Ivancevic_Applied-Diff-Geom

12 Applied Differential Geometry: A Modern Introductionregarded as pathological, so it’s common to add paracompactness to thedefinition of an n−manifold. Sometimes n−manifolds are defined to besecond–countable, which is precisely the condition required to ensure thatthe manifold embeds in some finite–dimensional Euclidean space. Note thatevery compact manifold is second–countable, and every second–countablemanifold is paracompact.Topological manifolds are usually required to be Hausdorff and second–countable. Every Hausdorff, second countable manifold of dimension nadmits an atlas consisting of at most n + 1 charts.1.1.3 Differentiable ManifoldsFor most applications, a special kind of topological manifold, a differentiablemanifold, is used. If the local charts on a manifold are compatible in acertain sense, one can define directions, tangent spaces, and differentiablefunctions on that manifold. In particular it is possible to use calculus ona differentiable manifold. Each point of an nD differentiable manifold hasa tangent space. This is an Euclidean space R n consisting of the tangentvectors of the curves through the point.Two important classes of differentiable manifolds are smooth and analyticmanifolds. For smooth manifolds the transition maps are smooth, thatis infinitely differentiable, denoted by C ∞ . Analytic manifolds are smoothmanifolds with the additional condition that the transition maps are analytic(a technical definition which loosely means that Taylor’s expansionTheorem 12 holds). The sphere can be given analytic structure, as can mostfamiliar curves and surfaces.In other words, a differentiable (or, smooth) manifold is a topologicalmanifold with a globally defined differentiable (or, smooth) structure. Atopological manifold can be given a differentiable structure locally by usingthe homeomorphisms in the atlas of the topological space (i.e., the homeomorphismcan be used to give a local coordinate system). The globaldifferentiable structure is induced when it can be shown that the naturalcomposition of the homeomorphisms between the corresponding openEuclidean spaces are differentiable on overlaps of charts in the atlas. Therefore,the coordinates defined by the homeomorphisms are differentiable withrespect to each other when treated as real valued functions with respect to12 The most basic example of Taylor’s Theorem is the approximation of the exponentialfunction near the origin point x = 0: e x ≈ 1 + x + x2 + x3 + · · · + xn . For technical2! 3! n!details, see any calculus textbook.