真实之路 宇宙法则的完全指南 (彭罗斯)8黎曼曲面和综.pdf

8 Riemann surfaces and complex mappings 8 1The idea of a Riemann surface There is a way of understanding what is going on with this analytic continuation of the logarithm function or of any other many valued function in terms of what are called Riemann surfaces Riemann s idea was to think of such functions as being deWned on a domain which is not simply a subset of the complex plane but as a many sheeted region In the case of log z we can picture this as a kind of spiral ramp Xattened down vertically to the complex plane I have tried to indicate this in Fig 8 1 The logarithm function is single valued on this winding many sheeted version of the complex plane because each time we go around the origin and 2pi has to be added to the logarithm we Wnd ourselves on another sheet of the domain There is no conXict between the diVerent values of the logarithm now because its domain is this more extended winding space an example of a Riemann surface a space subtly diVerent from the complex plane itself Bernhardt Riemann who introduced this idea was one of the very greatest of mathematicians and in his short life 1826 66 he put forward a multitude of mathematical ideas that have profoundly altered the course of mathematical thought on this planet We shall encounter some of his Fig 8 1The Riemann surface for logz pictured as a spiral ramp Xattened down vertically 135 other contributions later in this book such as that which underlies Ein stein s general theory of relativity and one very important contribution of Riemann s of a diVerent kind was referred to at the end of Chapter 7 Before Riemann introduced the notion of what is now called a Riemann surface mathematicians had been at odds about how to treat these so called many valued functions of which the logarithm is one of the simplest examples In order to be rigorous many had felt the need to regard these functions in a way that I would personally consider distaste ful Incidentally this was still the way that I was taught to regard them myself while at university despite this being nearly a century after Rie mann s epoch making paper on the subject In particular the domain of the logarithm function would be cut in some arbitrary way by a line out from the origin to inWnity To my way of thinking this was a brutal mutilation of a sublime mathematical structure Riemann taught us we must think of things diVerently Holomorphic functions rest uncomfort ably with the now usual notion of a function which maps from a Wxed domain to a deWnite target space As we have seen with analytic continu ation a holomorphic function has a mind of its own and decides itself what its domain should be irrespective of the region of the complex plane which we ourselves may have initially allotted to it While we may regard the function s domain to be represented by the Riemann surface associated with the function the domain is not given ahead of time it is the explicit form of the function itself that tells us which Riemann surface the domain actually is We shall be encountering various other kinds of Riemann surface shortly This beautiful concept plays an important role in some of the modern attempts to Wnd a new basis for mathematical physics most notably in string theory 31 5 13 but also in twistor theory 33 2 10 In fact the Riemann surface for log z is one of the simplest of such surfaces It gives us merely a hint of what is in store for us The function zaperhaps is marginally more interesting than log z with regard to its Riemann surface but only when the complex number a is a rational number When a is irrational the Riemann surface for zahas just the same structure as that for log z but for a rational a whose lowest terms expression is a m n the spiralling sheets join back together again after n turns 8 1 The origin z 0 in all these examples is called a branch point If the sheets join back together after a Wnite number n of turns as in the case zm n m and n having no common factor we shall say that the branch point has Wnite order or that it is of order n When they do not join after any number of turns as in the case log z we shall say that the branch point has inWnite order 8 1 Explain why 8 1CHAPTER 8 136 Expressions like1 z3 1 2 give us more food for thought Here the function has three branch points at z 1 z o and z o2 where o e2pi 3 see 5 4 7 4 so 1 z3 0 and there is another branch point at inWnity As we circle by one complete turn around each individ ual branch point staying in its immediate neighbourhood and for inW nity this just means going around a very large circle we Wnd that the function changes sign and circling it again the function goes back to its original value Thus we see that the branch points all have order 2 We have two sheets to the Riemann surface patched together in the way that I have tried to indicate in Fig 8 2a In Fig 8 2b I have attempted to show using some topological contortions that the Riemann surface actually has the topology of a torus which is topologically the surface of a bagel or of an American donut but with four tiny holes in it corresponding to the branch points themselves In fact the holes can be Wlled in unambiguously z z2 1 a Open Open c z2 z2 z2 z z2 z z z2 z z 1 1 1 1 1 b Fig 8 2 a ConstructingtheRiemannsurfacefor 1 z3 1 2fromtwosheets with branch points of order 2 at 1 o o2 and also 1 b To see that the Riemann surface for 1 z3 1 2is topologically a torus imagine the planes of a as two Riemann spheres with slits cut from o to o2and from 1 to 1 identiWed along matching arrows These are topological cylinders glued correspondingly giving a torus c To construct a Riemann surface or a manifold generally we can glue together patches of coordinate space here open portions of the complex plane There must be open set overlaps between patches and when joined there must be no non HausdorV branching as in the Wnal case above see Fig 12 5b 12 2 Riemann surfaces and complex mappings 8 1 137 with four single points