Smooth Manifolds - Core Definitions

Smooth manifolds form the foundation of differential geometry, providing the natural setting for calculus on curved spaces. They generalize the familiar notion of surfaces to arbitrary dimensions while maintaining the structure necessary for differentiation.

DefinitionTopological Manifold

A topological manifold $M$ of dimension $n$ is a Hausdorff, second-countable topological space such that every point has a neighborhood homeomorphic to an open subset of $\mathbb{R}^n$ . A chart is a pair $(U, \varphi)$ where $U \subset M$ is open and $\varphi: U \to \varphi(U) \subset \mathbb{R}^n$ is a homeomorphism.

The key innovation of smooth manifolds is the compatibility condition between charts. When two charts overlap, we require their transition maps to be smooth functions between Euclidean spaces.

DefinitionSmooth Atlas

An atlas $\mathcal{A} = \{(U_\alpha, \varphi_\alpha)\}$ is a collection of charts covering $M$ . Two charts $(U, \varphi)$ and $(V, \psi)$ are smoothly compatible if the transition map

\psi \circ \varphi^{-1}: \varphi(U \cap V) \to \psi(U \cap V)

is smooth (infinitely differentiable). A smooth atlas is one where all charts are pairwise smoothly compatible.

ExampleThe 2-Sphere

The unit sphere $S^2 = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\}$ is a smooth manifold. Using stereographic projection from the north pole $N = (0,0,1)$ and south pole $S = (0,0,-1)$ , we obtain two charts:

\varphi_N(x,y,z) = \left(\frac{x}{1-z}, \frac{y}{1-z}\right), \quad \varphi_S(x,y,z) = \left(\frac{x}{1+z}, \frac{y}{1+z}\right)

These charts cover $S^2 \setminus \{N\}$ and $S^2 \setminus \{S\}$ respectively, and their transition map is smooth.

DefinitionSmooth Manifold

A smooth manifold is a topological manifold $M$ together with a maximal smooth atlas, called the differentiable structure. Two smooth atlases are equivalent if their union is also a smooth atlas.

Remark

The condition of second-countability ensures manifolds have a countable basis, preventing pathological examples and guaranteeing the existence of partitions of unity. The Hausdorff property ensures limits are unique, which is essential for analysis.

The notion of smooth functions between manifolds arises naturally from the chart structure. A function $f: M \to N$ between smooth manifolds is smooth if for all charts $(U, \varphi)$ on $M$ and $(V, \psi)$ on $N$ , the composition $\psi \circ f \circ \varphi^{-1}$ is smooth wherever defined.