse inner product to ${\langle, \rangle_{{\rm g}(x)}: T_x M \times T_x M \rightarrow {\bf R}}$, which can be defined for instance by the formula

$\displaystyle \langle p_1, p_2 \rangle_{{\rm g}(x)^{-1}} = \langle \iota_{TM \rightarrow T^* M}^{-1}(p_1), \iota_{TM \rightarrow T^* M}^{-1}(p_2)\rangle_{{\rm g}(x)}.$

In coordinates, geodesic flow is given by Hamilton’s equations of motion

$\displaystyle \frac{d}{dt} x^i = {\rm g}^{ij} p_j; \quad \frac{d}{dt} p_i = - \frac{1}{2} (\partial_i {\rm g}^{jk}(x)) p_j p_k.$

In terms of the velocity ${v^i := \frac{d}{dt} x^i = {\rm g}^{ij} p_j}$, we can rewrite these equations as the geodesic equation

$\displaystyle \frac{d}{dt} v^i = - \Gamma^i_{jk} v^j v^k$

where

$\displaystyle \Gamma^i_{jk} = \frac{1}{2} {\rm g}^{im} (\partial_k {\rm g}_{mj} + \partial_j {\rm g}_{mk} - \partial_m {\rm g}_{jk} )$

are the Christoffel symbols; using the Levi-Civita connection ${\nabla}$, this can be written more succinctly as

$\displaystyle (\gamma^* \nabla)_t v = 0.$

If the manifold ${M}$ is an embedded submanifold of a larger Euclidean space ${R^n}$, with the metric ${{\rm g}}$ on ${M}$ being induced from the standard metric on ${{\bf R}^n}$, then the geodesic flow equation can be rewritten in the equivalent form

$\displaystyle \gamma''(t) \perp T_{\gamma(t)} M,$

where ${\gamma}$ is now viewed as taking values in ${{\bf R}^n}$, and ${T_{\gamma(t)} M}$ is similarly viewed as a subspace of ${{\bf R}^n}$. This is intuitively obvious from the geometric interpretation of geodesics: if the curvature of a curve ${\gamma}$ contains components that are transverse to the manifold rather than normal to it, then it is geometrically clear that one should be able to shorten the curve by shifting it along the indicated transverse direction. It is an instructive exercise to rigorously formulate the above intuitive argument. This fact also conforms well with one’s physical intuition of geodesic flow as the motion of a free particle constrained to be in ${M}$; the normal quantity ${\gamma''(t)}$ then corresponds to the centripetal force necessary to keep the particle lying in ${M}$ (otherwise it would fly off along a tangent line to ${M}$, as per Newton’s first law). The precise value of the normal vector ${\gamma''(t)}$ can be computed via the second fundamental form as ${\gamma''(t) = \Pi_{\gamma(t)}( \gamma'(t), \gamma'(t) )}$, but we will not need this formula here.

In a beautiful paper from 1966, Vladimir Arnold (who, sadly, passed away last week), observed that many basic equations in physics, including the Euler equations of motion of a rigid body, and also (by which is a priori a remarkable coincidence) the Euler equations of fluid dynamics of an inviscid incompressible fluid, can be viewed (formally, at least) as geodesic flows on a (finite or infinite dimensional) Riemannian manifold. And not just any Riemannian manifold: the manifold is a Lie group (or, to be truly pedantic, a torsor of that group), equipped with a right-invariant (or left-invariant, depending on one’s conventions) metric. In the context of rigid bodies, the Lie group is the group ${SE(3) = {\bf R}^3 \rtimes SO(3)}$ of rigid motions; in the context of incompressible fluids, it is the group ${Sdiff({\bf R}^3}$) of measure-preserving diffeomorphisms. The right-invariance makes the Hamiltonian mechanics of geodesic flow in this context (where it is sometimes known as the Euler-Arnold equation or the Euler-Poisson equation) quite special; it becomes (formally, at least) completely integrable, and also indicates (in principle, at least) a way to reformulate these equations in a Lax pair formulation. And indeed, many further completely integrable equations, such as the Korteweg-de Vries equation, have since been reinterpreted as Euler-Arnold flows.

From a physical perspective, this all fits well with the interpretation of geodesic flow as the free motion of a system subject only to a physical constraint, such as rigidity or incompressibility. (I do not know, though, of a similarly intuitive explanation as to why the Korteweg de Vries equation is a geodesic flow.)

One consequence of being a completely integrable system is that one has a large number of conserved quantities. In the case of the Euler equations of motion of a rigid body, the conserved quantities are the linear and angular momentum (as observed in an external reference frame, rather than the frame of the object). In the case of the two-dimensional Euler equations, the conserved quantities are the pointwise values of the vorticity (as viewed in Lagrangian coordinates, rather than Eulerian coordinates). In higher dimensions, the conserved quantity is now the (Hodge star of) the vorticity, again viewed in Lagrangian coordinates. The vorticity itself then evolves by the vorticity equation, and is subject to vortex stretching as the diffeomorphism between the initial and final state becomes increasingly sheared.

The elegant Euler-Arnold formalism is reasonably well-known in some circles (particularly in Lagrangian and symplectic dynamics, where it can be viewed as a special case of the Euler-Poincaré formalism or Lie-Poisson formalism respectively), but not in others; I for instance was only vaguely aware of it until recently, and I think that even in fluid mechanics this perspective to the subject is not always emphasised. Given the circumstances, I thought it would therefore be appropriate to present Arnold’s original 1966 paper here. (For a more modern treatment of these topics, see the books of Arnold-Khesin and Marsden-Ratiu.)

In order to avoid technical issues, I will work formally, ignoring questions of regularity or integrability, and pretending that infinite-dimensional manifolds behave in exactly the same way as their finite-dimensional counterparts. In the finite-dimensional setting, it is not difficult to make all of the formal discussion below rigorous; but the situation in infinite dimensions is substantially more delicate. (Indeed, it is a notorious open problem whether the Euler equations for incompressible fluids even forms a global continuous flow in a reasonable topology in the first place!) However, I do not want to discuss these analytic issues here; see this paper of Ebin and Marsden for a treatment of these topics.

Gauge theory” is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth.  But the underlying concept is really quite simple: a gauge is nothing more than a “coordinate system” that varies depending on one’s “location” with respect to some “base space” or “parameter space”, a gauge transform is a change of coordinates applied to each such location, and a gauge theory is a model for some physical or mathematical system to which gauge transforms can be applied (and is typically gauge invariant, in that all physically meaningful quantities are left unchanged (or transform naturally) under gauge transformations).  By fixing a gauge (thus breaking or spending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories), though the tractability of the resulting problem can be heavily dependent on the choice of gauge that one fixed.  Deciding exactly how to fix a gauge (or whether one should spend the gauge symmetry at all) is a key question in the analysis of gauge theories, and one that often requires the input of geometric ideas and intuition into that analysis.

I was asked recently to explain what a gauge theory was, and so I will try to do so in this post.  For simplicity, I will focus exclusively on classical gauge theories; quantum gauge theories are the quantization of classical gauge theories and have their own set of conceptual difficulties (coming from quantum field theory) that I will not discuss here. While gauge theories originated from physics, I will not discuss the physical significance of these theories much here, instead focusing just on their mathematical aspects.  My discussion will be informal, as I want to try to convey the geometric intuition rather than the rigorous formalism (which can, of course, be found in any graduate text on differential geometry).