The theory of Riemann surfaces has a geometric and an analytic part. The former deals with the axiomatic definition of a Riemann surface, methods of. Riemann Surfaces. Front Cover. Lars V. Ahlfors, Leo Sario. Princeton University Press, Jan 1, – Mathematics – pages. A detailed exposition, and proofs, can be found in Ahlfors-Sario , Forster Riemann Surface Meromorphic Function Elliptic Curve Complex Manifold.
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A Riemann surface is, in the first place, a surface, and its properties depend to a very great extent on the topological character of the surface. For this reason the topological theory of surfaces belongs in this book. There is a great temptation to bypass the finer deta.
The main demerit riemanh this approach is that it does not yield complete results. For instance, it cannot be proved by analytical means that every surface which satisfies the axiom of countability can be made into a Riemann surface. In other instances ahpfors analytical method becomes so involved that it no longer possesses the merit of elegance. For complete results this derivation must be based on the method of triangulation.
On the other hand, it is much easier to obtain superficial knowledge without use of triangulations, for instance, by the method of singular homology. It so happens that this superficial knowledge is adequate for most applications to the theory of Riemann surfaces, and our presentation is influenced by this fact. Since we strive for completeness, a considerable part of the first chapter has been allotted to the oombinatorial approach.
We have tried, however, to isolate this pq. This is done by formulating the combinatorial theory as a theory riemsnn triangulated surfaces, or polyhedrons.
Sutfaces the very end of the chapter it is then shown, by essential use of the Jordan curve theorem, that every surface which satisfies the second axiom of countability permits a. This elementary section has been included for the sakP of l”ompleteness and because beginning analysts are not always well prepared on this point. The section can of course be o. The fundamental group is introduced, and the notion of bordE. The chapter closes with the construction of a triangulation. If there arc no relations between the points, pure set theory exhausts all poRsibilities.
As soon as one wants to go beyond set theory to limits and continuity it becomes surfacces to introduce a topology, and the space becomes a topological space. The following conditions shall be fulfilled:. AI The union of any collldion of open sets is open. A2 The intersection of any finite coUection of open ahlvors is open.
According to usual conventions the union of an empty collection of sets is the empty set 0, and the intersection of an empty collection is the whole space 8. Hence AI and A2 imply that 0 and 8 are open.
In most a third requirement is added:. It is therefore convenient to introduce the notion of a basis for the open sets briefly: Such a ahlcors is a system fJI of subsets of 8 which satisfies condition. B Rieemann intersection of any finite collldion of sets in fJI is a union of sets in It is a Hausdorfl” topology 1f and only if any two distinct points belong to disjoint sets in A Migllborlwod of a set A c 8 is a set V c 8 which. Mostly, we consider only neighborhoods of points, and we use the notation V p to indicate that Vis a neigj borhood ofp.
An open set is a neighborhood of any subset, and a set is open if it contains a neighborhood of every point in the set.
A apace with more than one point can be topologized in different manners.
Springer : Review: Lars V. Ahlfors and Leo Sario, Riemann surfaces
A topology 9″ 1 is said to be weaker than the topology r 2 if r 1 c r ‘1. The strongest topology is the tliscrete topology in which every subset is open. Condition B is a consequence of 1 the triangle inequality, and Rtt is a Hausdorff apace. For the riemanm of the plane Jl2 we shall frequently use the complex notation The sphere 81, also referred to as the u: Uon of a finite oolleotion of closed seta are closed. The empty set and the Whole apace are simultaneously open and riemnan.
It exists, for it can be obtained as t lw intersection of all closed sets which contain P. The boundary of P is formed by all points which belong neither to the interior nor to the exterior. In other words, p belongs to the boundary of P if and only if every V p interaecta Pas well as the complement of P.
