It is rarely taught in undergraduate or even graduate curricula that the only conformal maps in Euclidean space of dimension greater than two are those generated by similarities and inversions in spheres. This is in stark contrast to the wealth of conformal maps in the plane. This fact is taught in most complex analysis courses. The principal aim of this text is to give a treatment of this paucity of conformal maps in higher dimensions. The exposition includes both an analytic proof, due to Nevanlinna, in general dimension and a differential geometric proof in dimension three. For completeness, enough complex analysis is developed to prove the abundance of conformal maps in the plane. In addition, the book develops inversion theory as a subject, along with the auxiliary theme of circle-preserving maps. A particular feature is the inclusion of a paper by Caratheodory with the remarkable result that any circle-preserving transformation is necessarily a Mobius transformation--not even the continuity of the transformation is assumed. The text is at the level of advanced undergraduates and is suitable for a capstone course, topics course, senior seminar or as an independent study text. Students and readers with university courses in differential geometry or complex analysis bring with them background to build on, but such courses are not essential prerequisites.

diameters which We have been considering and entirely contained in E. This figure transforms into a figure contained in E' from whose inspection we conclude that if we call y' the image of -y, the center of -y' is the image of the center of -y. 47 2. Linear Fractional Transformations 48 8. Preservation of Angles at the Origin. We take now the positive number r > 2 1/ 2 , (1) and consider the circle (2) y2 = r2 X2 and the circle y' 2 = r' 2 X' 2 (3) into which the circle (2) is

diameters which We have been considering and entirely contained in E. This figure transforms into a figure contained in E' from whose inspection we conclude that if we call y' the image of -y, the center of -y' is the image of the center of -y. 47 2. Linear Fractional Transformations 48 8. Preservation of Angles at the Origin. We take now the positive number r > 2 1/ 2 , (1) and consider the circle (2) y2 = r2 X2 and the circle y' 2 = r' 2 X' 2 (3) into which the circle (2) is

-ab -z maps y = a to y = b and is an isometry of the upper half plane. Noting also that (x ± ia, x + ib,x, oc) = ci , we see that two horocycles with the same boundary point are equidistant sets. 58 2. Linear Fractional Transformations We have discussed arcs of Euclidean circles lying in the disk and Euclidean circles tangent to the boundary, so now consider a Euclidean circle which lies entirely in the open unit disk. We have seen that two non-concentric circles can be inverted to concentric

is given by the Frenet equation 4; dc = KN , where T is the unit tangent and N the principal normal, which is defined to be the unit vector at each point that advances the unit tangent by i. Thus for normal sections at p we may identify the principal unit normal N and the surface normal n, the unit tangent at p being w. Then 6.1. Surface theory 101 If ki K2, let w1 , w2 be unit eigenvectors corresponding to the eigenvalues Ki K2 respectively. Writing a unit tangent vector w as w = cos Ow l ±

form a harmonic thus there exists a unique fourth point D such that set (Figure 1.7) if (AB,CD) = –1. We denote a harmonic set of points by H(AB,CD) and we say that C and D are harmonic conjugates with respect to A and B. A C B D Figure 1.7 In Exercise 3 below one sees that (AB,DC)= (AB ic D) and hence that the notion of harmonic conjugate is well defined: If D is the harmonic conjugate of C with respect to AB, C is the harmonic conjugate of D with respect to AB. If H(AB,CD), then the