Smooth Manifolds - Examples and Constructions

Smooth maps between manifolds are the morphisms in the category of smooth manifolds, preserving the differentiable structure. They generalize the notion of smooth functions from calculus to the setting of curved spaces.

DefinitionSmooth Map

Let $M$ and $N$ be smooth manifolds. A continuous map $f: M \to N$ is smooth (or $C^\infty$ ) if for every $p \in M$ and charts $(U, \varphi)$ containing $p$ and $(V, \psi)$ containing $f(p)$ , the composite map

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

is smooth as a map between open subsets of Euclidean spaces.

The key insight is that smoothness is a local property that can be checked using any charts. If the condition holds for one pair of charts around each point, it automatically holds for all compatible charts due to the smoothness of transition maps.

DefinitionDiffeomorphism

A smooth map $f: M \to N$ is a diffeomorphism if it is bijective and its inverse $f^{-1}: N \to M$ is also smooth. Two manifolds are diffeomorphic if there exists a diffeomorphism between them. This is the appropriate notion of equivalence for smooth manifolds.

ExampleDiffeomorphism of Euclidean Spaces

The map $f: \mathbb{R} \to (0, \infty)$ given by $f(t) = e^t$ is a diffeomorphism. Its inverse is $f^{-1}(s) = \ln s$ , which is smooth on $(0, \infty)$ . This shows that $\mathbb{R}$ and $(0, \infty)$ are diffeomorphic despite having different topologies as subsets of $\mathbb{R}$ .

Remark

Diffeomorphic manifolds are indistinguishable from the viewpoint of differential geometry - they have identical local geometric properties. However, a single manifold may admit multiple distinct smooth structures, as demonstrated by exotic spheres in dimension 7 and higher.

DefinitionLocal Diffeomorphism

A smooth map $f: M \to N$ is a local diffeomorphism at $p \in M$ if there exist neighborhoods $U$ of $p$ and $V$ of $f(p)$ such that $f|_U: U \to V$ is a diffeomorphism. The map $f$ is a local diffeomorphism if it is a local diffeomorphism at every point.

Local diffeomorphisms preserve local geometric structure but may not be global diffeomorphisms. The covering map $\mathbb{R} \to S^1$ given by $t \mapsto e^{2\pi i t}$ is a classic example - locally it looks like a diffeomorphism, but globally it wraps infinitely around the circle.

ExampleProjection Maps

The projection $\pi: \mathbb{R}^{n+k} \to \mathbb{R}^n$ given by $\pi(x^1, \ldots, x^n, y^1, \ldots, y^k) = (x^1, \ldots, x^n)$ is smooth. In coordinates, it's simply $(x, y) \mapsto x$ , which is clearly infinitely differentiable.

DefinitionSmooth Function

A smooth function on $M$ is a smooth map $f: M \to \mathbb{R}$ (or $f: M \to \mathbb{C}$ ). The set of all smooth real-valued functions on $M$ is denoted $C^\infty(M)$ .

The space $C^\infty(M)$ forms a commutative algebra under pointwise addition and multiplication, and plays a fundamental role in defining tangent vectors as derivations.