Elements of the Global Theory of Surfaces
Global theory of surfaces is interested in those properties which refer to wide pieces of the surface, not just to the neighborhood of each point. We study surfaces of revolution, ruled surfaces, developable surfaces. We study when two surfaces are just a
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Elements of the Global Theory of Surfaces
Up to now, most of the results concerning surfaces appearing in this book have referred only to what happens on some convenient neighborhood of a given point of the surface. This last chapter introduces some considerations which make sense only when one considers the surface globally. First, we study some special families of surfaces, obtained by letting a curve revolve around an axis in its plane (the surfaces of revolution), by letting a line move continuously along a curve (the ruled surfaces), or by rolling up a piece of the plane (the developable surfaces). We also pay special attention to the surfaces with constant Gaussian curvature and in particular, to the sphere. Of course, arriving at the end of this trilogy, we also “open some doors” to further fascinating developments of geometry. We achieve this by drawing the reader’s attention to some striking results whose proofs often rely on some deep topological results which are beyond the scope of this book (such as the Jordan curve theorem). We switch to the study of curve polygons drawn on a surface. Making clear which topological results we rely upon, we prove the famous Gauss–Bonnet theorem and we conclude with some first considerations on the Euler–Poincaré characteristics of a surface: an integer which gives information concerning the global shape of the surface.
7.1 Surfaces of Revolution A first class of surfaces of interest is given by the surfaces of revolution: Definition 7.1.1 A surface of revolution in three dimensional Euclidean space is one which can be obtained by revolving a plane curve C around a line contained in the plane of the curve, but not intersecting the curve (see Fig. 7.1). Of course, there is no loss of generality in choosing the axis of revolution to be the z-axis. Then F. Borceux, A Differential Approach to Geometry, DOI 10.1007/978-3-319-01736-5_7, © Springer International Publishing Switzerland 2014
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7 Elements of the Global Theory of Surfaces
Fig. 7.1 A surface of revollution
Proposition 7.1.2 In the xz-plane of three dimensional Euclidean space, consider a plane curve C of class C k represented by f : ]a, b[ −→ R2 , t → (x, z) = f1 (t), f2 (t) and not intersecting the z-axis, i.e. ∀t ∈ ]a, b[ f1 (t) = 0. Revolving the curve C about the z-axis yields a surface of revolution admitting the parametric representation of class C k g : R × ]a, b[ −→ R3 , (t, θ ) → (x, y, z) = f1 (t) cos θ, f1 (t) sin θ, f2 (t) . This representation g is regular as soon as f is regular. Proof Of course, revolving C about the z-axis yields the “representation” g of the statement. Trivially when f is of class C k , so is g. To have a parametric representation g of a surface, we must still prove that g is locally injective. But g(t, θ ) lies in the plane πθ containing the z-axis and making an angle θ with the xz-plane; moreover, g(t, θ ) is never on the z-axis, because f1 (t) = 0 for all t. Thus two points g(θ1 , t1 ) and g(θ2 , t2 ), for two “close distinct values” θ1 , θ2 , lie in tw
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