Differential geometry begins by separating the intrinsic space from the coordinates used to calculate on it. A manifold is locally Euclidean, while compatibility of charts makes differentiation independent of the chosen coordinates.
Topological and smooth manifolds
Definition. An \(n\)-dimensional topological manifold is a Hausdorff, second-countable space \(M\) for which every \(p\in M\) has a neighborhood \(U\) homeomorphic to an open subset of \(\mathbb R^n\). A chart is such a pair \((U,\varphi)\), with coordinate map \(\varphi:U\to\varphi(U)\subset\mathbb R^n\).
Hausdorffness guarantees uniqueness of limits. Second countability rules out pathological disjoint unions and implies paracompactness, hence the existence of partitions of unity.
Definition. Charts \((U,\varphi)\) and \((V,\psi)\) are smoothly compatible if \[ \psi\circ\varphi^{-1}:\varphi(U\cap V)\longrightarrow\psi(U\cap V) \] is a \(C^\infty\)-diffeomorphism. A smooth atlas is a covering by pairwise compatible charts. A smooth structure is a maximal smooth atlas.
Proposition. Every smooth atlas is contained in a unique maximal smooth atlas.
Proof. Add every chart smoothly compatible with all charts in the given atlas. Two newly added charts are compatible: on an overlap, insert any original chart around each point and write their transition as a composition of known smooth transitions. The resulting atlas is maximal, and any maximal extension must contain exactly these compatible charts.
Smooth maps and diffeomorphisms
Definition. A map \(F:M\to N\) is smooth at \(p\) if, in charts \(\varphi\) about \(p\) and \(\psi\) about \(F(p)\), \[ \psi\circ F\circ\varphi^{-1} \] is smooth near \(\varphi(p)\). A bijection with smooth inverse is a diffeomorphism.
Coordinate independence. Replacing \(\varphi,\psi\) by other charts composes the displayed map on both sides with smooth transition maps. The chain rule therefore preserves smoothness.
Basic constructions
Example (sphere). Stereographic projection from the north and south poles gives two charts on \(S^n\). Their transition on \(\mathbb R^n\setminus\{0\}\) is \[ x\longmapsto \frac{x}{|x|^2}, \] which is smooth. Thus \(S^n\) is a smooth \(n\)-manifold.
Open subsets of \(\mathbb R^n\), products \(M\times N\), disjoint unions, regular level sets, and suitable quotients all inherit smooth structures. The real projective space \(\mathbb{RP}^n=S^n/(x\sim -x)\) illustrates a quotient that is locally Euclidean although no single global coordinate system exists.
Partitions of unity
Theorem. If \((U_\alpha)\) is an open cover of a smooth manifold \(M\), there are smooth functions \(\rho_\alpha:M\to[0,1]\), locally finitely many nonzero, such that \[ \operatorname{supp}\rho_\alpha\subset U_\alpha,\qquad \sum_\alpha\rho_\alpha=1. \]
Partitions of unity turn local constructions into global ones. They build Riemannian metrics, patch local differential forms, and permit integration without choosing a global chart. Their existence is the practical reason that paracompactness is built into the manifold convention.