Smooth Manifold

Definition

A smooth manifold (or differentiable manifold) is a topological space that locally resembles Euclidean space and on which calculus can be performed.

Formally, a smooth manifold of dimension $n$ is a Topological Space $M$ together with a collection of charts (homeomorphisms from open subsets of $M$ to open subsets of ${\mathbb{R}}^n$) such that:

• The charts cover $M$
• The transition maps between overlapping charts are smooth (infinitely differentiable)

Structure

A smooth manifold consists of:

• An underlying topological space that is Hausdorff and second-countable
• An atlas: a collection of compatible charts
• A maximal atlas defining the smooth structure

Examples

• Euclidean space ${\mathbb{R}}^n$ itself
• The $n$-sphere $S^n$
• Lie groups (groups that are also smooth manifolds)
• The torus $T^2=S^1\times S^1$
• Real projective space ${\mathbb{RP}}^n$

Smooth Maps

A map $f:M\to N$ between smooth manifolds is smooth if in local coordinates it is represented by smooth functions. Smooth maps are the morphisms in the category Man (Category) of smooth manifolds.

Tangent Space

At each point $p\in M$, there is a Tangent Space $T_{p}M$, a vector space of dimension $n$ consisting of tangent vectors at $p$.

Properties

• Smooth manifolds admit partitions of unity
• Every smooth manifold can be embedded in some Euclidean space (Whitney embedding theorem)
• Smooth manifolds support differential forms and integration
• The tangent bundle $TM={\bigcup }_{p\in M}{T}_{p}M$ is itself a smooth manifold

Related Concepts

• Topological Space: The underlying structure
• Man (Category): The category of smooth manifolds
• Differential Geometry: The study of smooth manifolds
• Tangent Space: Vector spaces attached to points
• Smooth Map: Morphisms between smooth manifolds
• Lie Group: Groups with smooth structure
• Vector Bundle: Generalizes tangent bundles