Covered are the riemann mapping theorem as well as some basic facts about riemann surfaces. A geometric construction known as stereographic projection gives rise to a onetoone correspondence between the complement of a. We recall the formula for stereographic projection from the riemann sphere to c, and we derive a formula for its inverse. Mobius transformations complex analysis l marizza a bailey. In projective geometry, the xyplane is supplemented by adding an extra line and homogeneous coordinates are introduced to cope with this line at infinity.
We now look at the action of some simple mobius transformation. The two main references for this paper are complex variables and applications by james brown and ruel churchill and schwarzchristo. Mobius notes on mobius transformations september 29, 2006 6. Stereographic projection let a sphere in threedimensional euclidean space be given. The plane with a point at infinity appended is called the riemann sphere after the 18th century mathematician bernhard riemann although strictly speaking the riemann sphere is the complex plane with infinity appended see here for more on complex numbers. Nonlinear cauchyriemann equations and liouville equation for. I am reading the book computational conformal geometry by xianfeng david gu and shingtung yau. The riemann sphere is only a conformal manifold, not a riemannian manifold. Riemann sphere and mobius transformation cuttheknot. Hyberbolic lines as circles on the riemann sphere c 4 4. Orbits of quaternionic mobius transformations arxiv. The geometry of mobius transformations john olsen university of rochester spring 2010. Just take all the arguments in the previous section and put the word real before each occurance of the word mobius transformation.
Since these maps are rational functions, we can and often do regard them as analytic functions c. Stereographic projection can be used to characterize mobius transformations in a distinctly. Jun, 20 any riemann surface is the quotient of the complex plane, or the upper halfplane, or the riemann sphere by a suitable group of moebius transformations isomorphic to the fundamental group of the. In fact the mobius transformation preserves the riemann sphere fig. These transformations map the riemann sphere to itself, preserving the angles and orientation. Jan 10, 2011 this animation depicts a disk of the complex plane as it is acted upon by a range of polynomial equations. Every mobius transformation is a bijective conformal map of the riemann sphere to itself. Having in mind this observation and the previous deliberations, we can summarize theorem 5.
Is any homeomorphism from riemann sphere to riemann sphere. The images below show the projection in the riemann sphere of the sequences of concentric circles, rays, as well as, the effect in the riemann sphere namely, translations, rotations along the axis, and dilation via mobius transformation. Chapter 7 riemann mapping theorem 3 iii if ad bc6 0, then the function. Indeed, every such map is by necessity a mobius transformation. This extended complex plane can be thought of as a sphere, the riemann sphere, or as the complex projective line. Chapter 3 examples of functions obvious is the most dangerous word in mathematics. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. However, it is not obvious what the holomorphic automorphisms of the sphere look like, and it takes some e. The pictures are not reproduced here, but most of them are contained in the handout classi. Interpretation of symmetric and antipodal points on both, the riemann sphere and the riemann pseudosphere, are given. The mobius transformation is a bijective conformal mapping of the extended complex plane or the riemann sphere.
Finally, in chapter 6, several examples are given, as well as variants of the schwarzchristo. Thus a mobius transformation is always a bijective holomorphic function from the riemann sphere to the riemann sphere. Every mobius transformation can be constructed by stereographic projection of the complex plane onto a sphere, followed by a rigid motion of the sphere and projection back onto the plane, illustrated in the video mobius transformations revealed. The extended complex plane c is also called the riemann sphere, and we have defined t to be a map from the riemann sphere to itself. The riemann sphere rs, also know as the extended plane, was a breakthrough in complex analysis, introduced in b. Mobius transformations are isometries of a sphere math. Books and references for mobius transformation, hyperbolic. Loxodromic loxodromic moebius transformations with fixed points zero and infinity.
Classical kleinian groups are discrete subgroups of mobius transformations which act on the riemann sphere with. I talked about the continuity, and the topology of the riemann sphere via the stereographic projection. The latter is called the sphere of complex numbers or the riemann sphere. Constructing mobius transformations with spheres pdf paperity. Mobius transformations complex analysis l marizza a bailey the riemann sphere basis scottsdale march, 2015 a mobius transformation is a complex function on the riemann sphere, 1, that is of the form az b fz cz d where a, b, c, and d are complex constants. Riemann mapping theorem and riemann surfaces stephan tillmann these notes are compiled for an honours course in complex analysis given by the author at the university of melbourne in semester 2, 2007. Mobius transformations complex analysis l marizza a bailey a. There is a part in the book which i dont understand and i would like to ask for books and references. In c 1, f dc has a value, since fractions with a zero denominator exist, and are 1. A geometric construction known as stereographic projection gives rise to a onetoone correspondence between the complement of a chosen point a on the sphere and the points of the plane z.
Jul 23, 2011 in short, this is a proof without words that the mobius transformations are in correspondence with rigid motions of the unit sphere in. Transformations of this form, under inverse stereographic projection to the riemann sphere, can be shown to map each pair of antipodal points to another pair of. Mobius transformations and circles brown university. Moebius transformations make up fundamental groups of riemann. The set of all mobius transformations forms a group under composition. Every m obius transformation can be constructed by stereographic projection of the complex plane onto a sphere, followed by a rigid motion of the sphere and projection back onto the plane, illustrated in the video m obius transformations revealed. Notes on mobius transformations eecs at uc berkeley. These functions can be extended to the riemann sphere c 1, the complex numbers with the \in nity point 1. The open plane c may be identi ed with snn, the sphere with the north pole deleted. These happen to be bijective functions from the riemann sphere to itself. On the riemann sphere, m obius transformations are bijective, and their inverses are also m obius transformations. Geometrical properties of stereographic projection continued 1. Mobius transformations and stereographic projection.
One dimensional projective complex space pc2 is the set of all onedimensional subspaces of c2. Oct 01, 20 the two can be treated as one and the same thing. This 11 mapping is precisely riemanns stereographic projection. I also highlighted some properties of stereographic projection.
A visual point of view of different topics related to complex analysis. Let c1, c2 be two disjoint circles on the riemann sphere. Pdf this expository article considers noncircular ellipses in the riemann sphere, and the action of the group of mobius transformations find, read and cite all the research you need on. Mobius notes on mobius transformations september 29. However, if one needs to do riemannian geometry on the riemann sphere, the round metric is a natural choice with any fixed radius, though radius 1 is the simplest and most common choice. Then fcan be expressed as a composition of magni cations, rotations, translations. Namely, we identify the complex plane with the plane x3 0 in r3, and map it to the unit sphere by inverse stereographic projection from the north pole. M obius transformations and stereographic projection.
961 1584 266 368 762 1598 1001 1158 632 539 309 60 1642 1106 1319 924 433 1411 1326 1494 494 610 48 315 1311 13 845 1497 1420 1395 507 813 573 322