and the resulting Riemann surface then has exactly the topology of a torus 8 2 Riemann s surfaces provided the Wrst instances of the general notion of a manifold whichisaspacethatcanbethoughtofas curved invariousways butwhere locally i e inasmallenoughneighbourhoodofanyofitspoints it looks like a piece of ordinary Euclidean space We shall be encountering manifoldsmoreseriouslyinChapters10and12 Thenotionofamanifoldis crucial in many diVerent areas of modern physics Most strikingly it forms an essential part of Einstein s general relativity Manifolds may be thought of as being glued together from a number of diVerent patches where the gluingjobreallyisseamless unlikethesituationwiththefunctionh x atthe end of 6 3 The seamless nature of the patching is achieved by making sure thatthereisalwaysanappropriate open set overlapbetweenonepatchand the next see Fig 8 2c and also 12 2 Fig 12 5 In the case of Riemann surfaces the manifold i e the Riemann surface itself is glued together from various patches of the complex plane corres ponding to the diVerent sheets that go to make up the entire surface As above we may end up with a few holes in the form of some individual points missing coming from the branch points of Wnite order but these missing points can always be unabiguously replaced as above For branch points of inWnite order on the other hand things can be more compli cated and no such simple general statement can be made As an example let us consider the spiral ramp Riemann surface of the logarithm function One way to piece this together in the way of a paper model wouldbetotake successively alternatepatchesthatarecopiesof a thecomplexplanewiththenon negativerealnumbersremoved and b the complex planewith the non positivereal numbers removed Thetophalf of each a patch would be glued to the top half of the next b patch and the bottom half of each b patch would be glued to the bottom half of the next a patch see Fig 8 3 There is an inWnite order branch point at the origin and also at inWnity but curiously we Wnd that the entire spiral ramp is equivalent just to a sphere with a single missing point and this point can be unambiguously replaced so as to yield simply a sphere 8 3 8 2Conformal mappings When piecing together a manifold we have to consider what local struc ture has to be preserved from one patch to the next Normally one deals with real manifolds and the diVerent patches are pieces of Euclidean space 8 2 Now try 1 z4 1 2 8 3 Can you see how this comes about Hint Think of the Riemann sphere of the variable w logz see 8 3 8 2CHAPTER 8 138 a b of some Wxed dimension that are glued together along various open overlap regions The local structure to be matched from one patch to the next is normally just a matter of preserving continuity or smoothness This issue will be discussed in 10 2 In the case of Riemann surfaces however we are concerned with complex smoothness and we recall from 7 1 that this is a more sophisticated matter invoving what are called the Cauchy Riemann equations Although we have not seen them expli citly yet we shall be coming to them in 10 5 it will be appropriate now to understand the geometrical meaning of the structure that is encoded in these equations It is a structure of remarkable elegance Xexibility and power leading to mathematical concepts with a great range of appli cation The notion is that of conformal geometry Roughly speaking in con formal geometry we are interested in shape but not size this referring to shape on the inWnitesimal scale In a conformal map from one open region of the plane to another shapes of Wnite size are generally distorted but inWnitesimal shapes are preserved We can think of this applying to small inWnitesimal circles drawn on the plane In a conformal map these little circles can be expanded or contracted but they are not distorted into little ellipses See Fig 8 4 To get some understanding of what a conformal transformation can be like look at M C Escher s picture given in Fig 2 11 which provides a conformal representation of the hyperbolic plane in the Euclidean plane as described in 2 4 Beltrami s Poincare disc The hyperbolic plane is very symmetrical In particular there are transformations which take the Wgures in the central region of Escher s picture to corresponding very tiny Wgures that lie just inside the bounding circle We can represent such a transformation as a conformal motion of the Euclidean plane that takes Fig 8 3We can construct the Riemann surface for logz by taking alternate patches of a the complex plane with the non negative real axis removed and b the complex plane with the non positive real axis removed The top half each a patch is glued to the top half of the next b patch and the bottom half of each b patch glued to the bottom half of the next a patch Riemann surfaces and complex mappings 8 2 139 Conformal Non conformal the interior of the bounding circle to itself Clearly such a transformation would not generally preserve the sizes of the individual Wgures since the ones in the middle are much larger than those towards the edge but the shapes are roughly preserved This preservation of shape gets more and more accurate the smaller the detail of each Wgure that is being is exam ined so inWnitesimal shapes would indeed be completely unaltered Per haps the reader would Wnd a slightly diVerent characterization more helpful angles between curves are unaltered by conformal transformation This characterizes the conformal nature of a transformation What does this conformal property have to do with the complex smoothness holomorphicity of some function f z We shall try to obtain an intuitive idea of the geometric content of complex smoothness Let us return to the mapping viewpoint of a function f and think of the relation w f z as providing a mapping of a certain region in z s complex plane the domain of the function f into w s complex plane the target see Fig 8 5 We ask the question what local geometrical property characterizes