CHAPTER I. Manifolds and Vector Fields 1. Differentiable Manifolds 1.1. Manifolds. A topological manifold is a separable metrizable space M which is locally homeomorphic to Rn. So for any x G M there is some homeomorphism u : U u(U) C Rn, where U is an open neighborhood of x in M and u(U) is an open subset in M71. The pair (J7,16) is called a c/iart on M. One of the basic results of algebraic topology, called 'invariance of domain', conjectured by Dedekind and proved by Brouwer in 1911, says that the number n is locally constant on M if n is constant, M is sometimes called a pure manifold. We will only consider pure manifolds and consequently we will omit the prefix pure. A family (f/a, ua)aeA of charts on M such that the Ua form a cover of M is called an atlas. The mappings Uap := Ua o u^1 : up(Uap) - ua(Uap) are called the chart changings for the atlas (Ua), where we use the notation Uaf3 := Ua PI C^. An atlas (t^a?^a)a€ A for ^ manifold M is said to be a Ck-atlas, if all chart changings uap : up(Ua/3) ua(Uap) are differentiable of class Ck. Two Cfc-atlases are called Ck-equivalent if their union is again a Cfc-atlas for M. An equivalence class of Cfc-atlases is called a Ck-structure on M.
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