An Invitation to Cauchy-Riemann and Sub-Riemannian Geometries John P. D’Angelo and Jeremy T. Tyson certain amount of complex analysis, in both one and several variables, com- bines with some differential geometry to form the basis of research in diverse parts of mathematics. We first describe Athat foundation and then invite the reader to follow specific geometric paths based upon it. We discuss such topics as the failure of the Riemann mapping theorem in several variables, the geometry of real hypersurfaces in complex Euclidean space, the Heisenberg group, sub-Riemannian manifolds and their metric space structure, the Hopf fibration, sub-Riemannian geodesics on the three-sphere, CR mappings invariant under a finite group, and finite type conditions. All of these topics fall within the branch of mathematics described by the title, but of course they form only a small part of it. We hope that the chosen topics are representative and appealing. The connections among them provide a fertile ground for future research. We modestly hope that this article inspires others to develop these connections further. To this end, our refer- Figure 1. Georg Friedrich Bernhard Riemann. ence list includes a diverse collection of books and accessible articles. Given Riemann’s contributions to geometry, it There Is No Riemann Mapping Theorem in is not surprising that his name occurs twice in the Higher Dimensions title. It is more surprising that the uses of his name In an attempt to understand complex analysis here come from different parts of mathematics. We in several variables, especially questions revolv- find it delightful that contemporary mathematics ing around the failure of the Riemann mapping is forging connections that even Riemann could theorem, many mathematicians have focused at- not have anticipated. tention on geometric properties of the boundaries John P. D’Angelo is professor of mathematics at the Uni- of domains and how those geometric properties versity of Illinois. His email address is [email protected]. influence the complex analysis on the domain. edu. Such investigations have led to the fields of CR Jeremy T. Tyson is associate professor of mathemat- geometry and sub-Riemannian geometry. ics at the University of Illinois. His email address is We begin with the Riemann mapping theo- [email protected]. rem. Let Ω be an open, connected, and simply 208 Notices of the AMS Volume 57, Number 2 connected proper subset of C. Then there is a is no biholomorphic map from any polydisc to bijective holomorphic mapping f from Ω to the unit any ball. One classical proof of this statement disc D. The inverse mapping is also holomorphic. involves showing that the groups of biholomorphic We say that Ω and D are biholomorphically equiva- automorphisms of the ball and of the polydisc lent. The entire plane must of course be excluded, have different dimensions. because a bounded holomorphic function on C re- A second proof (see [D1, page 14]) proceeds in the duces to a constant. This theorem has applications following manner. First consider a biholomorphic throughout both pure and applied mathematics. In (or even proper) image Ω of the Cartesian product of many of these applications, such as uniform fluid bounded domains; one shows that bΩ must contain flow, the upper half plane substitutes for the unit complex analytic sets. Then an easy computation, disc. The explicit linear fractional transformation given in Lemma 6, shows that the definiteness of f (the Cayley transformation) defined by the Levi form precludes the existence of positive- i − w dimensional complex analytic subsets of b . This (1) z = f (w) = Ω i + w computation helps unify several of the ideas in this maps the upper half plane biholomorphically to article. For example, higher order commutators of the unit disc. complex vector fields can determine obstructions We mention in passing one of the most funda- to the existence of complex analytic sets in a real mental geometric generalizations of the Riemann hypersurface. Such obstructions lead to geometric mapping theorem, although we will not head in finite type conditions for subelliptic estimates. In that direction. The uniformization theorem states two complex dimensions a subelliptic estimate (see that the universal cover of a compact Riemann page 217) holds if and only if the bracket-generating surface must be one of three objects: the unit disc property (see page 212) holds. D, the complex plane C, or the Riemann sphere C. Consider a domain Ω with smooth boundary As a reference for analysis in one complex variable, bΩ. In order that geometric results about bΩ be we need only