this mapping as being holomorphic There is a striking answer Holomor phicity of f is indeed equivalent to the map being conformal and non reXective non reXective or orientation preserving meaning that the small shapes preserved in the transformation are not reXected i e not turned over see end of 12 6 The notion of smoothness in our transformation w f z refers to how the transformation acts in the inWnitesimal limit Think of the real case Wrst and let us re examine our real function f x of 6 2 where the graph of y f x is illustrated in Fig 6 4 The function f is smooth at z planew plane f Fig 8 4For a conformal map little inWnitesimal circles can be expanded or contracted but not distorted into little ellipses Fig 8 5The map w f z has domain an open region in the complex z plane and target an open region in the complex w plane Holomorphicity of f is equivalent to this being conformal and non reXective 8 2CHAPTER 8 140 some point if the graph has a well deWned tangent at that point We can picture the tangent by imagining that a larger and larger magniWcation is applied to the curve at that point and so long as it is smooth the curve looks more and more like a straight line through that point as the mag niWcation increases becoming identical with the tangent line in the limit of inWnite magniWcation The situation with complex smoothness is similar but now we apply the idea to the map from the z plane to the w plane To examine the inWnitesimal nature of this map let us try to picture the immediate neighbourhood of a point z in one plane mapping this to the immediate neighbourhood of w in the other plane To examine the imme diate neighbourhood of the point we imagine magnifying the neighbour hood of z by a huge factor and the corresponding neighbourhood of w by the same huge factor In the limit the map from the expanded neighbour hood of z to the expanded neighbourhood of w will be simply a linear transformation of the plane but if it is to be holomorphic this must basically be one of the transformations studied in 5 1 From this it follows by a little consideration that in the general case the transformation from z s neighbourhood to w s neighbourhood simply combines a rotation with a uniform expansion or contraction see Fig 5 2b That is to say small shapes or angles are preserved without reXection showing that the map is indeed conformal and non reXective Let us look at a few simple examples The very particular situations of the maps provided by the adding of a constant b to z or of multiplying z by a constant a as considered already in 5 1 see Fig 5 2 are obviously holomorphic z b and az being clearly diVerentiable and are also obvi ously conformal These are particular instances of the general case of the combined inhomogeneous linear transformation w az b Such transformations provide the Euclidean motions of the plane without reXection combined with uniform expansions or contractions In fact they are the only non reXective conformal maps of the entire complex z plane to the entire complex w plane Moreover they have the very special property that actual circles not just inWnitesimal circles are mapped to actual circles and also straight lines are mapped to straight lines Another simple holomorphic function is the reciprocal function w z 1 which maps the complex plane with the origin removed to the complex plane with the origin removed Strikingly this transformation also maps actual circles to actual circles 8 4 where we think of straight lines as being 8 4 Show this Riemann surfaces and complex mappings 8 2 141 particular cases of circles of inWnite radius This transformation to gether with a reXection in the real axis is what is called an inversion Combining this with the inhomogeneous linear maps just considered we get the more general transformation 8 5 w az b cz d called a bilinear or Mo bius transformation From what has been said above these transformations must also map circles to circles straight lines again being regarded as special circles This Mo bius transformation actually maps the entire complex plane with the point d c removed to the entire complex plane with a c removed where for the transformation to give a non trivial mapping at all we must have ad 6 bc so that the numerator is not a Wxed multiple of the denominator Note that the point removed from the z plane is that value z d c which would give w 1 correspondingly the point removed from the w plane is that value w a c which would be achieved by z 1 In fact the whole transformation would make more global sense if we were to incorporate a quantity 1 into both the domain and target This is one way of thinking about the simplest compact Riemann surface of all the Riemann sphere which we come to next 8 3The Riemann sphere Simply adjoining an extra point called 1 to the complex plane does not make it completely clear that the required seamless structure holds in the neighbourhood of 1 the same as everywhere else The way that we can address this issue is to regard the sphere to be constructed from two coordinate patches one of which is the z plane and the other the w plane All but two points of the sphere are assigned both a z coordinate and a w coordinate related by the Mo bius transformation above But one point has only a z coordinate where w would be inWnity and another has only a w coordinate where z would be inWnity We use either z or w or both in order to deWne the needed conformal structure and where we use both we get the same conformal structure using either because the relation between the two coordinates is holo morphic In fact for this we do not need such a complicated transformation between z and w as the general Mo bius transformation It suYces to consider the particularly simple Mo bius transformation given by 8 5 Verify that the sequence of transformations z 7 Az B z 7 z 1 z 7 Cz D indeed leads to a bilinear map 8 3CHAPTER 8 142 i 0 11 i i 1 1 iw 1 z z planew plane Fig 8 6Patching the Riemann sphere from the complex z and w