By M. Boucetta, J.M. Morvan

The purpose of this quantity is to provide an creation and evaluate to differential topology, differential geometry and computational geometry with an emphasis on a few interconnections among those 3 domain names of arithmetic. The chapters provide the history required to start study in those fields or at their interfaces. They introduce new examine domain names and either previous and new conjectures in those diverse topics express a few interplay among different sciences just about arithmetic. subject matters mentioned are; the foundation of differential topology and combinatorial topology, the hyperlink among differential geometry and topology, Riemanian geometry (Levi-Civita connextion, curvature tensor, geodesic, completeness and curvature tensor), attribute periods (to affiliate each fibre package deal with isomorphic fiber bundles), the hyperlink among differential geometry and the geometry of non soft items, computational geometry and urban purposes comparable to structural geology and graphism.

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Already in the definition we see that X is a topological surface as in Definition 2, since r defines a homeomorphism ϕU : U → V . The last two conditions make sense if we use the implicit function theorem (see Appendix 1). This tells us that a local invertible change of variables in R3 “straightens out” the surface: it can be locally defined by x3 = 0 where (x1 , x2 , x3 ) are (nonlinear) local coordinates on R3 . For any two open sets U, U , we get a smooth invertible map from an open set of R3 to another which takes x3 = 0 to x3 = 0.

Which is holomorphic. In the right coordinates this is the sphere, with ∞ the North Pole and the coordinate maps given by stereographic projection. For this reason it is sometimes called the Riemann sphere. 2. Let ω1 , ω2 ∈ C be two complex numbers which are linearly independent over the reals, and define an equivalence relation on C by z1 ∼ z2 if there are integers m, n such that z1 − z2 = mω1 + nω2 . Let X be the set of equivalence classes (with the quotient topology). A small enough disc V around z ∈ C has at most one representative in each equivalence class, so this gives a local homeomorphism to its projection U in X.

We have EG − F 2 = f (u)2 (1 + f (u)2 ) so the area is b b f (u) 1 + f (u)2 dudv = 2π a f (u) 1 + f (u)2 du. a If a closed surface X is triangulated so that each face lies in a coordinate neighbourhood, then we can define the area of X as the sum of the areas of the faces by the formula above. It is independent of the choice of triangulation. 3 Isometric surfaces Definition 19 Two surfaces X, X are isometric if there is a smooth homeomorphism f : X → X which maps curves in X to curves in X of the same length.