 By A. F. Beardon

Best geometry books

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Comparison Theorems in Riemannian Geometry

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Geometry of Cauchy-Riemann Submanifolds

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Extra info for A Primer on Riemann Surfaces

Example text

By taking B f are open and g open, we see that ifhis open then so is h then so is g. Finally, = gf. Exercise 2,7 1, Let generated by x that the map g G (x) b- exp(ix) G x + 2irand let of X/G into W be the group of translations of the real line f : X -► X/G be the natural map. Show = iz : |z| = l) defined by is properly defined and 1-1. Show that the composite map h : x h- exp(ix) is open, continuous and surjective and deduce that is homeomorphic to X/G W. 8 A BASE FOR A TOPOLOGY * Let B be a class of subsets of of those subsets of X of sets chosen from B.

It is a theorem (much later) and not an assumption that these conditions hold for a Riemann surface: this example merely serves to emphasize this result and is of no other importance. 6. First, define f : JR^ ‘►JR'* (x,y,z) if y < O? (x+^-,y,0) if y > O . f (x,y,z) Next, partition JR into the horizontal planes = i(x,y,z) e JR and define : z = t) by 38 S = f(IR3) = {(x,y,z) elR3 : y < 0} U PQ . Observe that f restriction of induced by so A f ffc. is acts on S to and let We give T-open as the identity map.

If the charts S S is a surface and has a natural concept of (for example, if a : S -*■ S is a smooth preserve angles (in sense and magnitude) then the same is true of all transition functions. 2) and so S is actually a Riemann surface. To summarise, a surface is a Riemann surface if angles can be defined on the surface and if the charts a are angle-preserving. We obtain even greater flexibility if we allow the angles at points region U^. x provided that each x a If this is so, the only transition function defined at is the identity (= to distort lies in only one parametric and so the distortion of angles at x a (x) cannot violate the requirement for a Riemann surface.