mention the classical text by Ahlfors meaningful for complex analysis on Ω, one requires [Ahl]. results concerning the boundary smoothness of Let us now consider complex Euclidean space Cn biholomorphic maps. Two early references in this of arbitrary dimension. We equip Cn with the usual area are [Bel] and [Fef]; we do not attempt to list any P Hermitian inner product given by hz; wi = zj w j of the many newer references. In general, smoothly and the corresponding norm given by jzj = hz; zi1=2. bounded domains, topologically equivalent to The unitary group U(n) consists of linear mappings the unit ball Bn, are seldom biholomorphically preserving the inner product. The unit ball Bn is equivalent to it. On the other hand, Fridman [Fri] the set of z for which jzj < 1. has established a wonderful approximate Riemann In dimension at least two the unit ball is not mapping theorem: given any topological ball and a biholomorphically equivalent to a half space. There positive , there is a biholomorphic mapping to a are many possible proofs. One approach takes into domain whose boundary is always within distance account the geometry of the boundaries. When the of the unit sphere. boundaries are smooth manifolds (as they are in this case), the Levi form plays a crucial role. The The Unit Sphere and the Heisenberg Group Levi form for a real hypersurface in Cn (for n ≥ 2) The quintessential principle connecting CR geom- is the complex variable analogue of the second etry and sub-Riemannian geometry arises from n fundamental form for a real hypersurface in R . the correspondence between the unit sphere bBn We define and discuss the Levi form on page 211. and the Heisenberg group, which bounds the For now we simply note that the Levi form is unbounded Siegel upper half space Hn. We will n−1 a Hermitian form defined on part of the tangent obtain a useful identification of bHn with C × R, spaces of the boundary. There is no Levi form in on which we will impose a nonabelian group law one dimension, because the tangent space is one arising from the biholomorphic automorphisms of real dimension and the part on which the Levi form Hn. The equivalence between Hn and Bn generalizes acts does not exist. For a half plane in dimension the one-dimensional Cayley transformation. at least 2 the Levi form is identically zero. For The Siegel upper half space Hn is the set of w the unit sphere it is positive definite. As a result in Cn such that the boundary geometries are so different that no n−1 biholomorphic mapping can exist. By contrast, X 2 (2) Im(wn) > jwj j : in one dimension, different boundary curves are j=1 indistinguishable from this point of view. A polydisc is a Cartesian product of discs. For The following elementary but useful identity is the 2 key to discovering the biholomorphic map: example, the subset of C defined by jz1j < 1 and 2 2 jz2j < 1 is a polydisc. It is also not biholomor- i + ζ i − ζ (3) Im(ζ) = − : phically equivalent to B2. In fact, for n ≥ 2 there 2 2 February 2010 Notices of the AMS 209 With ζ = wn we plug (3) into (2) and rewrite the CR Structures result to obtain We next discuss the geometry of a general real n 2 n−1 2 hypersurface M in C . The beautiful interplay i + wn X i − wn (4) > jw j2 + : between real and complex geometry dominates the 2 j 2 j=1 discussion. See [BER], [D1], [DT], [Jac], [Tre] and their references for more about CR geometry. After dividing by j i+wn j2 and changing notation, 2 We wish to consider complex vector fields and inequality (4) becomes therefore start by considering the complexified n X tangent bundle 1 > jz j2: j n = n ⊗ j=1 (6) CT(C ) T(C ) C: 2wj i−wn As usual in complex analysis we define the complex Here zj = for j < n and zn = . The reader i+wn i+wn partial derivative operators should check that this transformation f : w , z is ! @ 1 @ @ biholomorphic from Hn to Bn and also compare it = − i with the transformation in (1). @zj 2 @xj @yj and ! The automorphism group of Hn @ 1 @ @ = + i : We alluded earlier to the biholomorphic automor- @zj 2 @xj @yj phism groups of the ball and the polydisc. For any Using these operators we decompose the exterior complex manifold the automorphism group is a derivative d as fundamental algebraic invariant. For example, the n n X @f X @f automorphism group of the upper half plane in (7) df = @f + @f = dz + dz : @z j @z k C is PSL(2; R). Its elements act on the (one-point j=1 j k=1 k compactification of the) boundary of the upper half 2 Since d2 = 0, it follows that @2 = @ = @@ + @@ = 0. plane as linear fractional transformations x , ax+b , cx+d A section of the bundle (6) is a complex vector − = for ad bc 1.