Smooth Manifolds

1.1 Topological Manifolds

An $n$-dimensional topological manifold $M$ is a topological space that is:

1. Hausdorff: distinct points have disjoint neighborhoods.
2. Second countable: the topology has a countable basis.
3. Locally Euclidean: every point $p \in M$ has a neighborhood $U$ homeomorphic to an open subset of $\mathbb{R}^n$.

A homeomorphism $\varphi : U \to V \subseteq \mathbb{R}^n$ is called a coordinate chart (or just chart), and $(U, \varphi)$ is a coordinate neighborhood.

1.2 Smooth Manifolds and Atlases

A smooth atlas on a topological $n$-manifold $M$ is a collection of charts $\{(U_\alpha, \varphi_\alpha)\}$ such that:

1. The $U_\alpha$ cover $M$.
2. For every pair of overlapping charts, the transition map $\varphi_\beta \circ \varphi_\alpha^{-1} : \varphi_\alpha(U_\alpha \cap U_\beta) \to \varphi_\beta(U_\alpha \cap U_\beta)$ is a smooth diffeomorphism.

Two atlases are compatible if their union is also a smooth atlas. A smooth structure on $M$ is a maximal smooth atlas.

Example 1. $\mathbb{R}^n$ with the identity chart is a smooth manifold.

Example 2. $S^n = \{x \in \mathbb{R}^{n+1} : \|x\| = 1\}$ is a smooth manifold. Use stereographic projection or the $2(n+1)$ hemisphere charts.

Example 3. The general linear group $GL_n(\mathbb{R}) = \{A \in M_n(\mathbb{R}) : \det(A) \neq 0\}$ is an open subset of $\mathbb{R}^{n^2}$, hence a smooth manifold of dimension $n^2$.

Example 4. The real projective space $\mathbb{RP}^n$ is a smooth manifold of dimension $n$.

1.3 Smooth Maps and Diffeomorphisms

A map $f : M \to N$ between smooth manifolds is smooth if for every $p \in M$, there exist charts $(U, \varphi)$ near $p$ and $(V, \psi)$ near $f(p)$ with $f(U) \subseteq V$, such that $\psi \circ f \circ \varphi^{-1}$ is smooth as a map between open subsets of Euclidean spaces.

A diffeomorphism is a smooth bijection with smooth inverse. If $M$ and $N$ are diffeomorphic, we write $M \cong N$.

Proposition 1.1. Diffeomorphism is an equivalence relation on the class of smooth manifolds.