2 CHAPTER I. Manifolds and Vector Fields From differential topology we know that if M has a C -structure, then it also has a Cfl-equivalent C°°-structure and even a C^-equivalent C^-structure, where Cu is shorthand for real analytic see [84]. By a C^-manifold M we mean a topological manifold together with a Ck- structure and a chart on M will be a chart belonging to some atlas of the C^-structure. But there are topological manifolds which do not admit differentiate struc- tures. For example, every 4-dimensional manifold is smooth off some point, but there are such which are not smooth see [195], [62]. There are also topological manifolds which admit several inequivalent smooth structures. The spheres from dimension 7 on have finitely many see [156]. But the most surprising result is that on R4 there are uncountably many pairwise inequivalent (exotic) differentiable structures. This follows from the results of [42] and [62] see [78] for an overview. Note that for a Hausdorff C°°-manifold in a more general sense the following properties are equivalent: (1) It is paracompact. (2) It is metrizable. (3) It admits a Riemann metric. (4) Each connected component is separable. In this book a manifold will usually mean a C°°-manifold, and smooth is used synonymously for C°° — it will be Hausdorff, separable, finite-dimensional, to state it precisely. Note finally that any manifold M admits a finite atlas consisting of dim M + 1 (not connected) charts. This is a consequence of topological dimension theory [168] a proof for manifolds may be found in [80, I]. 1.2. Example: Spheres. We consider the space R n+1 , equipped with the standard inner product (x, y) = ^2xlyl. The n-sphere Sn is then the subset {x G R n + 1 : (x,x) = 1}. Since f{x) = (x,x), f : R n + 1 -• R, satisfies df(x)y = 2(x, y), it is of rank 1 off 0 and by (1.12) the sphere Sn is a submanifold of R n+1 . In order to get some feeling for the sphere, we will describe an explicit atlas for 5 n , the stereographic atlas. Choose a G Sn ('south pole'). Let U + :=Sn\{a}, u + :U + ^{a}^, u+ (x) = x - I ^, U„:=Sn\{-a}, u _ : C / _ - , { a } ^ „_(*) = ^ f e $ .

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