Lars V. Ahlfors, L. Sario-Riemann Surfaces – PDF Drive
One of these is the process of relativization. Surfzces fact, Al – A2 arf’ ouvwusly satisfied; A3 holds if it holds on B. Ahlfos call this topology on S’ the relative topology induced by the topology on B. The open and closed seta on S’ are often referred to as relatively open or closed sets. We shall always understand that the topology on a subset is its relative topology. The sum of two topological spaces 81, Sa is their union 81 u Bz on which the open seta are those whose intersections with 81 and Sa are both open.
This definition has an obvious generalization to the case of an arbi- trary collection of topological spaces. The topological product 81 x Sa skrfaces defined as follows: A topology in the space of points P can be riemsnn by the agreement that a set is open if and only if the union of the corresponding seta Pis an open set in B.
Again, AlHA2 are trivially fulfilled. It must be observed, however, that the new space is not necessarily a Hausdorff space even if Sis one. The space obtained by identification can be referred to as the qvotientspau of S with respect to the equivalence relation whose equivalence classes are the sets P.
The construction of a quotient-apace is sometimes preceded by forming the sum of several spaces. This means that points can be identified which were not initially in the same space. Certain characteristic properties which may or may not be present in a topological space are very important not only in the’ general theory, but in particular for the study of surfaces.
Mii1Nr of which ia tloitl. This is the moat useful form of the definition for a whole category of proofs. If thiH is so, one of the seta must be empty, and we conclude that the property holds for all points or for no points. Examples of this type of proof will be abundant. The definition applies also to subsets in their relative topology, and we can hence apeak of connected and disconnected subsets. An open connected set is called a region, and the closure of a region is referred to as a cloM.
A closed connected set with more than one point is a conlinuum. The empty set and all seta with only one point are trivially con- nected. Denote the given seta by P. If 01 is not empty it meets at least one P.
They are relatively open with respect to P. From the connectedness of P. Hence 01 contains the l’ommon point of all Pa.
Lars V. Ahlfors, L. Sario-Riemann Surfaces
This reasoning applies equally well to Oz, and we find that 01 and 02 cannot both be nonempty and at the same time disjoint. But 01 is also relatively closed in Q. From Q c 01 it follows that Oa is empty, and hence Q is connected.
As a consequence of 3B every point in a topological space belongs to a maximal connected subset, namely the union of all connected sets which contnin the given point. This maximal connected set is called the romJmnPnl of the “l”H’I’ dt’l’l’llined hy the point. This property neither implies nor is implied by connectedness in the large. Every open subset of a locally connected space is itself locally connected, for it has a basis conaisting of part of the basis for the whole space.
We have already pointed out that the components are closed, independently of the local connectedness. Suppose that p belongs to the component 0, and let V p be a connected neighborhood. Then 0 u V p is connected, by 3B, and by the definition of components we obtain 0 u V p c 0 or V p c 0.
This shows that 0 is open. It shows that every open set in a locally connected space is a union of disjoint regions. Rn durfaces also locally connected. We proceed to the definition of compact spaces. It has been found most convenient to base the dario on the consideration of open coverings. An open covering is a family of open sets whose union is the whole space, and a covering is finite if the family contains only a finite number of sets.
For instance, a compact space can thus be covered by a finite number of from an arbitrary basis. A compact subset is of course one which is compact in the relative topology.
In the case of a aubaet surfaves is convenient to replace the relatively open seta of a covering by corresponding open seta of the whole apace. An open covering of a subset is then a family of open seta whose union contains the given subset, and if the subset is compact we can again select a finite aubcovering. Sometimes it is more convenient to express compactneea in terms of closed seta.
We shall say that a family of closed seta has the finite inter- M. Ction properly if any finite number of seta in the family have a non void interaection.
Evidently, this is equivalent to saying that the family of complements contains no finite covering. The following theorem is thus merely a rephrasing of the definition. Sario – Riemann Surfaces Alexandre row Enviado por: The following conditions shall be fulfilled: The requirements are interpreted to hold also for the empty collection.
In most a third requirement is added: