Smooth Map

Definition

A smooth map (or smooth function) is a function between smooth manifolds that is infinitely differentiable.

Let $M$ and $N$ be smooth manifolds. A function $f:M\to N$ is smooth if for every point $p\in M$, there exist charts $(U,\varphi )$ around $p$ and $(V,\psi )$ around $f(p)$ such that the coordinate representation:

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

is a smooth function between open subsets of Euclidean space (i.e., has continuous partial derivatives of all orders).

Properties

• The composition of smooth maps is smooth
• Smooth maps form the morphisms in the category Diff (Category) of smooth manifolds
• Local diffeomorphisms are smooth maps that are locally invertible with smooth inverse
• The identity map on a smooth manifold is smooth

Examples

• Any polynomial function ${\mathbb{R}}^n\to {\mathbb{R}}^m$ is smooth
• Exponential, trigonometric, and logarithmic functions (where defined) are smooth
• Constant maps between manifolds are smooth
• The inclusion of a submanifold into the ambient manifold is smooth

Special Cases

• A smooth map $f:M\to \mathbb{R}$ is called a smooth function on $M$
• A smooth map that is bijective with smooth inverse is a Diffeomorphism
• A smooth map with constant rank is called a submersion or immersion depending on whether it is surjective or injective on tangent spaces

Related Concepts

• Smooth Manifold: The domain and codomain of smooth maps
• Diff (Category): The category of smooth manifolds and smooth maps
• Continuous Map: A weaker notion requiring only continuity
• Diffeomorphism: A smooth map with smooth inverse
• Tangent Map: The derivative of a smooth map between manifolds
• Differential Geometry: The field studying smooth structures